Preface
The purpose of this handbook is to supply a collection of mathematical formulas and
tables which will prove to be valuable to students and research workers in the fields of
mathematics, physics, engineering and other sciences. To accomplish this, care has been
taken to include those formulas and tables which are most likely to be needed in practice
rather than highly specialized results which are rarely used. Every effort has been made
to present results concisely as well as precisely so that they may be referred to with a maxi-
maximum of ease as well as confidence.
Topics covered range from elementary to advanced. Elementary topics include those
from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics
include those from differential equations, vector analysis, Fourier series, gamma and beta
functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions
and various other special functions of importance. This wide coverage of topics has been
adopted so as to provide within a single volume most of the important mathematical results
needed by the student or research worker regardless of his particular field of interest or
level of attainment.
The book is divided into two main parts. Part I presents mathematical formulas
together with other material, such as definitions, theorems, graphs, diagrams, etc., essential
for proper understanding and application of the formulas. Included in this first part are
extensive tables of integrals and Laplace transforms which should be extremely useful to
the student and research worker. Part II presents numerical tables such as the values of
elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as
advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion,
especially to the beginner in mathematics, the numerical tables for each function are sep-
separated. Thus, for example, the sine and cosine functions for angles in degrees and minutes
are given in separate tables rather than in one table so that there is no need to be concerned
about the possibility of error due to looking in the wrong column or row.
I wish to thank the various authors and publishers who gave me permission to adapt
data from their books for use in several tables of this handbook. Appropriate references
to such sources are given next to the corresponding tables. In particular I am indebted to
the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S.,
and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their
book Statistical Tables for Biological, Agricultural and Medical Research.
I also wish to express my gratitude to Nicola Monti, Henry Hayden and Jack Margolin
for their excellent editorial cooperation.
M. R. Spiegel
Rensselaer Polytechnic Institute
September, 1968

CONTENTS
FORMULAS
Page
1. Special Constants 1
2. Special Products and Factors 2
3. The Binomial Formula and Binomial Coefficients 3
4. Geometric Formulas 5
5. Trigonometric Functions 11
6. Complex Numbers 21
7. Exponential and Logarithmic Functions 23
8. Hyperbolic Functions 26
9. Solutions of Algebraic Equations 32
10. Formulas from Plane Analytic Geometry 34
11. Special Plane Curves ., 40
12. Formulas from Solid Analytic Geometry 46
13. Derivatives 53
14. Indefinite Integrals 57
15. Definite Integrals 94
16. The Gamma Function 101
17. The Beta Function 103
18. Basic Differential Equations and Solutions 104
19. Series of Constants 107
20. Taylor Series 110
21. Bernoulli and Euler Numbers 114
22. Formulas from Vector Analysis 116
23. Fourier Series 131
24. Bessel Functions 136
25. Legendre Functions 146
26. Associated Legendre Functions 149
27. Hermite Polynomials 151
28. Laguerre Polynomials 153
29. Associated Laguerre Polynomials 155
30. Chebyshev Polynomials 157

CONTENTS
Page
31. Hypergeometric Functions 160
32. Laplace Transforms 161
33. Fourier Transforms 174
34. Elliptic Functions 179
35. Miscellaneous Special Functions 183
36. Inequalities 185
37. Partial Fraction Expansions 187
38. Infinite Products 188
39. Probability Distributions 189
40. Special Moments of Inertia 190
41. Conversion Factors 192
Sample problems illustrating use of the tables 194
1. Four Place Common Logarithms 202
2. Four Place Common Antilogarithms 204
3. Sin x (x in degrees and minutes) 206
4. Cos x (x in degrees and minutes) 207
5. Tan x (x in degrees and minutes) 208
6. Cot x (x in degrees and minutes) 209
7. Sec x (x in degrees and minutes) 210
8. Csc x (x in degrees and minutes) 211
9. Natural Trigonometric Functions (in radians) 212
10. log sin x (x in degrees and minutes) 216
11. log cos x (x in degrees and minutes) 218
12. log tan x (x in degrees and minutes) 220
13. Conversion of radians to degrees, minutes and seconds
or fractions of a degree 222
14. Conversion of degrees, minutes and seconds to radians 223
15. Natural or Napierian Logarithms loge ж or In ж 224
16. Exponential functions ex 226
17. Exponential functions e~x 227
18a. Hyperbolic functions sinh x 228
18b. Hyperbolic functions cosh x 230
18c. Hyperbolic functions tanh x 232

CONTENTS
Page
19. Factorial n 234
20. Gamma Function 234
21. Binomial Coefficients 236
22. Squares, Cubes, Roots and Reciprocals „ 238
23. Compound Amount: A + r)n 240
24. Present Value of an Amount: A + r)~n 241
25. Amount of an Annuity: ^ + r)" - 1 242
26. Present Value of an Annuity: - \±JLLL— 243
27. Bessel functions Jo(x) 244
28. Bessel functions Ji(x) 244
29. Bessel functions Y0(x) 245
30. Bessel functions Yi (x) 245
31. Bessel functions /0(x) 246
32. Bessel functions h{x) 246
33. Bessel functions K0(x) 247
34. Bessel functions Кi (x) 247
35. Bessel functions Ber (x) 248
36. Bessel functions Bei (x) 248
37. Bessel functions Ker (x) 249
38. Bessel functions Kei (x) 249
39. Values for Approximate Zeros of Bessel Functions 250
40. Exponential, Sine and Cosine Integrals 251
41. Legendre Polynomials Pn(x) 252
42. Legendre Polynomials Pn(cos в) 253
43. Complete Elliptic Integrals of First and Second Kinds 254
44. Incomplete Elliptic Integral of the First Kind 255
45. Incomplete Elliptic Integral of the Second Kind 255
46. Ordinates of the Standard Normal Curve 256
47. Areas under the Standard Normal Curve 257
48. Percentile Values for Student's t Distribution 258
49. Percentile Values for the Chi Square Distribution 259
50. 95th Percentile Values for the F Distribution 260
51. 99th Percentile Values for the F Distribution 261
52. Random Numbers 262
Index of Special Symbols and Notations 263
Index 265

Part I
FORMULAS

THE GREEK ALPHABET
Greek
name
Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta
Iota
Kappa
Lambda
Mu
Greek letter
l/ower case
a
P
У
8
t
?
V
9
i
к
A
Capital
A
В
Г
д
Е
Z
Н
©
I
К
л
м
Greek
name
Nu
Xi
Omicron
Pi
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega
Greek
Lower case
V
0
7Г
P
a
T
V
Ф
X
0}
letter
Capital
N
«
0
IT
p
2
T
Y
Ф
X
¦*
п

SPECIAL CONSTANTS
1.1 - = 3.14159 26535 89793 23846 ^^$Щ^
1.2 e = 2.71828 18284 59045 23536 0287... = 1™ A + -
— natural base of logarithms
1.3 л/2 = 1.4142135623 73095 04883..
1.4 \/3 = 1.73205 08075 68877 2935...
1.5 л/5 = 2.23606 79774 99789 6964. . .
1.6 1/2 = 1.25992 1050. . .
1.7 ^3 = 1.44224 9570...
1.8 ^2 = 1.14869 8355...
1.9 л^з" = 1.24573 0940...
1.10 e77 = 23.14069 26327 79269 006...
1.11 тте = 22.45915 77183 61045 47342 715...
1.12 ee = 15.15426 22414 79264 190. ..
1.13 log,02 = 0.30102 99956 639811952137389...
1.14 Iog103 = 0.47712 12547 19662 43729 50279...
1.15 log10e = 0.43429 44819 0325182765...
1.16 log10 it - 0.49714 98726 94133 85435 12683...
1.17 loge 10 = In 10 = 2.30258 50929 94045 684017991...
1.18 loge2 = In 2 = 0.69314 71805 59945 309417232...
1.19 loge3 = In 3 = 1.0986122886 68109 69139 5245...
1.20 у = 0.5772156649 01532 86060 6512... = Euler's constant
+l + \+ ¦¦¦ +-
2 3 n
1.21 e* = 1.78107 24179 90197 9852... [see 1.20]
1.22 Ve - 1.64872 12707 00128 1468...
1.23 лПг = r(i) = 1.77245 38509 05516 02729 8167. . .
where Г is the gamma function [see pages 101-102].
1.24 Гф = 2.67893 85347 07748...
1.25 Щ) = 3.62560 99082 21908...
1.26 1 radian = 180%r = 57.29577 95130 8232...°
1.27 1° - тг/180 radians = 0.01745 32925 19943 29576 92. .. radians

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
(x + yJ = x2
(ж - j/J = ж2 -
+ у2
(ж + уK = ж3 + Зх2у + Вху2 + уг
(ж — уK — х3 — Зж22/ + Ъху2 — у3
(ж + уL ~ ж4 + 4ж32/ + 6ж22/2 + 4ж2/3 + у4
(ж — уL = ж4 — 4ж3т/ + 6ж2?/2 — 4ж2/3 + 2/4
(ж + j/M = ж5 + 5ж42/ + 10ж32/2 + Юж2?/3 + Ьху4 + 2/5
(ж — у)ь — ж5 — 5ж42/ + 10ж32/2 — 10ж22/3 + Ъху4 — у5
(ж + уN = ж6 + 6ж52/ + 15ж4г/2 + 20ж32/3 + 15х2у4 + Ъху5 + у6
The results 2.1 to 2.10 above are special cases of the binomial formula [see page 3].
2mm n n i \ 1 \ \
• I ¦ j; ^ — \Л, у/\A n^ f/^
2.12 ж3-2/3 = (ж - y)(x2 + ху + у2)
2.13 x3 + y3 = (x + y)(x2 ~ ху + у2)
2.14 x4-y4 = (x - y)(x + y)(x2 + у2)
2.15 ж** — y^ ~ (ж — y){x4 ~\- x3y + x2y2 -\- xy3 + y4)
2.16 ж5 + 2/5 = (ж + г/)(ж4 - Ж32/ + x2y2 - ху3 + у4)
2.17 жб-2/6 = (ж _ ^(а; + 2/)(ж2 + Ж2/+ 2/2)(ж2 - Ж2/+ 2/2)
2.18 ж4 + ж22/2 + 2/4 = (ж2 + ху + 2/2)(ж2 -ху + у2)
2.19 ж4 + 4г/4 = (ж2 + 2ху + 2у2)(х2 - 2ху + 2,у2)
Some generalizations of the above are given by the following results where и is a positive integer.
2.20
2.21
2.22
= (х — 2/)(ж2п + х2п~1у +
= (ж — у) ( ж2 — 2ж?/ cos 7
ж2 — 2ху cos
= (ж + 2/)(ж2" - х2п~1у +
= (ж + 2/) ( ж2 + 2ж2/ cos
2И7Г
2w
+ уА л «¦
+ з/2
y2n)
2 ~
2n + 1
2w+ 1
ж2 + 2xy cos
2ия
2и + 1
г/2")
+ 2/2 J( ж2 + 2ху cos
+ г/2
2n + 1
+ У2
ж2" — у2п = (ж — 2/)(ж + 2/)(ж"~1 + хп~2у + хп~3у2 + • • ^(ж™ — хп~2у + хп гуг — ¦ ¦ •)
= (ж - у)(х + у) (ж2 - 2жг/ cos- + у2 )( ж2 - 2жг/ cos— + 2/2
ж2 — 2ху cos- — + yz
2.23 ж2" + у2" - ( х2 + 2ху cos^ + У2 )( ж2 + 2жг/ cos|^ + 2/2
Bи - 1)тт
9 1
ж2 +
Ь у

3
The BINOMIAL FORMULA
and BINOMIAL COEFFICIENTS
FACTORIAL n
If n = 1,2, 3, . . . factorial n or n factorial is defined as
3.1 n\ - 1-2-3 n
We also define zero factorial as
3.2 0! = 1
BINOMIAL FORMULA FOR POSITIVE INTEGRAL n
If и = 1, 2, 3, ... then
3 4 ,. 7iGi JL) o o Tl^Ti Л-/\1Ь ** f _„О О I I «
.л ^i-2/; x -г пх у -r 2\ x y ^ 3! л # -г -г ^
This is called the binomial formula. It can be extended to other values of n and then is an infinite series
[see Binomial Series, page 110].
BINOMIAL COEFFICIENTS
The result 3.3 can also be written
' n
3.4
where the coefficients, called binomial coefficients, are given by
3.5
k fe! к\(п-к)\ \п-к

THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS
PROPERTIES OF BINOMIAL COEFFICIENTS
_ , /n\ / n
36 U +U+i
This leads to Pascal's triangle [see page 236].
¦
39 U+{ n
/к + т\ /п + т + 1 \
{ п ) = { п+1 )
n\z n
3-13
o)(
3.14
+ B) (g) + C)(з) + ••• +
= «2»-i
3.15 A)A ) - B)B) + C)C] - .•.(-!)« "(«)(„ ) = О
MULTINOMIAL FORMULA
3.16
where the sum, denoted by 2, is taken over all nonnegative integers nu n2, . . ., np for which
щ + n2 + ¦ ¦ ¦ + np = n.

RECTANGLE OF LENGTH b AND WIDTH a
4.1 Area = ab
4.2 Perimeter = 2a + 26
6
Fig. 4-1
PARALLELOGRAM OF ALTITUDE h AND BASE b
4.3 Area = bh — ab sin в
4.4 Perimeter = la + 26
6
Fig. 4-2
TRIANGLE OF ALTITUDE h AND BASE b
4.5 Area = ^bh = ^ab sin в
= Vs(s — a)(s — 6)(s — c)
where s = ^(a + 6 + c) = semiperimeter
4.6 Perimeter = a + b + с
b
Fig. 4-3
TRAPEZOID OF ALTITUDE h AND PARALLEL SIDES a AND b
4.7
4.8
Area = ±h(a + 6)
Perimeter = a + b + h I —. 1—;
sin $ sin ф
= a + b + h(csc в + esc Ф)
b
Fig. 4-4

GEOMETRIC FORMULAS
REGULAR POLYGON OF n SIDES EACH OF LENGTH Ь
4.9
4.10
Area = ±nb2 cot — =
* и
Perimeter = nb
1mU9 cos (itIn)
* sin \ir/n)
4.15
4.16
Fig. 4-5
CIRCLE OF RADIUS r
4.11 Area = я-г2
4.12 Perimeter = lirr
Fig. 4-6
SECTOR OF CIRCLE OF RADIUS r
4.13 Area = ^r2e [в in radians]
4.14 Arc length s = re
r
Fig. 4-7
RADIUS OF CIRCLE INSCRIBED IN A TRIANGLE OF SIDES a,b,c
¦\/ф-а)(8-~Ь)(8-.с)
—fc ¦
where s = l(a + Ь + с) = semiperimeter
Fig. 4-8
RADIUS OF CIRCLE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,e
„ __ abc
where s =
4\/s(s — a)(s — b){s — c)
6 + c) = semiperimeter
Fig. 4-9

GEOMETRIC FORMULAS
REGULAR POLYGON OF n SIDES INSCRIBED IN CIRCLE OF RADIUS r
n 360°
4.17 Area = 4w2 sin — = inr2 sin
2 n 2 n
180°
4.18 Perimeter = 2nr sin — = 2nr sin
n n
Fig. 4-10
REGULAR POLYGON OF n SIDES CIRCUMSCRIBING A CIRCLE OF RADIUS r
4.19 Area = nr2 tan- = tir2tan
180c
180°
4.20 Perimeter = Inr tan — = 2nr tan
n n
Fig. 4-11
SEGMENT OF CIRCLE OF RADIUS r
4.21 Area of shaded part = ?r2 (в — sin в)
Fig. 4-12
ELLIPSE OF SEMI-MAJOR AXIS a AND SEMI-MINOR AXIS b
4.22 Area = тгаЬ
Г
4.23 Perimeter = 4а I
- /с2 sin2 в
= 2тг V|(a2 + 62) [approximately]
where к = Va2 ~ b2/a. See page 254 for numerical tables.
Fig. 4-13
SEGMENT OF A PARABOLA
4.24 Area = fab
4.25 Arc length ABC -
Fig. 4-14

GEOMETRIC FORMULAS
RECTANGULAR PARALLELEPIPED OF LENGTH a, HEIGHT I, WIDTH с
4.26 Volume = abc
4.27 Surface area = 2(ab + ac + be)
a
Fig. 4-15
PARALLELEPIPED OF CROSS-SECTIONAL AREA A AND HEIGHT h
4.28 Volume = Ah = abc sin в
Fig. 4-16
SPHERE OF RADIUS r
4.29 Volume = %г*
о
4.30 Surface area = A-a-r2
Fig. 4-17
RIGHT CIRCULAR CYLINDER OF RADIUS r AND HEIGHT h
4.31 Volume = irr2h
4.32 Lateral surface area = 2-rrrh
Fig. 4-18
CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT I
4.33 Volume = vr2h = ттгЧ sin в
4.34 Lateral surface area = %trl = —. = Zvrh esc в
sin в
Fig.4-19

GEOMETRIC FORMULAS
CYLINDER OF CROSS-SECTIONAL AREA A AND SLANT HEIGHT I
4.35 Volume = Ah = Al sin в
4.36 Lateral surface area — pi = -7-— = ph esc в
sin в
Note that formulas 4.31 to 4.34 are special cases.
Fig.4-20
RIGHT CIRCULAR CONE OF RADIUS r AND HEIGHT h
4.37 Volume = \-rrr2h
4.38 Lateral surface area = irr yV2 + h2 = -rr
Fig.4-21
PYRAMID OF BASE AREA A AND HEIGHT h
4.39 Volume = \Ah
Fig.4-22
SPHERICAL CAP OF RADIUS r AND HEIGHT h
4.40 Volume (shaded in figure) -
4.41 Surface area = 2vrh
- h)
Fig. 4-23
FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII a, b AND HEIGHT h
4.42 Volume = ^тгМа2 + ab + b2)
4.43 Lateral surface area = тг(о + 6) y/h2 + (b - aJ
Fig. 4-24

10
GEOMETRIC FORMULAS
SPHERICAL TRIANGLE OF ANGLES A,B,C ON SPHERE OF RADIUS r
4.44 Area of triangle ABC = (A + В + С - тг)г2
Fig.4-25
TORUS OF INNER RADIUS a AND OUTER RADIUS b
4.45 Volume = \жЦа + b)(b - aJ
4.46 Surface area - тгЦЪ2 - а2)
Fig. 4-26
ELLIPSOID OF SEMI-AXES a,b,c
4.47 Volume = \irabe
Fig. 4-27
PARABOLOID OF REVOLUTION
4.48 Volume =
Pig.4-28

DEFINITION OF TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE
Triangle ABC has a right angle (90°) at С and sides of length a, b, с The trigonometric functions of
angle A are denned as follows.
5.1
5.2
5.3
5.4
5.5
5.6
. a opposite
sine of A = sin A = — = т г
с hypotenuse
„ . . Ъ adjacent
cosine of A = cos A = — = ¦=-—^—
с hypotenuse
. a opposite
= tan A - ъ = ^-^
tangent oi A
. , „ . , . 6 adjacent
cotangent of A = cot A = — = r—
a opposite
„ . . с hypotenuse
secant of A = sec A = ¦=- = —r: r—
о adjacent
. „ . л с hypotenuse
cosecant of A = esc A = — = — гт—
a opposite
Fig. 5-1
EXTENSIONS TO ANGLES WHICH MAY BE GREATER THAN 90°
Consider an xy coordinate system [see Fig. 5-2 and 5-3 below]. A point P in the xy plane has coordinates
(x, y) where x is considered as positive along OX and negative along OX' while у is positive along OY and
negative along OY'. The distance from origin О to point P is positive and denoted by r = y/x2 + y2 ¦
The angle A described counterclockwise from OX is considered positive. If it is described clockwise from
OX it is considered negative. We call X'OX and Y'OY the x and у axis respectively.
The various quadrants are denoted by I, II, III and IV called the first, second, third and fourth quad-
quadrants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A
is in the third quadrant.
X
II
III
X
IV
X
II
X
IV
11
f

12
TRIGONOMETRIC FUNCTIONS
For an angle A in any quadrant the trigonometric functions of A are denned as follows.
5.7 sin A = y/r
5.8 cos A = x/r
5.9 tan A = y/x
5.10 cotA = x/y
5.11 sec A = r/x
5.12 cscA = r/y
RELATIONSHIP BETWEEN DEGREES AND RADIANS
A radian is that angle в subtended at center О of a circle by an arc
MN equal to the radius r.
Since Ътт radians = 360° we have
5.13 1 radian = 180%r = 57.29577 95130 8232...°
5.14 1° = Я-/180 radians = 0.01745 32925 19943 29576 92. . .radians
Fig.5-4
RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS
5.15 tan A
5.16 cotA
5.17 sec A
5.18 esc A
sin A
cos A
1
tan A
1
cos A
1
cos A
sin A
5.19 sin2A + cos2A = 1
5.20 sec2 A - tan2 A = 1
5.21 esc2 A - cot2 A = 1
sin A
SIGNS AND VARIATIONS OF TRIGONOMETRIC FUNCTIONS
Quadrant
I
II
III
IV
sin A
+
0 tol
+
1 toO
0 to -1
-1 toO
cos A
+
1 toO
0 to -1
-1 toO
+
0 tol
tan A
+
0 to м
-°° to 0
+
0 to м
-°° to 0
cot A
+
°° to 0
0 to -°o
+
°° to 0
0 to ^°=
sec A
+
1 to «
-co to —1
-1 to -»
+
« to 1
csc A
+
00 to 1
+
1 to °°
-» to -1
—1 to — °°

TRIGONOMETRIC FUNCTIONS
13
EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES
Angle A
in degrees
0°
15°
30°
45°
60°
75°
90°
105°
120°
135°
150°
165°
180°
195°
210°
225°
240°
255°
270°
285°
300°
315°
330°
345°
360°
Angle A
in radians
0
тг/12
тг/6
tt/4
тг/3
5tt/12
tt/2
7tt/12
2^/3
3W4
5W6
1W12
7Г
13тг/12
7tt/6
5тг/4
4тг/3
17W12
Зтг/2
19тг/12
5тг/3
7тг/4
1W6
23тг/12
2-
sin A
0
Х(\/б-\/2)
i
J-\/2
1а/з
i(\/6 + \/2)
1
1(\/б + \/2)
iA/3
^\/2
J(V6-a/2)
0
-i(\/6-\/2)
-i
-*а/2
-*а/з
-l(\/6 + \/2)
-1
-i(\/6 + V2)
-|\/2
-i
-1(\/б-\Я)
0
cos A
1
i(\/6 + \/2)
|\/2
l
i(\/6-\/2)
0
-J(a/6-a/2)
-l\/2
-l\/3
-i(\/6 + \/2)
-1
-i(\/6 + \/2)
-*а/з
-iA/2
-i(\/e-V^)
0
i(\/6-V2)
i
|\/2
iA/3
|(\/6 + a/2)
1
tan A
0
2-V3
iV3
1
A/3
2 + л/3
±00
-B + \/3)
-а/з
-1
-x^
-B-V3)
0
2 - \/з
J-x/з
1
а/з
2 + \/i$
±00
"B + а/З)
-A^
-1
-X\/3
-B - \/3)
0
cot A
2 + V3
а/з
1
A\/3
2 - \/3
0
-B-\/3)
-|\/3
-1
-а/з
-B + \/3)
zpco
2 + \/3
A^
1
iVs
2 - \/3
0
-B-а/З)
-Ja/з
-1
-а/з
-B + л/3)
sec A
1
Vi-\/2
А/2"
2
а/6 + \/г
±со
-(\/6 + \/г)
-2
-А/2
-(\/б-\/г)
-(\/6-а/2)
-|а/з
-А/2
-2
-(\/б + \/2)
ZJZ СО
\/б + а/2
2
А^
§а/з
\/б - \/2
1
esc A
СО
\/б + \/2
2
А^
За/з
\/б-\/2
1
\/б-л/г
|Уз
А/2
2
\/б + \/г
±ОО
-(\/б + А/^)
-2
-V2
-|V3
-(\/6-v^)
-|
-(у/в-у/2)
-|\/3
-А^
-2
-(\/б + \/2)
Ijl 00
For tables involving other angles see pages 206-211 and 212-215.

14
TRIGONOMETRIC FUNCTIONS
GRAPHS OF TRIGONOMETRIC FUNCTIONS
In each graph x is in radians.
5.22 у = sin a;
Fig. 5-5
5.24 у — tan ж
1
j
О
/
У
0
j
/
i/
f
1
X
i
i
Fig.5-7
5.26 у = sec x
0
Fig.5-9
5.23 у = cos x
1,
/
-1-
У
0
К
/ \ x
A \
-S 2 \^
Fig.5-6
5.25 у = cotx
У
О i\
2
7Г 3?
2
Fig.5-8
5.27
= CSC X
О
Fig.5-10
5.28 sin(-A) = -sin A
5.31 esc (—A) = —esc A
FUNCTIONS OF NEGATIVE ANGLES
5.29 cos(-A) = cos A
5.32 sec(-A) = sec A
2tt
5.30 tan(-A) = -tan A
5.33 cot(-A) = -cot A

TRIGONOMETRIC FUNCTIONS
15
ADDITION FORMULAS
5.34
5.35
5.36
5.37
sin (A±B) =
cos (A±B) =
tan (A ± B) -
cot (A±B) =
sin A cos В ± cos A sin В
cos A cos В + sin A sin В
tan A ± tang
1 =P tan A tan В
cot A cot.B + 1
cot В ± cot A
FUNCTIONS OF ANGLES IN ALL QUADRANTS IN TERMS OF THOSE IN QUADRANT I
sin
cos
tan
CSC
sec
cot
-A
— sin A
cos A
— tan A
— esc A
sec A
— cot A
90° ± A
cos A
+ sin /1
=p cot/1
sec A
=p csc/1
=p tan/1
180° ± A
ir ± A
=p sin A
— cos A
± tan A
4= CSC/1
— sec A
±cotA
270° ± A
4±A
— cos A
± sin/1
=p cot/1
— sec/1
± esc A
=p tan /1
fcC60°) ± A
2ктг ± А
к = integer
± sin/1
cos A
±tan/l
± esc A
sec/1
±cotA
RELATIONSHIPS AMONG FUNCTIONS OF ANGLES IN QUADRANT I
sin A
cos A
tan/1
cot A
sec A
esc A
sin A -
и
VT^
и1л[\~-
vt^t
l/Vi^
1/m
= M
?/m
~u~*
cos A = и
VT^
и
Vl - m2/m
м/Vl ~ m2
1/m
1/VI-m2
tan /1 — м
m/\
l/\
V"
/l + M2
/l + M2
M
1/M
+ m2/m
cot /1 = и
1/Vl + m2
m/VTTm2
1/и
M
vTT^/m
VTT^2
sec/1 =
V^T,
1/m
V^l
1/Vm2 -
M
M
M
I
I
csc/1
1/m
v^
I/Vm2"
v^
vtt
1/M

For extensions to other quadrants use appropriate signs as given in the preceding table.

16
TRIGONOMETRIC FUNCTIONS
DOUBLE ANGLE FORMULAS
5.38
5.39
5.40
sin 2A =
cos 2A =
tan 2A =
2 sin A cos A
cos2 A — sin2 A
2 tan A
1 - tan2 A
= 1-2 sin2 A = 2 cos2 A - 1
HALF ANGLE FORMULAS
5.41
5.42
5.43
. A
sm-
tan-
V
1 — cos/1
1 + cos A
1 — cos/1
1 + cos/1
+ if A/2 is in quadrant I or II
— if A/2 is in quadrant III or IV
+ if A/2 is in quadrant I or IV
— if A/2 is in quadrant II or III
+ if A/2 is in quadrant I or III
— if A/2 is in quadrant II or IV
sin A
1 + cos/1
1 — cos A
sin/1
= esc A — cot A
MULTIPLE ANGLE FORMULAS
5.44
5.45
5.46
5.47
5.48
5.49
5.50
5.51
5.52
sin 3/1
cos 3/1
tan 3/1
sin 4/1
cos 4/1
tan 4/1
sin ЪА
cos 5/1
tan 5/1
See also formulas 5.68 and 5.69.
3 sin A — 4 sin3/!
4 cos3 A — 3 cos A
3 tan A - tan3 A
1-3 tan2 A
4 sin A cos A — 8 sin3 A cos A
8 cos4 A — 8 cos2 A + 1
4 tan A — 4 tan3 A
1-6 tan2 A + tan4 A
5 sin A - 20 sin3 A + 16 sin5 A
16 cos5 A - 20 cos3 A + 5 cos A
tan5 A - 10 tan3 A + 5 tan A
1-10 tan2 A + 5 tan4 A
POWERS OF TRIGONOMETRIC FUNCTIONS
5.53 sin2A = i - icos2A
5.54 cos2 A = \ + icos2A
5.55 sin3 A = JsinA — I; sin ЗА
5.56 cos3 A = JcosA + ? cos ЗА
See also formulas 5.70 through 5.73.
5.57
5.58
5.59
5.60
sin4 A
cos4 A
sin5 A
cos5 A
— J cos 2A + ^ cos 4A
+ | cos 2A
cos 4A
= I- sin A — -fi^ sin ЗА + -^ sin 5A
о 16 1 fa
= ^ cos A + -fig cos ЗА + Tlg cos 5A

TRIGONOMETRIC FUNCTIONS 17
SUM, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIONS
5.61 sin A + sin В = 2 sin i(A + B) cos J-(A - B)
5.62 sin A - sin В — 2 cos |(A + B) sin i(A - B)
5.63 cos A + cos В = 2 cos J-(A + B) cos -|(A - B)
5.64 cos A - cos В = 2 sin i(A + B) sin i(B - A)
5.65 sin A sin В = Ucos (A - В) - cos (A + В)}
5.66 cos A cos В = l{cos (A - B) + cos (A + B)}
5.67 sin A cos В = i{sin(A — B) + sin (A + B)}
GENERAL FORMULAS
5.68 sinnA = sin A J B cosA)" - ( П ^ j B cos A)" + f П ^ 3 j B cos A)" - • • •
5.69 cosnA = ^<jB cosA)" - jB cosA)" + ~(n ~ 3 ) B cosA)"~4
и / и — 4
3 V 2 ;BcosA)"-e+ • .
5.70 sin2" A = (~21n)"~1 JsinBw-1)A - Bn ~ 1 j sin Bre- 3)A + ••• (-I)"!' ^J"^ ) sin A
Г/ \
_ „ „ 2n _2 - cos Bn - 1)A + f ~ JcosB«-3)A + ••• +
5.72 sin2"A = ^( ^) + b^T^os2wA - ( Г ) cosBn-2)A + ••• (-1)»-!^^ jcos 2A
5.73 cos2"A = ^n^j + 25^T|cos2«A + (J) cos B« - 2)A + ••• + (№2T1)cos2A
INVERSE TRIGONOMETRIC FUNCTIONS
If x = sin j/ then j/ = sin ж, i.e. the angle whose sine is x or inverse sine of x, is a many-valued
function of x which is a collection of single-valued functions called branches. Similarly the other inverse
trigonometric functions are multiple-valued.
For many purposes a particular branch is required. This is called the principal branch and the values
for this branch are called principal values.

18
TRIGONOMETRIC FUNCTIONS
PRINCIPAL VALUES FOR INVERSE TRIGONOMETRIC FUNCTIONS
Principal values
0
0
0
0
0
0
? sin' 1 ж
= cos x
? tan -J ж
< cot x
? sec x
< CSC X
for ж S 0
Й77/2
ё ,7/2
< 77/2
? 77/2
< 77/2
S 77/2
Principal
values
-77/2 S sin
,7/2 <
-тг/2 <
,7/2 <
тг/2 <
-77/2 i
С cos
С tan
С cot
; sec
S esc
for
ж
ж
ж
ж
ж
ж
а; < 0
< 0
^ 77
< 0
< 77
? тг
< 0
RELATIONS BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS
In all cases it is assumed that principal values are used.
5.74 sin ж + cos ж = я72 5.80 sin (—a;) = —sin ж
5.75 tan ж + cot ж = W2 5.81 cos (—ж) = 77 — cos ж
5.76 sec-1» + CSC-1* = W2 5.82 tan (-я) = -tan-]a-
5.77 esc ж = sin A/ж) 5.83 cot (—ж) = т — cot-1 ж
5.78 sec ж = cos A/ж) 5.84 sec (—ж) = Т7 — sec-1 ж
5.79 cot a- = tan A/ж) 5.85 esc (-ж) = -esc ж
GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS
In each graph у is in radians. Solid portions of curves correspond to principal values.
5.86
у = sin"' ж
-12
О i
-тг/2-
5.87 у = cos 1 ж
5.88 у = tan ж
I
-1
/
\
тг-
\
-т/2
—77
У
\
0
/
\
X
тг12
У
/ ж
О
Fig. 5-11
Fig. 5-12
Fig. 5-13

TRIGONOMETRIC FUNCTIONS
19
5.89 у - cot ж
5.90
У
0 *
-1
-Я-/2
I
/
о i\
5.91
.,
У
= csc~
эг
,/2
/
1/
0
^-
1
1 *
Pig. 5-14
Fig. 5-15
Fig.5-16
RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A PLANE TRIANGLE
The following results hold for any plane triangle ABC with
sides a, b, с and angles А, В, С.
5.92 Law of Sines
5.95
а
sin A sin В sin С
5.93 Law of Cosines
C2 = a2 + ft2 _ 2ab cos С
with similar relations involving the other sides and angles.
5.94 Law of Tangents
a + b = tan \(A + B)
=
a - b tan l(A - B)
with similar relations involving the other sides and angles.
sin A = T-Vs(s — a)(s — b)(s — c)
be
Fig.5-17
where s = ^(a + b + c) is the semiperimeter of the triangle. Similar relations involving angles
В and С can be obtained.
See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A SPHERICAL TRIANGLE
Spherical triangle ABC is on the surface of a sphere as shown
in Fig. 5-18. Sides a, b, с [which are arcs of great circles] are
measured by their angles subtended at center О of the sphere. A,B, С
are the angles opposite sides a., b, с respectively. Then the following
results hold.
5.96 Law of Sines
sin a sin b
sin A sin В sin С
5.97 Law of Cosines
cos a = cos b cos с + sin b sin с cos A
cos A = — cos В cos С + sin В sin С cos a
with similar results involving other sides and angles.
Fig. 5-18

20
TRIGONOMETRIC FUNCTIONS
5.98 Law of Tangents
tan ЦА+В) tan^(a+6)
tan \ (a - b)
tan J-(A - B) \
with similar results involving other sides and angles.
, A _ /sin s sin (;
2 v sin b sin с
(s-c)
5.99 cos
where s = ^(a + b + c). Similar results hold for other sides and angles.
5.100
*= V
cos (S - B) cos (S - C)
sin В sin С
where S = JL(A + В + С). Similar results hold for other sides and angles.
See also formula 4.44, page 10.
NAPIER'S RULES FOR RIGHT ANGLED SPHERICAL TRIANGLES
Except for right angle C, there are five parts of spherical triangle ABC which if arranged in the order
as given in Fig. 5-19 would be a, b,A,c,B.
co-B
co-A
co-c
Fig. 5-19
Fig. 5-20
Suppose these quantities are arranged in a circle as in Fig. 5-20 where we attach the prefix со
[indicating complement] to hypotenuse с and angles A and B.
Any one of the parts of this circle is called a middle part, the two neighboring parts are called
adjacent parts and the two remaining parts are called opposite parts. Then Napier's rules are
5.101 The sine of any middle part equals the product of the tangents of the adjacent parts.
5.102 The sine of any middle part equals the product of the cosines of the opposite parts.
Example: Since co-A =90° -A, co-B = 90° —B, we have
sin a, — tan 6 tan {co-B) or sin a — tan 6 cot В
sin (co-A) = cos a cos (co-B) or cos A = cos a sin В
These can of course be obtained also from the results 5.97 on page 19.

DEFINITIONS INVOLVING COMPLEX NUMBERS
A complex number is generally written as a + Ы where a and b are real numbers and i, called the
imaginary unit, has the property that i2 = —1. The real numbers a and Ь are called the real and imaginary
parts of a + bi respectively.
The complex numbers a + bi and a — bi are called complex conjugates of each other.
EQUALITY OF COMPLEX NUMBERS
6.1 a + bi = с + di if and only if a = с and b — d
ADDITION OF COMPLEX NUMBERS
6.2 (a + bi) + (c + di) = (a + c) + F + d)i
SUBTRACTION OF COMPLEX NUMBERS
6.3 (a + bi) - (c + di) = (a-c) + (b-d)i
MULTIPLICATION OF COMPLEX NUMBERS
6.4 (a + bi)(c + di) = (ac - bd) + (ad + bc)i
DIVISION OF COMPLEX NUMBERS
, 5 a + bi _ a + bi с — di _ ac + bd /be — ad\ .
c + di ~ c + di' c- di c* + d2 \c* + d2J%
Note that the above operations are obtained by using the ordinary rules of algebra and replacing i2 by
—1 wherever it occurs.
21

22
COMPLEX NUMBERS
GRAPH OF A COMPLEX NUMBER
A complex number a + Ы can be plotted as a point (a, b) on an
xy plane called an Argand diagram or Gaussian plane. For example
in Pig. 6-1 P represents the complex number —3 + 4г.
A complex number can also be interpreted as a vector from
О to P.
Fig. 6-1
POLAR FORM OF A COMPLEX NUMBER
In Pig. 6-2 point P with coordinates (x, y) represents the complex
number x + iy. Point P can also be represented by polar coordinates
(г, в). Since x = r cos в, у = г sin в we have
6.6
x + iy = r(cos в + г sin в)
called the polar form of the complex number. We often call r = ^/x2 + y2
the modulus and в the amplitude of x + iy.
О
ix,y)
(r,e)
Fig. 6-2
MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM
6.7
6.8
[rj(cos »! + i sin »t)] [r2(cos 92 + г sin e2)] — rjr2[cos (»х + 82) + i sin ($1 + e2)]
ri(cos »i + i sin »j)
;——
r2(cos в2 + г sm e2)
г,
= — cos
— e2) + г sin (в! - »2)
DE MOIVRE'S THEOREM
If p is any real number, De Moivre's theorem states that
6.9 [r(cos в + i sin 0)]" = rp(cos рв + i sin pe)
ROOTS OF COMPLEX NUMBERS
If p = 1/ti where и is any positive integer, 6.9 can be written
6.10 McosS + г sin e)]1/n - r1/n I cos" ' "'"" + i sin
[r(cos 9 + г sin в)]1/п — r1/n cos
2b
where fe is any integer. Prom this the n nth roots of a complex number can be obtained by putting
к = 0,1,2, . . .,ra- 1.

7
EXPONENTIAL and LOGARITHMIC
FUNCTIONS
LAWS OF EXPONENTS
In the following p, q are real numbers, a, b are positive numbers and m, n are positive integers.
7.1
7.4
7.7
а0 = 1, а
7.2
7.5
7.8
= I/a*
7.3
7.6
7.9
(a*)" = a""
In a", p is called the exponent, a is the base and a" is called the pth power of a. The function у = ax
is called an exponential function.
LOGARITHMS AND ANTILOGARITHMS
If a" = N where а Ф 0 or 1, then p = logaAT is called the logarithm of 2V to the base a. The number
N = aP is called the antilogarithm of p to the base a, written antiloga p.
Example: Since 32 = 9 we have log3 9 = 2, antilog3 2 = 9.
The function у = loga x is called a logarithmic function.
LAWS OF LOGARITHMS
7.10
7.11
7.12
loga MN = loga M + loga
bga "jy = bga M - loga
loga M» = p loga M
COMMON LOGARITHMS AND ANTILOGARITHMS
Common logarithms and antilogarithms [also called Briggsian] are those in which the base a — 10.
The common logarithm of N is denoted by log10 N or briefly log N. For tables of common logarithms and
antilogarithms, see pages 202-205. For illustrations using these tables see pages 194-196.
23

24 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
NATURAL LOGARITHMS AND ANTILOGARITHMS
Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e =
2.7182818. . .[see page 1]. The natural logarithm of N is denoted by logeN or \nN. For tables of natural
logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of x]
see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF LOGARITHMS
The relationship between logarithms of a number N to different bases a and b is given by
7.13 loga N = -.
a logb a
In particular,
7.14 logeN = inN = 2.30258 50929 94... log10N
7.15 logi0N = logN - 0.43429 44819 03... logeAT
RELATIONSHIP BETWEEN EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
7.16 eie = cos в + i sin в, e~ie = cos в — i sin в
These are called Eider's identities. Here i is the imaginary unit [see page 21].
7.17
7.18
7.19
sin в
C0S9
cot»
sec»
егв — е~1в
~ 2i
2
e« - e~«
2
eie + e-te
2i
е-»
7.20
7.21
7.22
PERIODICITY OF EXPONENTIAL FUNCTIONS
7.23 ег(в+2ктт) = еи> ь = integer
From this it is seen that ex has period 2vi.

EXPONENTIAL AND LOGARITHMIC FUNCTIONS 25
POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL
The polar form of a complex number x + iy can be written in terms of exponentials [see 6.6, page 22] as
7.24 x + iy = r(cos в + i sin в) = reie
OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM
Formulas 6.7 through 6.10 on page 22 are equivalent to the following.
7.25 (Г1е^)(г2еЩ = rir2e«e. + 92)
7.26 ^ 1±
г2егвг r2
7.27 (reie)v = гРе'рв [De Moivre's theorem]
LOGARITHM OF A COMPLEX NUMBER
7.29 In (re»9) = In r + ie + 2kvi к = integer

DEFINITION OF HYPERBOLIC FUNCTIONS
8.1
8.2
8.3
8.4
8.5
8.6
Hyperbolic sine of ж = sinh ж =
Hyperbolic cosine of a; = cosh x = g
Hyperbolic tangent of ж = tanh x —
Hyperbolic cotangent of x = coth x =
Hyperbolic secant of a; — sech ж =
Hyperbolic cosecant of ж = csch ж =
ex
ex
e*
ex
ex
е*
— e
2
+ e
2
+ e
+ e
— e
2
+ e
2
-x
...
— x
— x
— x
— x
8.7
8.8
8.9
8.10
8.11
8.12
8.13
RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS
sinh ж
coth ж
sech ж
csch x
cosh2 ж
sech2 x
coth2 ж
cosh ж
1
tanh ж
h-i
cosh ж
1
sinh ж
— sinh2 x =
+ tanh2 x =
— csch2 ж =
cosh ж
sinh ж
к-*
h-i
h-i
FUNCTIONS OF NEGATIVE ARGUMENTS
8.14 sinh (—ж) = —sinh ж
8.17 csch (—ж) = —csch ж
8.15 cosh (—ж) = cosh a;
8.18 sech (—ж) = sech ж
8.16 tanh (-ж)
8.19 coth (-ж)
-tanh a;
—coth»
26

8.29
HYPERBOLIC FUNCTIONS 27
ADDITION FORMULAS
8.20 sinh (x ± y) = sinh x cosh у ± cosh x sinh у
8.21 cosh (a; ±j/) = cosh a; cosh у ± sinh a; sinh у
8.22 tanh(»±y) = tanh s± tanh у
1 ± tanh ж tanh у
8.23 oofh(x±y) =
coth у ± coth x
DOUBLE ANGLE FORMULAS
8.24 sinh 2x = 2 sinh x cosh x
8.25 cosh 2x = cosh2 x + sinh2 ж = 2 cosh2 a; — 1 = 1 + 2 sinh2 x
2 tanh ж
8.26 tanh 2x =
1 + tanh2 x
HALF ANGLE FORMULAS
8.27 sinh| = ±JCOShg - [+ifx>0, - if * < 0]
8 qB i * / c°sh ж + 1
.28 cosh- =
sinh a; _ cosh x — 1
cosh x + 1 sinh ж
MULTIPLE ANGLE FORMULAS
8.30 sinh Sx = 3 sinh ж + 4 sinh3 ж
8.31 cosh За; = 4 cosh3 a; — 3 cosh ж
о r<n j. v. о 3 tanh ж + tanh3 ж
8.32 1апЬЗж = — —г—
1 + 3 tanh2 x
8.33 sinh 4a; = 8 sinh3 ж cosh ж + 4 sinh a; cosh a;
8.34 cosh 4ж = 8 cosh4 ж - 8 cosh2 ж + 1
4 tanh ж + 4 tanh3 x
8.35 tanh 4ж =
6 tanh2 x + tanh4 ж

28
HYPERBOLIC FUNCTIONS
POWERS OF HYPERBOLIC FUNCTIONS
8.36
8.37
8.38
8.39
8.40
8.41
sinh2 x =
cosh2 x =
sinh3 x =
cosh3 x =
sinh4 x =
cosh4 x =
i
i
i
I
3
к
cosh 2x —
cosh 2x +
sinh 3x —
cosh 3x +
4
a
— ^ cosh 2x
+ \ cosh:
2x
sinh x
cosh x
+ jj cosh 4x
+ i cosh 4x
SUM, DIFFERENCE AND PRODUCT OF HYPERBOLIC FUNCTIONS
8.42
8.43
8.44
8.45
8.46
8.47
8.48
sinh x + sinh у — 2 sinh ^(x + y) cosh ^{x — y)
sinh x — sinh у = 2 cosh ^(x + 2/) sinh i(x — j/)
cosh x + cosh 2/ = 2 cosh ^(x + 2/) cosh -|(x — 3/)
cosh x — cosh j/ = 2 sinh ^(x + y) sinh l(x — y)
sinh x sinh у — -|{cosh (x + y) — cosh (x — y)}
cosh x cosh 2/ = ^{cosh (x + y) + cosh (x — 2/)}
sinh x cosh у ~ -^{sinh (x + y) + sinh (x — 2/)}
EXPRESSION OF HYPERBOLIC FUNCTIONS IN TERMS OF OTHERS
In the following we assume x > 0. If x < 0 use the appropriate sign as indicated by formulas 8.14
to 8.19.
sinh x
cosh x
tanh x
coth x
sech x
csch x
sinh x = и
и
уТТм2
м/уТТм2
V'и2 + 1/м
i/Vi + м2
1/м
cosh х = и
yV-i
м
Vm2 - 1/м
и/\и2 — 1
1/м
i/V«2 -1
tanh x = и.
tt/Vl - м2
l/Vl - м2
м
1/м
VI-м2
Vl ~ м2/м
coth х = и
1/Vm2 - 1
и/\и2 — 1
1/и
м
Vm2 - 1/м
Vm2 - 1
sech x = м
Vl ~ м2/м
1/м
Vl-м2
l/Vl - м2
м
u/yl — v?
esch a; = и
1/м
Vl + м2/м
l/Vl + и2
Vl + M2
m/Vi + м2'
м

HYPERBOLIC FUNCTIONS
29
GRAPHS OF HYPERBOLIC FUNCTIONS
8.49
у — sinh x
О
Fig. 8-1
8.50
Fig. 8-2
8.51
у — tanh x
У
О
Fig. 8-3
8.52 у — coth x
О
Fig. 8-4
8.53
= sech x
8.54
Fig. 8-5
у — csch x
Fig.8-6
INVERSE HYPERBOLIC FUNCTIONS
V x = sinh y, then у — sinh ж is called the inverse hyperbolic sine of x. Similarly we define the
other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the
case of inverse trigonometric functions [see page 17] we restrict ourselves to principal values for which
they can be considered as single-valued.
The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic
functions expressed in terms of logarithmic functions which are taken as real valued.
8.55
8.56
8.57
8.58
8.59
8.60
smb» = In (x + л/х2 + 1)
x = In (x + Уж2 — 1)
+ 1
sech~* x = In r \ —s- — 1
— °° < x < <*>
x = 1 [cosh a; > 0 is principal value]
-1< x < 1
x > 1 or x < —1
0 < x s 1 [sech x > 0 is principal value]
x ?= 0

30
HYPERBOLIC FUNCTIONS
8.61
8.62
8.63
8.64
8.65
8.66
8.67
RELATIONS BETWEEN INVERSE HYPERBOLIC FUNCTIONS
csch x — sinh A/x)
sech~xa; = cosh^1 A/x)
coth^z = tanh-^l/a;)
sinh^1 (—x) — — sinh^'x
tanh (—x) — — tanh x
coth (—x) = — coth":»;
csch (—x) = — csch-1x
8.71
GRAPHS OF INVERSE HYPERBOLIC FUNCTIONS
8.68 у = sinh x
О
Fig. 8-7
у — coth ~' x
У
О
8.69 у = cosh x
Fig. 8-8
8.72 у - sech-'ж
О
8.70
— x
Fig. 8-9
8.73 у - csch x
О
Fig.8-10
Fig. 8-11
Fig. 8-12

HYPERBOLIC FUNCTIONS
31
RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS
8.74 sin (гх) = г sinh x
8.77 esc (ix) — — г csch x
8.80 sinh (ix) = i sin x
8.83 csch (гх) = — г esc x
8.75 cos (ix) — cosh x
8.78 sec (ix) — sech x
8.81 cosh (г'ж) = cos a;
8.84 sech (гх) = sec x
8.76 tan (ix) — i tanh x
8.79 cot (ix) = — г" coth ж
8.82 tanh (we) = i tan x
8.85 coth (гх) = —г cot ж
PERIODICITY OF HYPERBOLIC FUNCTIONS
In the following к is any integer.
8.86 sinh (x + 2kvi) = sinh x 8.87 cosh (x + 2kvi) = cosh x 8.88 tanh (x + kvi) = tanh x
8.89 csch (ж + 2kvi) = csch ж 8.90 sech (x + 2kvi) = sech a; 8.91 coth (x + kvi) = coth a;
RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS
8.92 sin-i (ix)
8.94 cos x -
8.96 tan-1^)
8.98 cot-Чгх)
8.100 sec x =
8.102 csc-Чгя)
= i sinn x
± г cosh x
= г tanh-1 ж
= — г coth-1 x
± г sech x
= —г csch" a;
= i sin x
8.93 sinh -1 (г:
8.95 cosh ж =
8.97 tanh (га;)
8.99 coth-Чгж)
8.101 sech a; =
8.103 csch-1 (га;) = —г esc a;
i cos * x
itanx
— i cot ж
= =t г sec x

QUADRATIC EQUATION: ax2 + bx + с = О
9.1
Solutions:
2a
If a, b, с are real and if D — b2 — 4ac is the discriminant, then the roots are
(i) real and unequal if D > 0
(ii) real and equal if D = 0
(iii) complex conjugate if D < 0
9.2
If x1,x2 are the roots, then xx + x2 = —Ь/a and
CUBIC EQUATION: x3 + a-ix2 + a2x + a3 = 0
Let
_
Q=
S =
3a2 —
R
— 27a3 —
T =
9.3
Solutions:
= S + Г - iai
ж2 = -?(S+Г) - ?a, + ?г\^3 (S-Г)
Г) - -i-ai - |iV? (S - Г)
If ax, a2, a3 are real and if D = Q3 + R2 is the discriminant, then
(i) one root is real and two complex conjugate if D > 0
(ii) all roots are real and at least two are equal if D = 0
(iii) all roots are real and unequal if D < 0.
If D < 0, computation is simplified by use of trigonometry.
9.4
Solutions if D < 0:
9.5 Xi + x2 + x3 = —a1;
where xlt x2, x3 are the three roots.
= 2\Z=Qcos(±»)
x2 = 24/^ cos (l« + 120°)
^ cos ($6 + 240°)
x2x3
where cos в - -R/^-
x1x2x3 = -a3
32

SOLUTIONS OP ALGEBRAIC EQUATIONS
33
QUARTIC EQUATION: r4 + a,x3 + a2x2 + asx + a4 = 0
Let j/i be a real root of the cubic equation
9.6 y3 - a2y2 + (axdg - 4a4)j/ + Da2a4 - a| - a^a4) = 0
9.7 Solutions: The 4 roots of z2 + ^{ax ± ya\ - 4a2 + 4yt }z + J-{?/i ± V v\ - 4a4 } = 0
If all roots of 9.6 are real, computation is simplified by using that particular real root which produces
all real coefficients in the quadratic equation 9.7.
9.8
x2 + x3 + ж4 = — ax
+ x2x3 + x3x4 + x4x1
+ + x1x2x4 +
k
= a2
where x1: x2, x3, x4 are the four roots.
к

10
FORMULAS from
PLANE ANALYTIC GEOMETRY
DISTANCE d BETWEEN TWO POINTS
AND
10.1
10.2
d =
*l) I
Fig. 10-1
SLOPE m OF LINE JOINING TWO POINTS Pi{xuyx) AND P2(^2,2/z)
3/2 ~ 3/1
m = = tan в
EQUATION OF LINE JOINING TWO POINTS Pi(xi,?/i) AND Р-г{хч,Уч)
10.3
10.4
3/ - 3/1 _ 3/2 ~ 3/i
= tn or У ~ Vi — m(x — Xi
where b = y1 — mx1 =
у = rnx + b
is the intercept on the у axis, i.e. the у intercept.
EQUATION OF LINE IN TERMS OF x INTERCEPT a?-0 AND у INTERCEPT
10.5
- + т =
a b
Fig.10-2
34

FORMULAS FROM PLANE ANALYTIC GEOMETRY
35
NORMAL FORM FOR EQUATION OF LINE
10.6 Ж COS a + у sin a — p
where p = perpendicular distance from origin О to line
and a = angle of inclination of perpendicular with
positive x axis.
Fig.10-3
GENERAL EQUATION OF LINE
10.7
Ax + By + С = 0
DISTANCE FROM POINT {xuy{) TO LINE Ax + By + С = 0
10.8
в2
where the sign is chosen so that the distance is nonnegative.
10.9
ANGLE ф BETWEEN TWO LINES HAVING SLOPES ж, AND m2
m2 —
Lines are parallel or coincident if and only if mx = m2.
Lines are perpendicular if and only if от2 = —1
Fig.10-4
AREA OF TRIANGLE WITH VERTICES AT (xuyi),(xs,y2),(x9,y3)
10.10 Area = ±j
. 1
X\ 3/i 1
#2 3/2 1
2 2/зх2 - 2/2*3 -
where the sign is chosen so that the area is nonnegative.
If the area is zero the points all lie on a line.
Fig.10-5

36
FORMULAS FROM PLANE ANALYTIC GEOMETRY
10.11
10.12
TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION
= x - x0
= x' + x0
or <
- У' + Уо {У' = У - Уо
where (x, y) are old coordinates [i.e. coordinates relative to
xy system], (x',yr) are new coordinates [relative to x'y' sys-
system] and (xQ, y0) are the coordinates of the new origin O'
relative to the old xy coordinate system.
V \У
I («о. г/о)
Ю'
О
Fig.10-6
TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION
X — X COS a — у sin a
у = x' sin a + y' COS a
x' = X cos a + у sin a
y' — У COS a — X sin a
where the origins of the old [xy] and new [x'y'] coordinate
systems are the same but the x' axis makes an angle a with
the positive x axis.
¦x'
.x'
Fig.10-7
TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION
10.13
x = x' cos a — y' sin a + x0
у — x' sin a + y' cos a + y0
x' = (x — x0) cos a + (y — y0) sin a
У' = (У — Уо> cos a - (x - x0) sin a
where the new origin O' of x'y' coordinate system has co-
coordinates (xQ, y0) relative to the old xy coordinate system
and the x' axis makes an angle a with the positive x axis.
¦ Л*
\
О \
\
\
Fig.10-8
POLAR COORDINATES (r,8)
A point P can be located by rectangular coordinates (x, y) or
polar coordinates (г, в). The transformation between these coordinates
10.14
X = Г COS в
у = г sin в
r = yxz + yz
в = tan^1 (y/x)
о
Fig. 10-9

FORMULAS FROM PLANE ANALYTIC GEOMETRY
37
EQUATION OF CIRCLE OF RADIUS R, CENTER AT (xo,yo)
10.15
10.16
Fig. 10-10
EQUATION OF CIRCLE OF RADIUS R PASSING THROUGH ORIGIN
r = 2R cos (в — а)
where (г, в) are polar coordinates of any point on the
circle and (R, a) are polar coordinates of the center of
the circle.
Fig. 10-11
CONICS [ELLIPSE, PARABOLA OR HYPERBOLA]
If a point P moves so that its distance from a fixed point
[called the focus] divided by its distance from a fixed line [called
the directrix] is a constant e [called the eccentricity], then the
curve described by P is called a conic [so-called because such
curves can be obtained by intersecting a plane and a cone at
different angles].
If the focus is chosen at origin О the equation of a conic
in polar coordinates (г, в) is, if OQ = p and LM = D, [see
Fig. 10-12]
10.17
P
eD
1-е COS в
The conic is
(i) an ellipse if e < 1
(ii) a parabola if e = 1
(iii) a hyperbola if с > 1.
1-е cos в
M
I 4
Directrix^
h— D
V
Focus
Fig. 10-12

38
FORMULAS FROM PLANE ANALYTIC GEOMETRY
ELLIPSE WITH CENTER C(xo,yn) AND MAJOR AXIS PARALLEL TO x AXIS
10.18 Length of major axis A'A — la
10.19 Length of minor axis B'B = 26
10.20 Distance from center С to focus F or F' is
10.21 Eccentricity = e = —
a
10.22 Equation in rectangular coordinates:
(x - x0J (y - j/oJ
„2 + 7,2 — 1
10.23 Equation in polar coordinates if С is at O:
Г
О
Fig.10-13
a2 sin2 в + 62 cos2 e
coordinates if С is on ж axis and F' is at О: г — — ^~
1-е cos в
10.24 Equation in polar
10.25 If P is any point on the ellipse, PF + PF' = 2a
If the major axis is parallel to the у axis, interchange x and у in the above or replace в by ^ir — в [or
90° - в].
PARABOLA WITH AXIS PARALLEL TO x AXIS
If vertex is at A(xo,yo) and the distance from A to focus F is a > 0, the equation of the parabola is
10.26 (y - y0J = 4a(x - x0) if parabola opens to right [Fig. 10-14]
10.27 (y-y0J = -4a(x-x0) if parabola opens to left [Fig. 10-15]
If focus is at the origin [Fig. 10-16] the equation in polar coordinates is
2a
10.28
у
(«о. Ун)
О
1 — cos i
У
О
У
о
Fig.10-14 Fig. 10-15 Fig. 10-16
In case the axis is parallel to the у axis, interchange x and у or replace в by ^тг — в [or 90° — fl].

FORMULAS FROM PLANE ANALYTIC GEOMETRY
39
HYPERBOLA WITH CENTER C(xo,y0) AND MAJOR AXIS PARALLEL TO r AXIS
F'
/у
У
ч
ч
\ х
А' у'
f у"
у
В
у
у
у
Су'
Ч А
Ч
ч
ч
В'
0
у/
(
F
Fig. 10-17
10.29 Length of major axis A'A = 2a
10.30 Length of minor axis B'B - 2b
10.31 Distance from center С to focus F or F' = с = v/a2 +
10.32 Eccentricity е = — =
/а2 + Ъ2
a a
10.33 Equation in rectangular coordinates: % p = 1
10.34 Slopes of asymptotes G'H and GH' = ±-
Co
10.35 Equation in polar coordinates if С is at О: r2 — -p= = —
b2 cos2 в — a2 sin2 в
10.36 Equation in polar coordinates if С is on X axis and F' is at О: г — —— —
1 - f cos в
10.37 If P is any point on the hyperbola, PF — PF' = ±2a [depending on branch]
If the major axis is parallel to the у axis, interchange x and у in the above or replace в by
[or 90° - e].

IEMNISCATE
11.1 Equation in polar coordinates:
r2 = a2 cos 2»
11.2 Equation in rectangular coordinates:
11.3 Angle between AB' or A'B and x axis = 45°
11.4 Area of one loop = \аг
В
CYCLOID
11.5 Equations in parametric form:
fx — а(ф — sin Ф)
\y = a(l — cos0)
11.6 Area of one arch = 3va2
11.7 Arc length of one arch = 8a
This is a curve described by a point P on a circle of radius
a rolling along x axis.
Fig.11-2
HYPOCYCLOID WITH FOUR CUSPS
11.8 Equation in rectangular coordinates:
Ж2/3 _|_ y2/3 = o2/3
11.9 Equations in parametric form:
x = a cos3 в
11.10 Area bounded by curve = §тг<г2
11.11 Arc length of entire curve — 6a
This is a curve described by a point P on a circle of radius
a/4 as it rolls on the inside of a circle of radius a.
Fig.11-3
40

SPECIAL PLANE CURVES
41
CARDIOID
11.12 Equation: r — <z(l + cos в)
11.13 Area bounded by curve = \тта2
11.14 Arc length of curve = 8a
This is the curve described by a point P of a circle of radius
a as it rolls on the outside of a fixed circle of radius a. The
curve is also a special case of the limacon of Pascal [see 11.32].
CATENARY
11.15 Equation: у = ^
= a cosh -
This is the curve in which a heavy uniform chain would
hang if suspended vertically from fixed points A and B.
THREE-LEAVED ROSE
11.16 Equation: r — a cos 39
The equation r = о sin 3e is a similar curve obtained by
rotating the curve of Fig. 11-6 counterclockwise through 30° or
я-/6 radians.
In general r — a cos ne or r = a sin ne has n leaves if
n is odd.
Fig.11-4
Fig.11-5
Fig. 11-6
FOUR-LEAVED ROSE
11.17 Equation: r = a cos 20
The equation r = a sin 2e is a similar curve obtained by
rotating the curve of Fig. 11-7 counterclockwise through 45° or
jt/4 radians.
In general r = a cos ne or r = a sin ne has In leaves if
n is even.
Fig.11-7

42
SPECIAL PLANE CURVES
EPICYCLOID
11.18 Parametric equations:
b) cos в — b cos
\y = (a + 6) sin в — b sin
a + b
b
a + b
This is the curve described by a point P on a circle of
radius b as it rolls on the outside of a circle of radius a.
The cardioid [Fig. 11-4] is a special case of an epicycloid.
GENERAL HYPOCYCLOID
11.19 Parametric equations:
x = (a — b) cos Ф + b cos
y = (o-b)sin0 -
This is the curve described by a point P on a circle of
radius b as it rolls on the inside of a circle of radius o.
If 6 = a/4, the curve is that of Fig. 11-3.
TROCHOID
Fig.11-9
11.20 Parametric equations:
x = аф — b sin ф
у = a — Ь cos ф
This is the curve described by a point P at distance b from the center of a circle of radius a as the
circle rolls on the x axis.
If b < a, the curve is as shown in Fig. 11-10 and is called a curtate cycloid.
If 6 > a, the curve is as shown in Fig. 11-11 and is called a prolate cycloid.
If b = a, the curve is the cycloid of Fig. 11-2.
У
Fig.11-10
Fig.11-11

SPECIAL PLANE CURVES
43
TRACTRIX
11.21 Parametric equations:
x — a(ln cot \ф — cos Ф)
у — a sin 0
This is the curve described by endpoint P of a taut string
PQ of length a as the other end Q is moved along the x
axis.
WITCH OF AGNESI
11.22 Equation in rectangular coordinates:
8a3
x2 + 4a2
11.23 Parametric equations:
x = 2a cot в
у = <z(l — cos 20)
In Fig. 11-13 the variable line OA intersects у = 2a
and the circle of radius a with center @, a) at A and В
respectively. Any point P on the "witch" is located by con-
constructing lines parallel to the x and у axes through В and
A respectively and determining the point P of intersection.
О
Fig.11-13
FOLIUM OF DESCARTES
11.24 Equation in rectangular coordinates:
x3 + у3 = Заху
11.25 Parametric equations:
3at
1 + fi
3a t2
1 + t3
11.26 Area of loop = — a2
11.27 Equation of asymptote: x
a — 0
\°
О
Fig.11-14
INVOLUTE OF A CIRCLE
11.28 Parametric equations:
' x — a(cos 0 + 0 sin 0)
у — a(sin ф — 0 cos 0)
This is the curve described by the endpoint P of a string
as it unwinds from a circle of radius a while held taut.
Fig.11-15

44
SPECIAL PLANE CURVES
EVOLUTE OF AN ELLIPSE
11.29 Equation in rectangular coordinates:
(axJ'3 + Fj/J/3 = («2 - 62J/3
11.30 Parametric equations:
' ax - (a2 - b2) cos3 в
by = (a2 - b2) sin3 в
This curve is the envelope of the normals to the ellipse
x2/a2 + y2/b2 = 1 shown dashed in Fig. 11-16.
Fig.11-16
OVALS OF CASSINI
11.31 Polar equation: r* + a4 - 2a2r2 cos 2e = b*
This is the curve described by a point P such that the product of its distances from two flxed points
[distance la apart] is a constant b2.
The curve is as in Fig. 11-17 or Fig. 11-18 according as b < a or b > a respectively.
If b = a, the curve is a lemniscate [Fig. 11-1].
У
О
Fig. 11-17
\ -a 0
ч ^
У
P
.% «^
\ >v
\ \
V 1
a 1
Fig.11-18
LIMACON OF PASCAL
11.32 Polar equation: r — b + a cos в
Let OQ be a line joining origin О to any point Q on a circle of diameter a passing through O. Then
the curve is the locus of all points P such that PQ = 6.
The curve is as in Fig. 11-19 or Fig. 11-20 according as b > a or b < a respectively. If b = a, the
curve is a cardioid [Fig. 11-4].
У
0
У
f
\
\
^\ \
\ 1
/ 1
/ 1
Fig.11-19
Fig.11-20

SPECIAL PLANE CURVES
45
CISSOID OF DIOCLES
11.33 Equation in rectangular coordinates:
2a - x
11.34 Parametric equations:
Г x — 2a sin2 в
1 2a sin3 в
cos
This is the curve described by a point P such that the
distance OP = distance RS. It is used in the problem of
duplication of a cube, i.e. finding the side of a cube which has
twice the volume of a given cube.
Fig.11-21
SPIRAL OF ARCHIMEDES
11.35 Polar equation: r = ae
Fig.11-22

12
FORMULAS from SOLID
ANALYTIC GEOMETRY
DISTANCE d BETWEEN TWO POINTS Pi(xbyuZi) AND Р%{х%,%1,гг)
d =
A/2 - 2/iJ
Fig.12-1
DIRECTION COSINES OF LINE JOINING POINTS Pi(arbi/i,2i) AND Р2(ж2)t/г,z2)
12.2
_
— cos a —
d
, m = cos /? =
d
n ~ cos у =
d
where a, /?, у are the angles which line PXP2 makes with the positive x, y, z axes respectively and
d is given by 12.1 [see Fig. 12-1].
RELATIONSHIP BETWEEN DIRECTION COSINES
12.3
cos2 a + cos2 /3 + cos2 7 = 1 or I2 + m2 + rfl = 1
DIRECTION NUMBERS
Numbers L,M,N which are proportional to the direction cosines l,m,n are called direction numbers.
The relationship between them is given by
12.4
I =
M
N
2 + M2 + №
\JL2
N2'
+ M2 + №
46

FORMULAS FROM SOLID ANALYTIC GEOMETRY
47
EQUATIONS OF LINE JOINING Pt(xuyi,Zi) AND P2(x2,y2,z2) IN STANDARD FORM
12.5
x — xt _ У — Vi z — zx x ~ #! у — V\ z —
These are also valid if I, in, n are replaced by L, M, N respectively.
EQUATIONS OF LINE JOINING Pi(a;i, 1/1, z,) AND Р2(ж2,у%,z2) IN PARAMETRIC FORM
12.6 X = Хл + It, 3/ = 3/i + ?Wt, Z = Zj + И4
These are also valid if I, m, n are replaced by L, M, N respectively.
ANGLE ф BETWEEN TWO LINES WITH DIRECTION COSINES 1итиш AND h,m%n2
12.7
cos ф =
GENERAL EQUATION OF A PLANE
12.8
Ax + By + Cz + D = 0
[А, В, С, D are constants]
EQUATION OF PLANE PASSING THROUGH POINTS {xuyuzi), (x2,ybz2), (xs,y3,zs)
12.9
12.10
У - У! Z -
y2— У1 Z2 -
З/3 - 3/i z3 -
- 0
3/2 — v\ z2
3/3 - 3/1 z3
+
Z2 - Zi X2-
ZS - Zj XS-
C/ - 3/1)
«1 Vi~ 3/1
Xi 3/3 - 3/1
(z -Zi) = 0
12.11
EQUATION OF PLANE IN INTERCEPT FORM
where а, Ь, с are the intercepts on the x, y, z axes
respectively.
Fig.12-2

48
FORMULAS FROM SOLID ANALYTIC GEOMETRY
12.12
EQUATIONS OF LINE THROUGH (xo,yo,Zo)
AND PERPENDICULAR TO PLANE Ax + By + Cz + D = 0
- xo У - Vq
zn
В
or x = x0 + At, у = y0 + Bt, z = z0 + Ct
Note that the direction numbers for a line perpendicular to the plane Ax + By + Cz + D = 0 are
A,B,C.
DISTANCE FROM POINT (xo,yo,zo) TO PLANE Ax + By + Cz + D = 0
12.13
xa + By0
D
± Va2 + в2 + c2
where the sign is chosen so that the distance is nonnegative.
NORMAL FORM FOR EQUATION OF PLANE
12.14 Ж COS a + у COS /3 + Z COS У = p
where p = perpendicular distance from О to plane at
P and a, li, у are angles between OP and positive x, y, z
axes.
— У
Fig. 12-3
TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION
12.15
= x' + xt
V =
о
+ 3/o
+ z0
where (x, y, z) are old coordinates [i.e. coordinates rela-
relative to xyz system], (z',y',z') are new coordinates [rela-
[relative to x'y'z' system] and (x0, y0, z0) are the coordinates
of the new origin O' relative to the old xyz coordinate
system.
\z'
j
Fig.12-4

FORMULAS FROM SOLID ANALYTIC GEOMETRY
49
TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION
12.16
X = \\X' + 1>У' + l3z'
1 у = mxx' + m2y' + m3z'
z = nxx' + n2y' + n3z'
x' = l^x
y' = hx
Z1 = 13X + 1П3У + 713Z
m2y
where the origins of the xyz and x'y'z' systems are the
same and I1,m1,n1; I2,m2,n2; ls,m3,ns are the direction
cosines of the x', y', z' axes relative to the x, y, z axes
respectively.
Fig. 12-5
TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION
12.17
x = lxx' + l2y' + l3z' + x0
у = mxx' + m2y' + m3z' + y0
z = щх' + n2y' + n3z' + z0
x'
= lx(x - x0)
= I2(x - x0)
z' = I3(x - x0)
m^y - y0) + nx(z - z0)
m.2(y - y0) + n2(z - z0)
ms(y - y0) + n3(z - z0)
where the origin 0' of the x'y'z' system has coordinates
(xo,yo,zo) relative to the xyz system and I1,ml,n1;
I2,m.2,n2; ls,m3,n3 are the direction cosines of the
x', y', z' axes relative to the x, y, z axes respectively.
CYLINDRICAL COORDINATES {r,Q,z)
\
О')
0,!/(). zo)
Fig.12-6
A point P can be located by cylindrical coordinates (r,e,z)
[see Fig. 12-7] as well as rectangular coordinates (x, y, z).
The transformation between these coordinates is
X — Г COS в
12.18 i у = г sin в
z = z
0Г
Fig.12-7

50
FORMULAS FROM SOLID ANALYTIC GEOMETRY
SPHERICAL COORDINATES {г,в,ф)
A point P can be located by spherical coordinates (r,
[see Fig. 12-8] as well as rectangular coordinates (ж, у, z).
The transformation between those coordinates is
12.19
\x — r sin в cos <
\y = r sin в sin (
\ z = r cos в
\r = V x2 + y2 + z2
= tan (ylx)
Fig. 12-8
EQUATION OF SPHERE IN RECTANGULAR COORDINATES
12.20 (x - xo)t + (y - j/0J + (Z - zoJ = R*
where the sphere has center (x0, y0, z0) and radius R-
Fig. 12-9
EQUATION OF SPHERE IN CYLINDRICAL COORDINATES
12.21 r2 - 2r0r cos (в - e0) + r2 + (z-z0J = R2
where the sphere has center (r0,60, z0) in cylindrical coordinates and radius R.
If the center is at the origin the equation is
12.22 r2 + г2 = R2
EQUATION OF SPHERE IN SPHERICAL COORDINATES
12.23 r2 + r2 - 2r0r sin й sin й0 cos (^ - 0O) = R2
where the sphere has center (r0, в0, Фо) in spherical coordinates and radius R.
12.24
If the center is at the origin the equation is
r = R

FORMULAS FROM SOLID ANALYTIC GEOMETRY
51
EQUATION OF ELLIPSOID WITH CENTER {xo,yo,Zo) AND SEMI-AXES a,b,c
(<y> — /y- \2 (if — -j/ ^2 B — 2*л^2
- ___ ~j- — — - . -(- -— J_
Fig. 12-10
ELLIPTIC CYLINDER WITH AXIS AS г AXIS
12.26
where a, 6 are semi-axes of elliptic cross section.
If 6 = a it becomes a circular cylinder of radius a.
Fig.12-11
12.27
ELLIPTIC CONE WITH AXIS AS z AXIS
L + ± =
a2 b2 c2
Fig. 12-12
HYPERBOLOID OF ONE SHEET
12.28
«i + i?_ *i - 1
a2 + 62 c2
Fig. 12-13

52
FORMULAS FROM SOLID ANALYTIC GEOMETRY
HYPERBOLOID OF TWO SHEETS
12.29
,.2 *.2 ~2
о2" Ь2 с2 *
Note orientation of axes in Fig. 12-14.
12.30
ELLIPTIC PARABOLOID
_
a2
Fig. 12-14
Fig. 12-15
HYPERBOLIC PARABOLOID
12.31
Note orientation of axes in Fig. 12-16.
Fig. 12-16

DEFINITION OF A DERIVATIVE
If у — f(x), the derivative of у or f(x) with respect to x is defined as
dx
f(x + h)- f(x) = Ит f(x + Ax) ~ f(x)
h дх-»о Ax
where h = Ax. The derivative is also denoted by y', df/dx or f'(x). The process of taking a derivative is
called differentiation.
GENERAL RULES OF DIFFERENTIATION
In the following, u, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828. . .
is the natural base of logarithms; In и is the natural logarithm of и [i.e. the logarithm to the base e] where
it is assumed that и > 0 and all angles are in radians.
13.2 -fie) = 0
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
xn~x
d . . du . dv dw
-r(u±V±W ± ¦ ¦¦) = — ± -Г- ± -7—
dx dx dx dx
d . du
; (uv) — U
dx
dv
du
dx
d , . dw , dv , du
-r—(uvw) = uv—. h uw— h vw-r—
dx dx dx dx
_d_ fu\ _ v(du/dx) — ujdv/dx)
dx
dx
= nun
. du
dx
13.,1 dj, = d^du (chabrule)
dx du dx
13.12 ^ = *
dx dx/du
13.13 ^
dx
dy/du
dx/du
53

54
DERIVATIVES
DERIVATIVES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS
лл d .
.14 —- sin и —
dx
du
-т—
dx
,« . c d du
13.15 — cos м = — sinM-r-
dx dx
11 1 ?.
13.16
-T-
dx
= sec2M
du
dx
13
13
13
.17
.18
.19
d
dx
d
dx
d
dx
cot к =
sec и =
esc и =
-csc2m —
U dx
, du
sec и tan и —r-
dx
— esc и cotM-r-
da;
1 ¦> on d . , 1 du
13.20 ¦3-sin~1M = -r-
dx
dx
7Г - . ,
~2 Bm"
13.21
X7 [°
13.22 -±
da;
-^< tan-iu < ^
2 2
13.23 A
13.24 j— sec-1M =
13.25 ^
|м| ум2 - 1
dx
±1 du
-1 dx
-1 du
,Л,2_1 dx
uy/u2 -1
mVm2-!
+ if 0 < sec-1?* < тг/2
- if я-/2 < sec~lu < г
— if 0 < csc-!w < я-/2
+ if -я-/2 < esc?* < 0
DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS
13.26 -r-logau = 3
dx a и dx
dtt
-3- а #0,1
dx
l ^ 0-7 ^ 1 d 1
13.2/ -5— In м = -5— logeM =
<ta da; 6e м
5
da;
13.28
13.29
dx
13.30 ^-м» =
da;
du
dx
DERIVATIVES OF HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIONS
13.31 -f-sinhw
dx
, du
cosh м -r—
da;
13.34 -?- cothM =
dx
— csch2 м 3—
da;
13.32 ^-
da;
. , du
sinh и j—
dx
13.35 -5— sech и =
dx
, , , du
— sech и tanh м -г~
da;
13.33 -j— tanh и
dx
sech2 м t—
dx
13.36 -7— csch м =
dx
— csch м coth и -j—
da;

DERIVATIVES 55
13t37 ~-z— sinh~* v, — —
dx Vm^+I dx
_ ±1 du Г+ ^ cosh-1!* > 0, и > 1"]
W ~ JU2 _ 1 dx \_- if cosh-1?* < 0, и > lj
13.39 ?
13.40 ^
13 41 -^hi т1 dM Г-if eech-iu>0, 0<«<n
cto
Г-if eech-iu>0, 0<«<n
L+ >f sech-iM < 0, 0 < и < lj
13.42 -fcsch-iu = ~=^ = ,Tl ^ [-if u>0, +if u<0]
HIGHER DERIVATIVES
The second, third and higher derivatives are defined as follows.
13.43 Second derivative = ^ (a) = Ix* = /"(ж) = г/"
13.44 Third derivative = ± (g) = g = /»'(«) = /
13.45 nth derivative = 4(^Щ) = 4
da; \dxn~1 dxn
LEIBNITZ'S RULE FOR HIGHER DERIVATIVES OF PRODUCTS
Let Dp stand for the operator -т-g- so that Dpu = -~ = the pth derivative of u. Then
13.46 D*(uv) = M/Jnv + (n\{Du){Dn-lv) + (П) (D2u)(D"-2v) + ¦¦¦ + vDnu
where ( j, ( j, ... are the binomial coefficients [page 3].
As special cases we have
1. .T d2 , d?v „ du dv , d2u
13-47 !^{UV) = ul^+2lid+v-d^
13.48 -ftiuv) = м^|+3^Й+з||^
da;3 da;3 da; da;2 da;2 da;
DIFFERENTIALS
Let у = f(x) and Aj/ = f(x + bx) - f(x). Then
13.49 M = Zfe + A»)-/^) = ,(a) + 6 = |/ + e
Да; дж v da;
where e -> 0 as Да; -* О. Thus
13.50 Д3/ = f'(x)Ax + eAx
If we call Дж = da; the differential of x, then we define the differential of у to be
13.51 dy = f'(x)dx

56 DERIVATIVES
RULES FOR DIFFERENTIALS
The rules for differentials are exactly analogous to those for derivatives. As examples we observe that
13.52 d(u±v±w± •¦¦) = du ± dv ± dw ± ¦ ¦ ¦
13.53
13.55
13.56
13.57
d(uv) —
\vj
d(un) -
d(sin m)
d(cos u)
udv + v du
vdu — udv
v2
пи"-Ыи
= cos и du
= — sin и du
PARTIAL DERIVATIVES
Let f(x, y) be a function of the two variables x and y. Then we define the partial derivative of f(x, j
with respect to x, keeping у constant, to be
13.58 ^- = lim f(x + Ax' y^ ~ f(x'v)
dx Ax->0 AX
Similarly the partial derivative of f(x, y) with respect to y, keeping x constant, is defined to be
1359 3/ _ Ит f(x, y + by) - f{x, y)
By дв-*о Ay
Partial derivatives of higher order can be defined as follows.
s2/ a /s/\ sy a /a/
13<6° ax2 8x\dxJ' ay2 sy\dy
13 61
= (\ = {
дхду 8x\dyJ' дудх ду\Вх
The results in 13.61 will be equal if the function and its partial derivatives are continuous, i.e. in such
case the order of differentiation makes no difference.
The differential of f(x, y) is defined as
13.62 df = ^dx+fydy
where dx = Ax and dy = Ay.
Extension to functions of more than two variables are exactly analogous.

DEFINITION OF AN INDEFINITE INTEGRAL
ctit
If ~r~ — f(x), then у is the function whose derivative is f(x) and is called the anti-derivative of f(x)
or the indefinite integral of f(x), denoted by j f(x) dx. Similarly if у = I f(u) du, then -~ = /(«).
Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant.
For the definition of a definite integral, see page 94. The process of finding an integral is called
integration.
GENERAL RULES OF INTEGRATION
In the following, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated;
e = 2.71828... is the natural base of logarithms; In и denotes the natural logarithm of и where it is assumed
that и > 0 [in general, to extend formulas to cases where и < 0 as well, replace In и by In |w|]; all angles
are in radians; all constants of integration are omitted but implied.
14.1 j adx = ax
14.2 Г af(x)dx = а С f(x)dx
14.3 \ (u ± v ± w ± • ¦ •) dx = \ udx ± \ vdx ± I w dx ± • ¦ •
14.4 I M<ii) = uv — I vdu [Integration by parts]
For generalized integration by parts, see 14.48.
14.5 Г f(ax)dx = ^Cf(u)du
14.6 j F{f(x)}dx = J F(u)^du = J J^du where и =/(*)
14.7 I undu - M" , n#-l [For n=-l, see 14.8]
14.8 Г— = lnw if w>0 or ln(-tt) if и < О
= In |m|
14.9 J eudu - eu
— С С ри\па «u
14.10 I a"du = I eulnadM = -. = ~, a>0, a ?= 1
J J In a In a
57

58 INDEFINITE INTEGRALS
14.11 I sin и du = —cosm
14.12 I cosm^m = sin и
14.13 J tan и du = In sec и = — In cos и
14.14 j cot и du = In sin и
J/
sec и du = In (sec и + tan u) — In tan I — + j
14.16 I cscwdM = In (esc и — cot u) = In tan —
14.17 I sec2 и du = tan и
14.18 I esc2 м du = — cot и
14.19 I tan2 м du — tanw — и
14.20 I cot2Mdw = —cotu — u
l л 11 Г • о j M sin 2m .
14.21 I sm2Mdw = — -A— = i(!t-smucosJt)
т
14.22 Jcos2MdM = | + ^L?H = ,|(M + sinMC0SM)
14.23 I secMtanwdM = secw
14.24 j съе и cut и du = — cscw
14.25 I sinh и du = cosh и
14.26 I coshwdw = sinhw
14.27 I tanh и du — In cosh и
14.28 I coth и du = In sinh и
14.29 j sechMdw = sin (tanh u) or 2 tan eu
14.30 I cschMcut = In tanh- or — coth^ie"
14.31 I sech2MdM = tanh и
14.32 j csch2MdM = -cothw
14.33 I tanh2w.dM = и — tanhw

INDEFINITE INTEGRALS
59
14.34
14.35
14.36
14.37
14.38
14.39
14.40
14.41
14.42
14.43
14.44
14.45
14.46
14.47
14.48
coth2 и du = и — coth и
аи —
sinh 2м и
и *, . ,
— = -|(sinh и cosh и — и)
/
I sinh2 м
,, , sinh 2м и ,, . ,
cosh2 udu = — h — = |(sinh и cosh и + и)
I sech и tanh и du = — sech м
I csch м coth и du = — csch м
С du 1 ,м
I о i—о — — tan 1 —
j w л- а- а а
Jdu 1 /u — a\ 1 , ,n , . „
~5 9 = 7Г In ) = — coth— u2 > a2
u2 -a2 2a \u + a] a a
J. i / i \ i
a2-M2 2a " [а- и) ~ a
Jdw . _ и
_^_^ = sin *—
tanh"" m2 < a2
7=== = ln(M + VM2 + a2) or sinh-1-
lu2 + a2 a
/2 - „2
= In (m + V«*2 - a2 )
(' du 1
•— = - sec
•^ mVm2- a2
dw
= --In
a
= /<»-i>«r - /Cn-zj^' + f(n-z)g» _ ... (-1)» J*
This is called generalized integration by parts.
IMPORTANT TRANSFORMATIONS
Often in practice an integral can be simplified by using an appropriate transformation or substitution
and formula 14.6, page 57. The following list gives some transformations and their effects.
14.49 { F(ax + b)dx = - Г F(u) du
J a J
14.50 I F(y/ax + b) dx = — I uF{u)du
J a J
14.51 I F(y/ax + b)dx = — I un~1F(u)du
J a J
14.52 I ^(Va2 - x2) dx = a j F(a cos u) cos и du
where и — ax+ b
where и = -\/ax + b
n.
where и = у ах + b
where x = a sin и
14.53 I F{\Jx2 + a2)dx — a j F(a sec u) sec2 и du where x — a tan и

60 INDEFINITE INTEGRALS
14.54 I F(V«2 — a2 ) dx = a j F(a tan u) sec и tan и du where x — a sec и
14.55 Г F(ea*)dx = - C^^du where м = eax
J a J и
14.56 j F(\nx)dx = j F(u) eu du where и - In ж
14.57 I J1 ( sin- ) dx = a ^(м) cosm du where и = sin —
J \ a) J a
Similar results apply for other inverse trigonometric functions.
14.58 j F(smx,cosx)dx = 2 / F( Г+V'TlH?) ГТ1? where u = tan I
SPECIAL INTEGRALS
Pages 60 through 93 provide a table of integrals classified under special types. The remarks given on
page 57 apply here as well. It is assumed in all cases that division by zero is excluded.
INTEGRALS INVOLVING ax + Ъ
14.59 Г d* = -\n(ax + b)
J ax + b a v '
i л ?.п С xdx x b
14-6° J ^TJ = a-V*
J ax + b
2Ь(ах
2a3 as
14.62
гьцах
ax + b за4 2a4
x
14'63 f x(aT+b) = &1п
14.64 I -ir. —гг = — 7 h тя In
J ж2(аж + 6) bx b2
.65 J
- . , c f da; 2ax — 6a2,/ ж
14.65 J ^a>, , ,, = „t,_., + t^ In ( -
ж3(ааг+ Ь) 262аг2 б3 ^аж + 6
14.66 j **
(ах + бJ а(ах + 6)
. . ... С х dx Ь 1 , , ,.
14.67 I -. rrz = —ir. г-rz + —5 In (ax + 6)
J (ax + бJ аЦах + Ь) а2
...» Г x2dx ах+Ь Ь2 26, . ,.
14'69 J (аж + ЬJ - ~2^ S + «4(аж + 6) +~аТ
, „ —n Г dx 1 , 1 , / ж
14.70 j r_, .. , t40 = T7-—r-TT + т^ In (-
ж(аж + 6J 6(aa; + 6) 62 V аж + ь
I ff^lrfY -r* ni^
—a _ __1 2a / аж + 6

INDEFINITE INTEGRALS
61
14.72
14.73
14.74
14.75
14.76
14.77
14.78
14.79
14.80
14.81
14.82
Jdx
хЦах + bJ
Г dx
J (аж + 6K
Jxdx
<
Г x3 dx
X
(ax + bJ 3a(ax + b) asx За2, (ax + b
— ГТ-: ^ - -+- — — In
64
)
26%2
Ь*х
ЬЦах
-1
2(аж + 6J
-1
(аж + бK а2(ах + Ъ) 2a2(ax + bJ
x2dx _ 2b __ 62 + A\ ( +h)
\x + 6K a?(ax + 6) 2cv^(ax + 6J a3
_ JL _ 3 b2 b3 36 .
(аж + 6K а3 аЦах + 6) + 2а4(аЖ + 6J »4 In (аЖ + 0)
x(ax + 6K
dx
_1
б3
J хЦах + bK
X *
263(аж + бJ 63(аж + 6) б3
—a 2a
ax + Ъ
1 За, (ax + b
1 » л
63(аж+6) 63ж 6*
4а3ж _ (аж + бJ _
x3(ax + VK 2V>(ax + bJ ЪЦах+Ь) 2bW
б5
In
ах±Ь
dx =
j (ax+b)n
J X X ~ (n + 2)a2
If n = -1, -2, see 14.60, 14.67.
; = —1, see 14.59.
(w+l)a2
14.83
)dx
Ml ~ (« + 3)а3 (п+2)а3
If n = -1,-2,-3, see 14.61, 14.68, 14.75.
(п + 1)а3
X
xm(ax + b)n dx —
6)"
m + n + 1
(m + n+
(п+1)Ь
n j xm(ax + b)n~1dx
n + 2 С
nw J
(и+
INTEGRALS INVOLVING v/ar-I-
14.84 J
14.85 j
14.86 /
14.87
dx
I ax + b
xdx
/ax + b
x2dx
2yax + b
a
2(ax - 26)
3a2
Зд2ж2 - 4аЬж + 862)
15a3
X
dx
xv ax + b
¦ tan
ax + b +
ax + b
-6
¦ 4.88 J.
dx
fax +
26 J
dx
xv ax + 6
[See 14.87]

62
INDEFINITE INTEGRALS
14.89
14.90
14.91
14.92
14.93
14.94
14.95
14.96
14.97
14.98
14.99
14.100
14.101
14.102
14.103
14.104
J
s-
x-
ax + b
myjax + b dx
ax + b
(m — \)bxm
_ Bm - 3)д Г
-» Bm -2N J xm~
dx
dx
dx =
ax + b
dx
ax + b
dx =
fy/ax
J —
J
j x(ax + b)m'2 dx>
J хЦах + b)m'2 dx
Г (ax +
J a
» Г
(m — l)xm~l "^ 2(пг — 1) J xm-\
-(ax + by'2 Bm- 5)a Г y/ax+b
(m-Vibx™-1 Bm-2N J a;1"
_ 2(аж + bI
xm~ly/ax + b
ax+b
a(m + 2)
a2(m + 4)
a2(m + 2)
262(аж +
а3(т
а3(т
2(ax + b)^'2 b С (ах +
-dx
Г (ax +
к/ Л
X:
b)m/2
2
dx
dx = -¦
bx
2
ma
26
С (ах + 6)"
dx
ж(аж -
INTEGRALS INVOLVING ax + b AND px + q
14.105 Г
14.106
14.107
dx
1
14.108
14.109
/
/
J
(ю +
(ax +
(ax +
(ax +
b)(px + q)
xdx
6) (pa; + q)
dx
bJ(px + a)
xdx
bJ(px + q)
x2dx
In
+ b
bp — aq
т— \-In (ax + b) - - In (px + q)
bp - aq \a V
1 p I px + q
bp — aq ]ax + b bp — aq \ax + b
. fax + b\
In . .. , .. -
bp — aq \bp — aq \px + qj a(ax + 6)
(ax + bJ(px + q) (bp - aq)a2(ax + b) (bp-aqJ [p
In (px + q) +
b(bp - 2aq)
In (ax + b)

INDEFINITE INTEGRALS
63
14.110
14.111
dx
-1
(ax + 6)™(рж + g)" (и - l)(bp - aq) 1 (еж + Ь)™-Црх + g)"
a(m + n-2)
аж + 6 , _ ax bp — aq
s
dx
(ax + b)m(px + a)"
px + q
dx — h
In (px + q)
14.112
(ax + b)m
-1
— iWy; + (n-m-2)af
(ax + b)m
(рж + д)"-1
1 dx
(рж + g)"
(и-1)р
C
J
INTEGRALS INVOLVING д/«^ + 6 AND px + q
14.113
14.Г
I ax 4- 6
da;
За2
(px + а) у ax + b
6) — Vbp - aq\
/bp — aq yfp \y'p(ax + b) + y'bp — aqt
2 '
, / р(аж 4
- tan-U|—
'"T V ад —
14.115
рж + g
I aq —
Zyax + Ъ y'bp — aq jy'p(ax + b) — y'bp
'p(ax + b) + ybp —
iy'ax + b 2y'aq — bp
pVp
tan
Vp(ax -
aq —
+ 6)
bp
... C
.116 J
14
14.117
14.118 /
dx
fax
Bи - S)a С
n'1 2(n - l)(aa - bp) J
(px + g)"
. /аж 4- 6
14.119 | , .. , ._ dx
Bn 4- \)a
—V'иж 4- 6
(рж 4- g)"
(w — 1)р(рж
2(и-1)(ад-6р)
2и(ад — 6p) С (рх 4- g)"^
Bп4-1)а J
2(«-l)pJ (рж + д)»-1\/«ж+Ь
da;
INTEGRALS INVOLVING yJax + Ъ AND \/px + q
14.120
14.121
dx
—=r In (ija(px + q) + -\/p(ax + b) )
b)(px + g)
>—ар
у1 (ах + b)(px + q)
bp + aq С
ар
dx
2ap J V(ax + b)(px + q)
dx
q)n

64 INDEFINITE INTEGRALS
Mi.)» С г. . , w ;—; , 2apx 4- bp 4- aq n nxi 1—\ (bp — aqJ С dx
.122 | у (ax + b)(px + q) dx = —? :—? - V (ax + b)(px + q) ?rr: I =
b)(px 4- q)
{px + q) ^/(ax + b)(px + q) (aq — bp)
INTEGRALS INVOLVING x2 + a2
лл юс С dx 1 ,%
14.125 I -5——5- = -tan —
J x2 + a2 a a
14.126
14.127
x* + a* a
14.128 f fdXa2 = ~- ~
14>129 J a;(a;24-a2) = 2a2 ln \Ж+
14.130 J; dX - X X
14.131 J ¦ „;..o , _„ч = — г.„о„«) — 7Г^ In I
2a2a;2 2a« '" \^a;2 4- »2
14.132 J /^2+^2J = 2a.g г+-Цип-1|
14 133 Г—^^__ - —1
J (a-2 4-«2J 2(a;24-a2)
14.134
14.135
14 136 Г dx = 1 4 1 1- / <*_ \
J а;(а;2 4- a^J 2»2(a;2 4- a2) 2a" \a;2 4- a2 j
Г
J
14 138 f__^__ - 1 1 1 .
J ;3(a;2 4»2J 2а%2 2«B 42) «
?
4-a2J а4ж 2а4(ж2 4-а2) 2a»
14139 f «to = i 2.rL_-_3_ f
J (a;24-a2)" 2(те-I)a2(a;24-a2)"-1 Bw-2)a2J
14 140 Г xdx — ~X
J (a;2 4-a2)" ~ 2(те-1)(а;2 + a2)n-i
?f* = 1 _i_ JL 1 _
a;(a;2 4-a2)n 2{n - l)a*(x2 + a2)»"! a2 J ж
14.142 1"^^. = J^'^-.-^Jg^
a2)"
da;
14143 С dx = If da; 1_ Г
J «"(a;2 4" »2)n a2 J j;».(a;2+(,2)»-l a2 J ^.m-
2(ж2 4- a2)"

INDEFINITE INTEGRALS 65
INTEGRALS INVOLVING ж2-a2, x2 > a2
лл ллл С dx 1 .(х — а\ 1 , х
14-144 j -j--^ = -m^-j or --coth-i-
14.145
M,., f x2dx a Ix —a
. 146 I -s 5- = a; + 7Г In —¦—
J x2 - a2 2 \ж + а
.. .._ Г Xя dx _ Ж2 «?!/¦>_¦
t/ X Or1 di Z
14.148 Г ^ = J_
" J X(X2. — a2) 2a2
14.149
J
,л ,cn I *? 1 1 , / ж2
14.150 I = ln '
a2) 2a2x2 2a4
14.151 C—^L— = 2~Ж_ а -
14.152 f Ж{гж = Zl
J (a;2-a2J 2(a;2-a2)
14-153 f~~w =
2(x2-a?) ' 4a"'\x + a
J x(x2-
14 155 Г rfa; - -1 1 1 i
J B?J 22B 2) 2«
a2)
14.156 f da; - 1 « 3 ,_fx-a
_ а2J а4ж 2а4(ж2 _ a2)
14 157 f__^__ = 1 1 1 I,.
J жЗ(а;2а2J 2а4ж2 24B 2) a8
=1 ,
жЗ(а;2_а2J 2а4ж2 2а4(ж2 - a2) a8 \x2 - a2
14 158 Г d^ _ zze — 2n — 3 С dx
J (ж2-»2)" 2(w-l)a2(a;2-a2)"-i Bn-2)a2J (ж2-а
14.159 I **e - "I
Г da; = -1 ! f
J x{x2-a2y 2(и-1)а2(ж2-а2)™-1 a2J
14 160
141A1 f ж"» da; Г ж"» da; 2 Г a;"»-2^
J (a;2-a2)n J (ж2_а2)п-1 J (ж2 _ агу
14IAO f d^ = If dx If dx
J a;m(z2 - a2)" a2 J a;m-2(a;2 - a2)" a2 J а;т(ж2_ a2)n-l

66 INDEFINITE INTEGRALS
INTEGRALS INVOLVING a2 - ж2, ж2 < a2
14.163 (-^-2 = f ln(t+) or I
J 2 - ж2 la \a — x J a
= -| In <»«
Mi ac I *¦ «¦* . a. la + x
•165 I -5 -5- = -ж + vrln'
ж3 <te _ _x?_ _ a?
r 2 2
14.166 f^
J a-5 -
14167 f,(/!,2) = 2Ь1п(^
a;2
14.168 J—-g—- = __L + J_ln
14 169 I ¦ — =
' J ж3(а2-ж2)
14.170 Г^ — = x
J (»2-ж2J 2а2(а2-ж2)
14 171 С _x_dx _ 1
J (а2-ж2J 2(а2-а;2)
т^ In
14.172 Г -r?^r= = x . -±.\n(a + x
J {a2 — xly 2(a2 — a;2) 4a \ a — x
14.173
14.174 Г
da;
ж(а2 - ж2J 2а2(а2 - х2) 2а* \а2 - х2
14 175 Г dx = -1 + ж | 3 in (а + ж
J ж2(а2-а;2J а<ж 2а4(а2-ж2) ^ 4а» \a-x
14 176 Г d« _ -1 . 1 , j_, / ж2
J ж3(а2 - ж2J 2а«ж2 2а4(а2 - ж2) а6 \^а2 - х2
14177 Г dx = « 2п-3 Г
J (а2-ж2)" 2(п-1)а2(а2-ж2)"-1 B%-2)«2J (а2-
14.178 Г жсга; = I
J (а2-ж2)п 2(№-1)(а2-ж2)"-1
14 179 Г dx = 1 + — Г
J ж(а2-а;2)" 2(л- 1)а2(а2- a;2)n-i ^ а2 J
da;
14 180 Г *mda; = а2 Г ^m~2da; Г ^m^
J (а2-ж2)" J (a2-a;2)« J (а2_ж2)п-1
14181 Г dx = If «fa ¦ 1 Г d*
J a;m(a2 - a;2)" a2 J жт(а2- ж2)« a2 J жт-2(а2 - a;2)"

INDEFINITE INTEGRALS
67
14.182
14.183
14.184
14.185
14.186
14.187
14.188
14.189
14.190
14.191
14.192
14.193
14.194
14.195
14.196
14.197
14.198
14.199
14.200
14.201
14.202
'x2 + a2
xdx
/x2 + a2
x2dx
/ж2 + а2
x3 dx
/x2 + a2
dx
INTEGRALS INVOLVING л/ж2 + a2
= In (x + y/x2 + a2) or sinh —
a
x\lx2 + a2
- -5- In (ж + V*2 + a2)
/ж2 + а2
dx
x2y/x2 + a2
dx
1, / a-
= - - In
a
lx2 + a2
ln
a +
X
X:
/v
X;
/
J xs\/x2 + a*
Г \/x2 + a2 dx =
{xVx^+^dx = (*2 + K>2
J 3
/„ /—« : ог(у2 + «2\з/2 a2xy/x2 + a2 a*
x2\lx2 + a2 dx — x[x + a > - %¦ In (x
4 8 8
/
/-
/¦
X
^/ж2 + a2
a2x
/x2 + a2
2a2x2
\Jx2 + a2
y ln (x +
а2)
x*y/x2 + a?dx = (ж2 + f
о
ж
/ж^+a2
ж2
/жг+a2
da; =
- a In
dx = —¦
+ ln (x + V«2 + a2)
dx = —
/ж2 + a2
2ж2
2а
f dx
J (я2 + а2
Г xdx
x
J (
Г
x2 + a2K'*
x2dx
a2Va;2 + a2
-1
V<»2 + a2
—a;
In (ж 4
а2
(ж2 + а2K/2
/x2 + a2
+ а2K'2
dx
a2y/x2 + a2
y/x2 + a2
1 . /о +
х In
X
/
к2(ж2 + а2)з/2
dx
a*x
-1
a*\/x2 + a2
3
3 / а + /ж2 + а2
= + ;г^ In

68
INDEFINITE INTEGRALS
14.203 Г (ж2 + »2)з/2 dx =
14.204 Г Ж(Ж2 + а2K/2 d:
14.205 Г х2(х2 + а-
14.206 Г
14.207 Г
14.208 j
14.209
~(r2 о. „213/2 3a2xVx2 + а2 о ,
Щ A ' + + -|a4 In (x + Уж2 + a2 )
(x2 + a2
.,.. =
(О I O\fv/O О / О i О\ О /О
Ж "Т* О/} CL СС\йС ~т~ CL ) '
~ 6 24
Д2M/2
X2 + a2 6
! + а2
а + у/х2 + а2\
х )
In (ж +
'а + У*2 + а2\
^Л/Г2 4- „2 - -„ In 1
1х2 + а2)
INTEGRALS INVOLVING л/х2 - а2
14.210
da;
- „2
- \п(х + у/х2 - а2),
xdx
,4.211 f^^?L
¦^ Уж2 —
14.212 Г
XV х*-а*
а2
+ -=- In (х + Vic2 - а2)
= (X2- «2K/2
- a2
14.213
Г da? 1 ,
I ——=г- = — sec
14.214 Г
14.215 J
- a2
.. Уж2-»2
14.216 Г Уж2 - а2 dx =
14.217 Г жУж2 - a2 dx -
14.218 I ж2Уж2 — а2 da;
14.219
2а2ж2
sec
2
2 r
~ In (ж + V»2 -
13/2 а2ж
2жуж2 — a2
14.220 j — dж = Уж2 - а2 - а sec-1
•/ ж
j;2-»2
14.221
14.222 j
14.223 frH^
¦ dx = — ¦
+ In (ж + ^x2 - a2)
! - а2 Уж2 - а2
-5 dж = —
-a2K'2
а~~9— + тг~ sec
2ж2 2а
_ „2

INDEFINITE INTEGRALS
69
, 4.224 fr^
J (хг — а-
х dx
,2K/2
Т4.225 fj-fi^.
J (ж2 — a-
x3 dx
x2dx
,2K/2
Ix2 - a2
X
«2 _ „2
+ In (x + Уж2 - a2 )
14.226 fjp=%
14.227 /
14.228 f
dx
- 75 sec
dx
14.229
J;
;2 - a2K/2
dx
a4x
1
sec"
14.230 Г (ж2-a
14.231 j
14.232
14.233
14.234 Г(xi ~^2K/2 rfa;
- a2K/2
!-a2 , 3 ., .
1- ^a4 In (ж
- a2
a2)
Z4
lb
(Х2 - „2K/2 _ _
i —Л— — а2ух2 — а2 + a3 sec x
3
14.235
14.236
Г (ж2 - a2
J x2
Г (жа - а2
-dx =
(ж2-а2K/2 зжУж2-а2 3 „. . , г-; j,
1 —a2 In (ж + у я2 — а2)
(Ж2 _ а2K/2 3V»2 - а2
+
INTEGRALS INVOLVING v/a2 - x2
Moo-» Г йж
•237 I
^ Уа2 -
14.238
14.239
14.240 f-g!ig-
14.241
+ — sin * —
2 a
g2 -
14.242 Г ^
J x2^Ja2 - x2
14.243 Г ^
J a;3Va2 - x2
2 ~ x2
/a*-x
/a2 ~ x2
2a2x2
2a3'
a + Va2

70
INDEFINITE INTEGRALS
xy/a2 — x1 a2 . , x
+ — sin-1-
Z 2. a
14.244 Г V»2 - х2 d
14.245 Г ж\/а2 - ж2
14.246 /жУ^Ма; = _«^-«Уд +
14.247 J жЗ^^Г^
14.248
•^ ж
8 a.
a2 -
dx — ya2 — x2 — a In
a + ycfl — x2
14.249
14.250
ж2
dx = — •
da; =
— sm-
2aln
xdx
2K/2
,4.251 J(^|
14253 ha
14.254 f- 2^..2
14.255
x
x2^»2 - ж2
1
/a2 - ж2
14.256
-a;2K
Г da;
Г dx
J x2(a2-x2)s'2
14 257 Г ^
J xs(a2 — x2)s/2
14.258 J (о*-;
14.259 J«(a2-:
14.260 Г ж2(а2 - ж2)з/2 dx
14.261
— sin —
/a2 - x2
-]ь
a*\/a2 —
- ж2K
—/
1
-a4sin-i-
Ж(а2-а;2M/2 а2а;(а2 - «2K/2
_ + _
в
14.262 Г<°2-*2>3%ж =
^ ж
14.263 I 5 йж =
J a;2
14.264 f(a2-fS/2dx =
7
j2 - ж2K^
3 +
(a2 - x2F2
— a3 In
— — a* sin"
(a2 _ ж2K/2 3\/и2 - ж2 3 , / a + Va2 - ж2
2x~2 2 + 2alnV x

INDEFINITE INTEGRALS
71
INTEGRALS INVOLVING ax2 + bx + с
14.265
/;
dx
2 . . 2ax + Ъ
: tan
Uac - b2
4ac - 62
In
If 62 = 4ac, aa;2 + Ьх + с
results on page 64. It a or с —
У»2-4ас \2ax + b + У&2 - 4ac/
= a(x + b/2aJ and the results on pages 60-61 can be used. If 6 = 0 use
0 use results on pages 60-61.
14.266 J
J
*d*
14.267
14.268
14.269
14.270
aa;2 + 6a; + с
ж2 da;
aa;2 + 6a; + с
™dx
+ bx + с
da;
= ^- In (ax2 + bx + c) -
2a ^ '
ij i
da;
a; ?>
~ ~ n~? In (ax + bx + c)
a 2a2 ч '
2a J ax2 + bx + e
62 - 2ac С dx
•S;
f—^
J ax2 4
к
2a2 J ax2 + bx + с
m~2dx 6 Г xm^1dx
(aa;2 + bx + c)
J xH
dx
x2(ax2 + bx + c)
14 271 Г —
J xn(ax2 +bx + c)
14.272 J dx
(m — l)
2c
_ _6_
2c2
i _ с Г xm~2dx _ Ь Г
)a a J ax2 + bx + с a J
x2 + bx + с
_ _6 Г dx
2c J ax2 + ba
6a; + с
+bx + c
ex
ex 2c2
dx
J ax2
dx
+ 6a; + с
= 1 b Г dx а С
(№-1)ct"-1 cj xn~l(ax2 + bx + c) cj xn~
dx
~2(ax2 + bx + c)
2aa; + 6
2а
(ax2 +bx + cJ
2 J
da;
14 273
14 274
Г
J {ax2
xdx
{ax2 +bx + сJ
Dac - 62)(aa;2 + bx + c) 4ac - 62 J ax2 + bx + с
bx + 2c b Г dx
Г
2 J i
/x-
(ax2 +
/
Dac _ 62)(aa;2 + bx + e) 4ac - b2 J ax2 + bx + с
dx
2dx _
bx + cJ aDac - ЬЩах2 + bx + c)
- 2ac)x + be 2c
(ах2+Ъх + с)"
-т-
(rt -
'±_ С
- 62 J ax2 + bx + с
J
Bn-m-l)aJ (aa;2 + 6a; + c)"
xm~1dx
14 276 Г
J
a;2" da;
n~2dx
(ax2+bx + c)n
14 277
14.278
14 279
s
da;
(ax2 + bx + cJ
dx
(ax2 +bx + cJ
n-m)b С x^_
Bn — m — l)a J (ax2 + bx + c)"
1 С x2n~3dx _ с С a;2"-3 da; 6 С х2п'
aj (ax2 + bx + c)n~1 aj (ax2 + bx + c)n aj (ax2 + bx + c)n
1 — Г — — Г ^x
2cJ (ax2 + bx Л-сJ с J \
2c(ax2 + bx + c)
1
(aa;2 + ba; + cJ
da;
За Г dx 26f_
с J (ax2 + bx + сJ с J x<
a;(aa;2 + 6a; + c)
da;
—
xm(ax2 + bx + c)"
cx(ax2 + bx + c) с J (ax2 + bx + cJ с J x(ax2 + bx + cJ
1 (m + 2n — S)a С dx
2 + bx + c)"
-2)Ъ
(m-l)c
Г
J жт-!
dx

72
INDEFINITE INTEGRALS
INTEGRALS INVOLVING л/ах2 + Ъх + с
In the following results if b2 = 4ac, \/ax2 + bx + с = -/a (x + b/2a) and the results on pages 60-61 can
be used. If 6 = 0 use the results on pages 67-70. If a = 0 or с = 0 use the results on pages 61-62.
14.280
/
dx
14.281 Г
14.282 Г
bx + с
/4ас - 62
2aJ y/ax2 + bx + с
bx + c
14.283 С —
J х\
14.284 Г —
J X2
14.285 Г у[у
( х
dx
+ bx + с
1_ jn
V
bx
bx + 2c
Vc
x\ л/4ас -
+ bx + с
-If
dx
Baa; + 6) V~tta;2 + bx + c 4ac - Ь2
-. +
4a 8a
ху/ах2 + bx + с
14.286
_ bBax + 6)
За 8а2
_ ЬDас - Ь2) Г dx
14.287 Г
14.288 Г
14.289 Г
14.290 ?
14.291 Г
bx + cdx -
24a2
(aa;2 + bx + с)з/2
4ас
16а2
— + с
bx + e
dx
bx + с
dx
2Bаа; + Ь)
a I
J
bx + с
dx
bx + e dx
dx
y/ax2 +
+
xy/ax2 + bx + с
6 /* da;
Г
I
J
x\/ax2 + bx + с
(ax2 + bx + с)*'2 Dac-b2)y/ax2+bx
a; da; _ 2(fta; + 2c)
,4292
/i
B62 - 4ac)a; + 2bc
(ax2 + bx
14 293 Г
1 Г
c) x
l /'
a J
dx
dx
bx
b С dx
2c J (аж2 + 6а; +
14 294
14.295
Г
J
2bx + c
ЗЬ Г
2c2 J
4-
dx
2ac
Г
J
x\]ax2 + 6a; + с
Bи + l)Dac - б2) С . „ , , , . ..„ ,
n / , ,ч * I (аж + bx + c)n~l'2dx
aa(n+1) J

INDEFINITE INTEGRALS 73
14.296 f*(e*s + b* + c)»+i/2d* = ("**+b* + g"
J a{Zn+ 6)
14 297 f — = 2Bax + 6)
J B + b )n + i'2 ()B)B
14.298
8a{n ~ 1) С dx_
Bи — l)Dac — Ь2)./ (аж2 + Ьа; +
da; __ 1
da;
_1 Г dx Ъ_ Г
cj х(ах2 + Ьх + с)"-1'2 2с J (аж2 +
Ьх
INTEGRALS INVOLVING ж3 + а3
Note that for formulas involving a;3 — a3 replace a by —a.
14 299 f
dx 1 , (x + aJ 1 2a; — a
:tan~x —-
l a "inn f x dx 1 ж2 — ax + a2 , 1 , 2ж — а
14-300 J = ln +tan-1
14'301 J^fe = ><* + "> 14'302
i л -ъм С dx 11, x2 - ax + a2 1 2ж - а
14.303 I „ = = т-j In —, ,2— tan —-
J a;2(a;3 + а3) а3а; 6а4 (ж + а)г а4\/з а\/3
da; x 1 , (ж + aJ . 2
J
^' ж da; _
1 (ж + а) 2
95ln x2 аж + a2 + ^
1 a;2 - aa; + a2 1 _t 2ж - a
ln + tan
_t
J4 306 f x2dx
-Li
36 m
Г ^^ _
J a;(a;3 + a3J -
Г ^ = 1- - x2 _ J_ С xdx [See 14.300]
j 3.2(^3 + аЗJ авж Зав(хз + а3) 3a*J ж3 + аз L J
14.309
14.310
2 J a;3 + as
da;
Г ^ж - -1 1 Г
J a;"(a;3 + a3) ~ ацп-х)хп-1 asJ а;"-з
INTEGRALS INVOLVING X* ± a*
14.311 Г ¦ ^ ai = -J-zln^^-
, . .,_ С xdx 1 . .a;2
14.312 J iTT^r = ^Jtan-i^
14.313
14.314

74
INDEFINITE INTEGRALS
14.315
14.316
14.317
14.318
14.319
14.320
14.321
14.322
14.323
С dx __ 1 / х4
¦а4)
1 /х2 - ахф. + а2
In
4a5\/2
2a5\/2
tan"
ax
\/2
J хЦх* + а*)
1 1 . . ж2
~ № tan ^
/;
dx _ 1
4а2
1 ,
a;4 — a4
In
ж3 da;
t«,i
2a a
J:
J x(x4-a4) ~ 4а4
ах _ 1 1 . / х — а \ 1
/y2//v-4 — fiA\ п^т Ап*> \ т А- а I 9G
At \л> LL ) XX, X Ч\Лг \ А, П^ tb / tuUi
V
х*- а4
ж4
_ , tan-i-
2а5 а
f_^L_ = _^_ + _L_ln/x2-a2
x2+ a
INTEGRALS INVOLVING xn ± a"
14.325
J
dx
1
In-
x(xn + а") теа" "* xn + a"
14.326
14.327 Г ni^nw
14.329 С—Й=
14.330 Г-г-^—г
J x(xn-an)
\xn + a")r
J^ С dx 1^ Г dx
anj xm(xn + an)r~x anj xm~n(xn + a")r
/ж» + a" - \/a"\
.
a;n + а" + л/а"/
_J_ . /ж" — a"
теа" l
Jrn—1 /7r I
14.331
14-332 jw^iw =
14.ззз J__^__
14.334 ( ,dX
n Г a;"-" fa Г xm~ndx
a J (xn-an)r J (xn-an)r-1
_ 1 Г dx If dx
anj xm-n(xn — an)r a"J жт(ж" - a»)"— i

INDEFINITE INTEGRALS
75
14 335 f x"-ldx
* "" J я2™ + o*™
,,-ti WlzDsz tstr,-i (x + a cos №k ~~ 1)'r/2m]
a sin fBfc — l)ir,
tan
Bfc - 1)ртг
where 0 < p f= 2m.
ln
14.336 (-^~^,
fc>r
a sin (ктг/т)
where 0 < p ^ 2тп.
f-vp— 1 Ant
Bm + l)a2m-p
_t /x + a cos [2WBw + 1)]
2m + 1 аП ^ a sin [2fcr/Bm + 1)]
x + 2ax
+
where 0 < p S 2m + 1.
14-338 fsJT-tL + i
Bm +
Bm + I)a2
-2 S .
I)a2m~p + 1 fc = iSm 2m
ж - acos[2fcWBm+l)]
1)]
Bm+l)a?m~p+ik = i 2m+ 1
ln (ж — a)
Г
2m + 1
+
where 0 < p = 2m + 1.
INTEGRALS INVOLVING sin ax
cos ax
xcosax
2x
si
14.339 i sin аж йж =
14^340 I a; sin aa; dx =
14.341 |'x2si
14.342 Г жз sin аж dx = ( ^ - -^ ) sin ax +
Ы*., /^ sin aa; ,
.343 I dx =
14.344
2x /2 x\
sin ax dx — —5- sin act + ( -j cos ax
a2 ^a3 a /
o? a
cos ax
5*5!
J X
[see 14.373]
14.345
f'_I^_ ^ i In (esc ax-cot ax) = -In tan ^
J sm aa; a a 2
14.347
Г
J
dx = ?- sin 2aa:
2 4a

76
INDEFINITE INTEGRALS
14.348
14.349
14.350
14.351
14.352
14.353
14.354
14.355
14.356
14.357
14.358
14.359
14.360
fxSin2axdx = ^ - x sin 2ax - cos2ax
J 4 4a 8a2
Jsir
sin3 ax dx =
cos ax , cosJ ax
sm*axdx = 3* _
a 3a
sin 2ax sin <
32a
J si:
da; 1 ,
~s = cot ax
г ах а
Г dx
, * sin3 ax
la sin2 ax
sin px sin gx
dx 1 / тг . ax
: = - tan — H
— sin ax a \ 4 2
a; da;
I г
Г dx
J 1 + si:
S
J A + sin axJ
2(p - q)
2(p
= ±
sra ax
x dx
- sin ax
dx
4 2
x , / я* ax \ , л , . I тг ах
= tan T - — H = In sin 7 + -—
a \4 2 / a2 \4 2
1
-^ , у i. , ax \ , 1 . ¦> / тг , aa
A-sin axJ = 2^tanU + Ty + 6^tan U + T
dx
p + q sin ax
tan^1
p tan iax + g
п2-„2
aVpz-gz
1
In
14.361
14.362
14.363
14.364
14.365
14.366
14.367
14.368
p tan J-ax + g — Vg2 — p2
P2 "" \p tan lax + g + V<72 - P2/
If p = ±g see 14.354 and 14.356.
dx _ я cos ax
J
(p + g sin aiJ a(p2 — g2)(p + g sin aa;)
If p = ±q see 14.358 and 14.359.
p С dx
P2 — g2 J p + g sir
da;
w p2 + g2 sin2 <
tan
VP2 + 92 tan ax
da;
xm sm aa; da; — —
/p2 + g2
1 _ Vp2 "~ 92 tan ax
tan"
sin ax
$
Jsin с
xn
I sin"
J sin"
P
/g2 — p2 tan ax + p
2apVg2 — P2 VV92 — P2 tan ax — p
xmcosax mxm^1smax m(m —
In
dx = -¦
a
sin aa;
m-l) Г ^.
a2 J
^_ Г cos ax dx [See 14.395]
axdx = _
sin""'ax cos ax
dx
"~a
dx
a-re
cos ax
— I sin"
 ax dx
+
w-2 Г dx
те - 1 J sin"
1
I —1) sin™ аж а2(те — 1)(и —2)
lax а(и —1) sin"^1 ax
Г x dx _ —x cos ax 1 _j_ ^ — 2 Г
J sin" ax ~ a(m-l) sin™ аж ~ а2(те - l)(w — 2) sin"-2ax и —lj :
x da;

INDEFINITE INTEGRALS 77
INTEGRALS INVOLVING cos ax
sin ax
14.369 Г cos ax dx = ^
J a
14.370 Г x cos ax dx = cos aa: + x sin '
.y a2 a
14.371 I
J
, 2x fx2 ¦ 2\ .
cos aa: dx = —5- cos aa: + 5 I sin ax
a2 \a a3/
5 cos aa: + 5
a2 \a a3
,. „. f , , [Zx2 б\ , /ж3 6оЛ .
14.372 I xJ cos ax ax = I — cos ax + I -I sin aa:
J у a2 a4/ \a a3/
14.373 I da; = In x — ^——. + -—— — — -I- • • ¦
14.374 ( cos"-x. dx = —cos ax - a I aX dx [See 14.3431
X2 X J X ' J
14'375 /c^ = i In (sec a, + tan ax) = J
J cosaa; a2[ 2
Ы37А Г ж<гж = 1 ((a»J 1 («L 5(ax)«
J cosaa; a2[ 2 8 144
( 1
a2[ 2 8 144 B% + 2)Bю)!
14.377 С
J
cos* ax dx = g + sin2aa:
2 4o
14.378 Г * cos2az dx = ^ + x sin 2ga: + cos2aa;
J 4 4a 8a2
14.379 Г cosSaa; da; = sin ax - si aa:
»/ a 3a
Moon Г д j За: . sin 2aa: sin 4ax
.380 I cos4 ax da: = — H h
8 4a 32a
14 481 С dx __ tan ax
j cos2 аж
a
,4.382
>s3 aa: 2a cos2 aa: 2a ^4 2
.. nQo Г j sin (a — p)x sin(a+p)x гт„ ,_
14.383 I cos aa; cos px dx = 7 — h —тгг i-r~ If «• — *?,
J 2(a - p) 2(a + p)
14.384
14.385
14.386
14.387
14.388
14.389
J
/
/
/
/
1
1
1
1
<!
A
da:
— cos ax
x dx
— cos ax
dx
+ cos aa;
a: da;
+ cos aa:
dx
— cos aa:J
da;
+ cos axJ
1 , ax
cot —
a 2
^cot^
a 2
1 tan аж
tan
a 2
a tan 2
1
2a c0
1 ax
" 2atan 2
+ —5 In sin
a2
2 , aa
a2 S 2
2 6a
1 fn_3
ba
ол;
2
с
ax
' 2
aa;
2

78
INDEFINITE INTEGRALS
14.390
14.391
14.392
14.393
14.394
14.395
14.396
14.397
14.398
dx
J p + q cos ax
dx
(p + q cos axJ
dx
a-V p2 - q2
1
tan V(P - q)l(p + q) tan \ax
I tan Aax ¦
In ?
[If p = ±q see
14'384 and 14'386-]
- p* yt;
g sin ax
cos ax)
92 - P2 J P
dx
[If p = ±q see
+ qr cos aa; 14.388 and 14.389.]
J p
Г dx
J p2 - q2 cc
Г xm
cos2 ax
cosax dx =
apypi — (
1
:ln
2apyq2 — p2 \p tan aa;
p tan aa; — V<72 ~ P2
_ пфп-1) С x
a2- J
J
cos"aa;
an
sin ax
B^lJ Г со,,-!
n J
и - 2 Г dx
n-lj соэп^2а
Г xdx
J cosna
а(и — 1) cos"-1 ax
x Sin aa; i
a(n — l) cos*-1 ax a2(n ~ l)(w - 2) cos"-2 ax
INTEGRALS INVOLVING sinew AND
xdx
w-2 Г x_d
n-lj cos"~
ax
14.399 I sin ax cos ax dx =
sin2 ax
2a
2(p + g)
[If те = -1, see 14.440.]
l ji jaa i • j cos (?) — o)a;
14.400 I sin px cos qx dx = — ———
J 2(p-q)
14.401 I sin" oa; cos aa; da; =
14.402 Г cos" oa; sin aa; dx = -cos" + \ttic [If n = -1, see 14.
J (n + l)a
14.403 I sin2 aa; cos2 aa; da;
429.
8 32a
14.404 f^
J si
da;
= - In tan aa;
14.405 Г-
J si
14.406
sin aa; cos aa; а
л. , , i v , ax \ 1
= - In tan tH 1 —
da;
sin2 aa; cosaa; a \^4 2 у a sin aa;
= Ilntanf
2
Г . dx 2 = Ilntanf + -^—
J sin ая cos2 ox а 2. a cos аж
14.407
/
dx
2 cot lax
sin2 aa; cos2 aa;

INDEFINITE INTEGRALS
79
14.408
14.409
14.410
14.411
14.412
14.413
14.414
14.415
14.416
14.417
14.418
14.419
14.420
J ci
sin2 асе , sin асе . 1 , , I ax тт
dx = — + - In tan \- —
, sin ax , 1 , ,
dx — — + - In tan + 7
cos ax а а у 2 4
f COS
J sii
J
' ax j cos ax , 1 , , ax
dx = -I In tan —
асе а а. 2
dx
cos ажA ± sin асе)
dx
sin асеA ± cos ax)
dx
sin ax ± cos ace
sin ace dx
sin ace ± cos ace 2 """ 2a
1
2a(l ± cos ace)
1 Intanff ±±
1
+ к~ In (sin ax ± cos ax)
2a
ax
ax
Y
J-
r—
cos ax dx a; 1 . . .
— = ±— + ¦=- m sin ax ± cos ax)
sin ax ± cos аж 2 2a
sin ax dx
q cos аж
cos ace dx
In (p + q cos ax)
/cos ace dee 1 .
—: ; = — In (p + q sin ace)
p + q Sin ax aq>
J;
sin ax dx
(p + gcosace)" ag(n - l)(p + g cos ax)"
/cos ace dee _ —1
(p + q sin ax)" ~ aq{n - l)(p + q sin ace)"-1
dee 1
J p si
sin ax + q cos ax
J p si
dec
sin ace + q cos ax + r
: tan~
_j /p + (r — q) tan (аж/2)
-p2-q2
In
p — Vp2 + q2 - r2 + (r - q) tan (ax/2)\
a\/p2 + q2 - r2 \p + Vp2 + q2 - r2 + (r - q) tan (ax/2) /
If r = q see 14.421. If r2 = p2 + q2 see 14.422.
14.421
14.422
14.423
14.424
14.425
J V si
dee _ J_.
sin act; + g(l + cos ace) ap
dx
p sin ax + q cos ace ±
p tan — )
2 /
a\/p2 + <72
J p2 si
dee
_ 1 tan-i /ptanaceX
sin2 ace + g2 cos2 ax apq \ 4 J
J p2 sii
dx
In
sin2 ace — g2 cos2 ace 2apg V p tan ax + q
p tan ax — q
sinm аж cos" ax dx =
sinm~1ax cos" + 1 ax m —
a{m + n) m +
sinm + 1 ace cos" ace
a(m + n)
Jsinm~2 аж cos" ax dx
Jsinm ace cos"~2 ax dx

80
INDEFINITE INTEGRALS
14.426
14.427
14.428
Jsinm
cos"
ax
dx =
Jcosm
sinn<
dx =
dx
sin™ aa; cos" ax
sinm~l ax _ m — 1 С sin 2 ace
a(« —1) cos" ax и —lj cos"-2 ax
t - то + 2 Г sin"
/t x ,y cos
n — \ С sin"*-2 a
re — то J cos" ax
sinm + 1 ax
a(n — 1) cos" 1 ax
— sin™ — 1 ax
. a(m — те) cos" ax
dx
-cos-1 ax _ m - 1 Г cosm-2
о(те- 1) sin"-1 аж n-lj sin"
aa;
ax
i ax _ m — то + 2 С cos™ act
а(те- 1) sin"-^ ~ и - 1 J sin"-2 ax dx
*-1m W, — 1 С COS"
— то J sir
-cos
cos ax
m —
. a(m — и) sin""J ax тог
sm-2
ax
sin" ax
dx
1
m
a(n — 1) sinm 1 ax cos" ax
=
la(m — 1) sin1" ax cos" ax
+ и-2 Г
те — 1 J si
то- 2 Г_
- 1 J si
sinm ax cos"~z ax
m + n —
m
dx
gjnm-2 ax ,
14-434
14.436
INTEGRALS INVOLVING tan ax
tan ax dx = In cos ax = - In sec ax
a a
14.430 С tan* ax dx = ianax - x
J a
14.431 (tunPaxdx = Wrtax + -Incos as
J 2a a
14.432 Г tan»aa;sec2aa;da; = tan"+laa!
J (n + l)o
14.433 Г:
sec2 aa; , 1. ,
dx — — In tan ax
tan аж а
14.435 Cxtznaxdx = \
J 2
3 15 105
+ •¦• +
75
14.437 Г x tan2 аи da: =
14.438 f——^ =
J p + q tan аж
Bте+1)!
Bn -
+
In (q sin аж + p cos аж)
14.439 ftann
= tan
(то
" \ax - Г tan»-* oas «fe
- l)a J

INDEFINITE INTEGRALS 81
14.440 I cot ax dx — — In sin аж
14.441 Г
cot2 ax dx = -
а
cot ах
14.442 Г cots axdx = - S2*™L _ i in sin aa;
14.443
(и + l)a
esc2 аж , 1 ,
dx = --lncotaz
dx 1
~ In cos ax
14.444 Г
14.445 С
14 446 Г
J ~ . а2 ["" 9 225 Bте+1)!
14 447 fcotcta;dj; = 1 ах (ахK 22"Дп(ажJ"-'
J a; аж 3 135 Bw-l)Bn)!
14.448 j
cot аж а
* cotax
J_lnsina?C _
a2
+ lnsina?C
a a2 2
14.450
fcot»o*dr = -cot"-tax _ f^-,
J (те — l)a J
INTEGRALS INVOLVING sec ex
secaa;dx = - In (sec ax + tan ax) = ilntanf^
14.452 | Sec2ажdж = ^^
a
! dx
_ sec" aa;
14.453 J sec3 аж dж = sec аж^ tan аж + X_ ^ (gec m + ^ ^
14.454 j sec"ажtanаж
14.455 Г
14.456
_ sin аж
sec аж
,4.457
XX 1
a; sec2 аж dx = - tan ax -\ 5 In cos аж
а а2

82 INDEFINITE INTEGRALS
14 459 '«x x v i dx
dx _ x_ _ P [
q + p sec ax q q J P +
g cos ax
14.460 1 sec" ax da; = —— aa; an aa; _^ w ~ I sec" aedx
J o(rt -1) n — 1 J
INTEGRALS INVOLVING csc ax
/1 1 ПТ
csc aa; dx = — In (csc ax — cot aa;) = — In tan —
14.462 Г csc2 aa; da; = - cot ax
J a
14.463 Ceac»axdx = -csc ax cot ax
J 2a
14.464 f cBc»o*coto*cte = -csc"ax
J na
14.465 Г dx = _ cos ax
J
2a 2
Г
J
,,,,, С , if, («жK 7(аа;M 2B2n-i -1)? (axJn + i
14.466 I ж csc ar da; = -5 4 ax + "-z~- + * ' + ¦¦¦ + —-— +
J a2 [ 18 1800 Bn +1)!
14.468 f^csc^axdx = - ?_?^^ + J- in sin ax
+
a a
14.469 Г Ё* = ^-Pf—A [See 14.360]
J q + p csc ax q q J p + q sin ax l J
Г csc" ax <te = - <*с»-2аЕ cotas + n^l Ccscn-2ax dx
J a(n — 1) n - 1 J
14.470
INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
14.471 Г sin-1-da; = x sin - + V^2^
J a a
.472 I a; sin-da; - ( — - — ) sm - +
J a
2 a2\
—Ism
f
...„ f , . .ж, жЗ . , ж ,
14.473 I a;2sin-1-dx = -5- sin - H
J а 3 а
dx = 5 sin H
а 3 а 9
14 474 Г sin-i (ж/а) . = x (xlaY 1 • 3(x/a)s 1-3-5(х/аУ
J x a 2-3«3 2-4-5-5 2-4-6«7'7
14.475
x a \ x
2
14.476 ((sin-1-) dx = x (sin-1-) - 2x + 2\/a.2 - x2 sin -

INDEFINITE INTEGRALS
83
14.477
14.478
14.479
14.480
14.481
14.482
14.483
14.484
14.485
14.486
14.487
14.488
14.489
14.490
14.491
14.492
14.493
14.494
j cos-da; = ж cos - - Vo2- ж2
J a a
x cos 1 - dx = ( —
a
J
J
Г cos (x/a
x
s
2 - э;2
xv az — x
x2 cos -dx = —
a 3
dx = ^ .
[See 14.474]
cos
. cos (x/a) 1 j / a + Va2 - ж2
a; a
= ж ( cos - ) - 2ж - 2^
ay
-
a
= ж tan- - ^
a 2
jжtan^dж = 1(ж2 + а2) tan - -
J a z a
J X%
\~x - dx = — •
а 3
2
ж ax2 a3,
(x/a)
ж _ (ж/аK (ж/аM
а З2 52
dx = - -
Г tan (
J x2
\ cot-i-dx = ж cot- + ^1п(ж2 + а2
J a a 2
cot -
a
2 cot - da; = —-
а 3
/cot (ж/а)
ж2
= \ In ж - I
[See 14.486]
da; =
_ cot-1 (x/a) X i /ж2 + а2
ж 2а
ж sec - — а In (ж + у/х2 — а2) 0 < sec - < ^
Oj Oj 2i
ж sec - + а In (ж + у/х2 — а2) | < sec - < тг
/х sec — dx =
а
14.495
J
a;2sec-1-o!x
а
0 < sec-1 ^ < |
» < sec-1 - < тг
2 а

84
INDEFINITE INTEGRALS
14.496 Г sec 1 (х/а) dx
14.497 Г sec~^/a> dx
5 , 1 • 3 • 5(а/х)?
2-4-Б-Б + 2 • 4 • 6 • 7 • 7
0 < sec- < ?
а 2
¦x < sec - < я-
2 а
14.498 J
14.499
ж csc^1 h a In (ж + V*2 — а2)
0 <
c-1-«"ж =
а
, X тт
= a<2
ж esc a In (ж + V*2 — a2 ) ~Ъ < esc- < 0
а & <x
14.500 | xScsc-i-da; =
14.501
x axy/x2 — a2 a3 , , .
3- a--*-6 -!*<* +
0 < esc- < I
a 2
— -ц < esc - < 0
2 a
14.502
Г esc (ж/а)
Гж-Цх/а)
J ж2
_(a (а/жK 1 ¦ 3(й/жM 1« 3 • 5(а/жO
\ж 23'3 2«455 24677
da; =
esc (x/a)
esc-1 (x/a)
+
2 - „2
/xz-a:
ax
/a^a2
аж
О < esc - < 5
а 2
14.503 1 a;" sin1
m + 1
a2-x2
жт cos~l — dx
a
14.504 J
14.505
14.506 Г ж» cot x- dx =
J a
cos - +
m+1
x _ 1 С xm + 1
a ra + 1 J ya2_
1 Г ж" + :
: dx
da;
14.507 (
sec - dx
a
xm+l x а С Жт+1 ,
tan I „ , 2dx
m + 1 a m + 1 J x2 + aL
ЗГ7 ) T2 _i_ „2 dx
(x/g) _ а Г _ж
+ 1 J ,/:
+ 1J
^11 cot ? + -
m+1 a я
ra + 1
1 sec (ж/а)
mdx
ra + 1
Jx-
14.508 i j» esc
esc

то + 1
CSC
+
a;2-a2
xm dx
2 - „2
x^ — a
dx
_ci Г_ж^_Л
ra +
1 (x/a) _ а Г жт d
1 m + 1 J .^2 _
0 <
2<sec-<
0 < esc - < |
а 2
_|<cc-i|<0

INDEFINITE INTEGRALS
85
INTEGRALS INVOLVING еи
S
14.509 I e°xdx = —
pax I 1
xeax dx = f_ ( x _ A
a V a
14.510 J
14.511 J
14.512 fa;"e«*da; = хПеаХ - - Г ж"~
./ a « J
» I* o a2
1
14.513
14.515 f —
14.516 Г7-1
J (P +
dx
x 1
i
14.517 J-
14.518 J
14.519
qe~
e"*Smbxdx =
1 ln /e« - V=g7p
. 2a\/~pq \eax + yj—
сов Ь» + Ь Bin
!\ .. ... . .
if и = positive integer
/
14 520 Г же 'n 6 d — xepx(a s'n bx — b cos 6a;) _ enx{(o2 — 62) sin 6a; — 2o6 cos 6a;}
J a2 + b2 (a2 + 62J
14 521 Г ж aI 6 d = xeftx(a cos Ь^ + Ь sin 6a;) _ eax{(a2 — 62) cos 6a; + 2ab sin 6a;}
14.522
14.523 Г е«
14.524 Г е"
sin» 6ж da; =
(a sin 6ж - «6 cos 6c) +
a
cos" 6a; da; = <°tw" ^to (a cos 6a; + «6 sin 6») +
I e sin"~2 6ж dx
e cosn~2 6ж dx

86 INDEFINITE INTEGRALS
INTEGRALS INVOLVING In x
14.525 I Inxdx = x In x — x
xlnxdx - y (In x - 1)
14.527 j x™\nxdx = ^-jj-Ппа; - w^ J [If m =-1 see 14.528.]
14.528 Г
14.529 (±?
X
In ж , 1 , „
dx = - In2 x
x 2
14.530 Г
In2 a; dx = a; In2 x — 2x\nx + 2x
14.531 flH!^^ = Ij^llx [if M = -i see 14.532.]
J а; те + 1 l J
14'532
14-533
,4.534 f^ = In (in,) + (» + !) in« +
2*2! 3-3!
14.535 I \ппх dx — х \ппх — n I In":» dx
J, ч i /*
МОС In** ?C ** I
rpfty In ft ф л/v* ~-~ л-3-Х шл/ __ i ¦ /vTH I vifl'— 1 л* п/и
m + l m + lj
If m = -1 see 14.531.
14.537 Г In (я2 + a2) da; = * In (a;2 + a2) - 2a; + 2a tan"» -
*/ a
In (a;2 - a2) da; = a; In (a;2 - a2) - 2a; + a In ^-^
\ x a,
14.539 Г а;« In (a;2 ± a2) da; = «m + x Ь (ж2 ± a2) 2 Г _|
J m+1 m + 1J x2
INTEGRALS INVOLVING sinh
1 + 2
14.540 I sinh aa; da; =
J a
14.541 fa;sinhaa;dx = * cosh ax - sin\ax
J a a2
14.542 I a;2 sinh aa; da; — ( — + ~s ) cos^ ax f sinn ax

INDEFINITE INTEGRALS
87
14.543
14.544
14.545
14.546
14.547
14.548
14.549
14.550
14.551
14.552
14.553
14.554
14.555
14.556
14.557
14.558
14.559
14.560
14.561
Г sinh
/sinh
х-
dx = ax
sinh ax
. _ _
3-3! 5-5!
sinh ax
/" cosh ax , r^i -i . ^n^i
+ a I dx [See 14.565]
sinh ax
x dx
sinh ax
(axK 7(axM
18 1800
Bв + 1)!
Г sinhSax dx = sinh ax cosh ax _ ?
J 2a 2
fajsinhaeasdx = ^inh2ax cosh 2ax xj
J
si
8a2
eoth ax
sinh2 ax
f sinh ax sinh px dx =
sinh (a - p)s
2(a
For a = ±p see 14.547.
= ^ cosh ax sin px - p sinh ax cos px
a2 + p2
('sinhaxsinp^
«/
Г sinh ax cos pxdx = ^ cosh ax cos px + p sinh ax sin px
J a2 + p2
JV
dx
In
+ p - Vp2 + <72
q Sinh aX a^/p2 + q2 \qeax + p + ^p2 + q2 /
dx _ — q cosh ax , p С dx
J (p + q sinh axJ a(p2 + q2)(p + q sinh ax) P2 + q2 J P + q sinh ax
_1 tan \/<?2 — p2 tanh ax
P
/p + Vp2 — Ч2 tanh ааД
In
J P2
dx
+ q2 sinh2 ax
Г dx
J p2-q2 si
apyq2 — P2
l2apVp2-g2 \p - VP2-g2 tanhax/
p + \/p2 + g2 tanh aж^
I xm si
I sinh"
Jsinh
x»
sinh2 ax
sinh ax dx -
In
2ap\/p2 + 92 Vp - Vp2 + Ф tanh ax/
m cosh ax -
ax dx -
aX
sinhn-iaxcoshax
an
dx = —
(¦re
dx — cosh ax
sinh™ ax a(n — 1) sinh™'1 ax
xdx
/xd:
sinh"
cosh ax dx [See 14.585]
n — 1 С
I sinh"~2axdx
n J
h™i + —^— Г -0Snh_"a: dx [See 14.587]
to-2 С dx
n — 1 J sinh" ax
- 2 С xdi
, - 1 J sinh"-
— x cosh ax
x dx
a(w-l) sinh"'1 ax a2(n — l)(n — 2) sinhn'2ax

88 INDEFINITE INTEGRALS
INTEGRALS INVOLVING cosh ax
14.562 Г cosh ax dx = sinh ax
J a
14.563 fxcoshaxdx = x sinh ax - cosh ax
J a a2
14.564 J* *2 cosh ax dx = - 2xc°s2ha* + № + |) sinh ax
14.565 C™bl*dx = lnz + -<^+ i^+i^+ ...
J ж 2-2! T 4-4! ^ 6-6!
14.566 C™*?*dx = _eoshga; + a fsinh^^ [See 14.543]
14.567 f—^- = ^tan~ie«
./ cosh aa; a
14568 f j
J
__ = j _
coshaa; a2 \ 2 8 + 144 + ''' + Bn + 2)Bn)!
14.569
cosh aa;
f
14.570 f* cosh^a* «fa, = «! +
J 4
2a
«! + ж sinh 2cta: ?osh2a?
+ _ ?
4 4a 8a2
14 571 t dx _ tanh aa;
cosh2 aa; a
14.572 Г cosh aa; cosh px da; = sinh (ct ~ p)x + sinh (ct + p)x
J 2(a-p) 2(a + p)
14.573 f cosh aa; sin pa; dx = « »i"h a* gin ps - p cogh as cog p»
J a2 + p2
14.574 f cosh a* cos pa; d* = " sinh aa; cos px + p cosh aa; sin pa;
J a1 + pz
J cosh aa; — 1 a "'""" 2
. - ___ Г xdx x , ax 2 . , aa;
14.577 I —г r^r = — tanh — т In cosh -r-
J cosh ox + 1 a 2 a2 2
14.578 Г—j^—r = --coth^+ -^lnsinh^-
J cosh aa; — 1 a 2 a2 2
dx 1
i л etc l da; 1 , , ra
14.575 I —г -T-T = -tanh —
cosh aa; + 1 a 2
14.576 ' dx X " аж
T4.579
J
/^1Ч2 = f
(cosh aa; + IJ 2a
.580 I -—: ГГ5 = ТГ coth -= ^— coth3 -=-
J (cosh aa; — IJ 2a 2 6a 2
2
14.581
14.582
+ q cosh ax
tan -
aVq23^2 VV - P2
1 , /ge°* + p - Vp2-?2
In
/? _=
! — g2 \gea:c + p + Vp2"-'
d« _ g sinh aa; _ P Г da;
(p+ q cosh aa;J ~ a(q2 - p2)(p + q cosh aa;) q2 - p2 J P + Я cosh aa;

INDEFINITE INTEGRALS
89
14.583
Г
J p2 -
dx
14.584
h
q2 cosh2 ax
dx
1
2apv P2 — q2
-1
In
p tanh ax +
tan
p tanh аж —
p tanh аж
2 + q2 cosh2 ax
2apVp2 +
In
p tanh аж + Vp2 + <72
1 , p tanh аж
-===. tan ——=r
- apyp2 + g2 v P2 + <72
14.585
14.586
14.587
14.588
14.589
X
X
X
с
xm cosh аж
cosh" аж dx
cosh ax
ж"
dx
cosh" аж
a; da;
a;m sinh аж
= cosh" » ax sinh аж
— I x^1 sinhax dx [See 14.557]
n — 1 С
I cosh" аж da;
n J
sinhaa; .
-cosh ax ^_ Г sinh ax dx [See 14.559]
(n-l)xn~1 n~\J xn~l '
sinh ax n — 2 С dx
a(n — 1) совУ^-'аж n — 1 J coshn~
x sinh аж
аж a(n — 1) cosh
ьх 1 n — 2 Г xj.
n^1 aa; (n - 1)(и — 2)а2 coshn~2 aa; n - 1 J coshn
a; da;
ax
INTEGRALS INVOLVING sinh аж AND cosh
14.590
14.591
14.592
14.593
14.594
14.595
14.596
14.597
14.598
14.599
14.600
14.601
sinh aa; cosh ax dx = slnh2 ax
la
/-
sinh px cosh джсгж =
(p + д)ж cosh (p - g)»
аж cosh aa; da; = 8lnh" + 1<ta: [If n = -1, see 14.615.]
j sinhn
Jcosh" aa; sinh ax dx = cosh" + laa! [if и = _1( see 14.604.]
sinh2 aa; cosh2 aa;
sinh 4аж _ ж
32а 8
-——^—: = i In tanh aa;
sinh aa; cosh ax a
XiT
dX = _Itan-,sinha?c _ CSChO?
smh2 aa; cosh ax a a
dx
J si
J Cc
/COJ
si
X
sinh аж cosh2 ax
dx
а ' а 2
2 coth 2аж
sinh2 ax cosh2 аж
а
siTUfax dx = sinh as _ltan-lsinha:K
osh aa; a a
cosifax
sinh аж
dx =
= 1 In
2
= In Л
cosh ax A + sinh ax) 2a \ cosh ax I a
1 tan-i
a

90 INDEFINITE INTEGRALS
14'602 J sinh ax (cosh ax + 1) = ^lntanh У + 2a(cosh ax + 1)
J sinh ax (cosh ax - 1) 2a 2 2a(cosh aa; - 1)
INTEGRALS INVOLVING tanh
14.604 Г
14.605 f
tanh ax dx = — In cosh ax
a
dx = x - tanhax
14.606 I tanh3 ax dx = — In cosh aa; —
a
tanh2 ax
2a
14.607 1 tanh" ax sech2 ax dx =
14.608 CsechZax dx = i In tanh
J tanh aa; »
14.609 Г—^— = ilnsinhaa;
J tanh ax a
14.610 j
t . . lW (ox)»
x tanh aa; da; = -^ 1 —g^ jg- +
B»+l)!
14.611 Г x tanh2 aa; dx = ^ - * tanh аж + \ In cosh aa;
J 2 a a1
^dx = aa;__^-+^ B*-l)Bn)l
14.613 Г ^ = -^~ - . 2q „, In (q sinh a* + V cosh ax)
,7 p + q tanh aa; p2 — q2 a(p2 — q2)
14.614 ftanh»aa;da; = ~ta"h"~lcta! + Г tanh"-2aa; da;
J а(и -1) J
INTEGRALS INVOLVING colh ax
14.615 I coth aa; da; = - In sinh aa;
J a,
14.616 Г coth2 aa; da; = x - cothax
J a
14.617 f coth^ax da; = i In sinh ax - coth2aa!
a 2a
14.618
f cothnazcschSazda; = -«>^n+\*x
J (n+ l)a
14.619 rcsch2ax ^ = _llncothaa.
./ coth ax a
14.620 Г—^— = ilncoshax
J coth aa; а

INDEFINITE INTEGRALS 91
14.621
1
14.622 Г x coth2 axdx = ? _ «cotha» + 1 j ginh
J 2 a a2
1ЛА9Ч Г coth ax ^ _ 1 .ax (ax)*, (-l)*2P"Bn(ax)**-* ,
2 J ~T~ - ~ ax + 3 ~ "Ж" + '" ' Bn-l)Bn)! +
14.624 f da; = _??Ц - g In (p sinh a* + 9 cosh ax)
J p+ q coth ax p2 — g2 a(p2 - q2)
14.625
coth" ax da; = - ??Hi ~+ I coth"-2 ax dx
a(n - 1) J
INTEGRALS INVOLVING sech ax
14.626 j sech aa; da; = — tan e°*
14.627 Г sech2 aa; da; = tanhcta;
J a
ЫАОо С 1.1 j sech ax tanh аж . 1 , _, . ,
.628 I sech3 аж da; = h — tan 1 sinh аж
J 2a 2a
14.629 Г sechn аж tanh «
14.630 f.
J sech ах а
.,.„ f , , 1 I (axJ (axL
14.631 J x sech ax dж = -2 |-^ — -r -j^- -r ¦ ¦ ¦ Bи+-2)Bи),
14.632 Г ж sech2 аж dж = *1м1"и - -1 In cosh аж
J a a2
x = -sech"aa;
da; sinh aa;
sechax . _ , , (aa;J
^^rf* - In*- —
14.634 Г ^ = *-? Г *f [See 14.5811
J j + p sech аж g q J p + q cosh aa;
14.635 Г sechn аж da; = sech^ ax tanh ax n^2 Csechn-2axdx
J a(n-l) n-1 J
INTEGRALS INVOLVING csch ax
14.636 Г
csch аж da; = - In tanh ^r
а 2
14.637 Г csch2 aa; da; = - coth ax
J a
14.638 Г
14.639 Г
csch3 аж da, = - csch aa; coth ax _ 1 tanh «^
2a 2a 2
csch" аж coth o^dx - -csch"aa;
па

92
INDEFINITE INTEGRALS
d? = - cosh ax
csch ax a
14.640 Г
14.641 I a; csch aa; da; = —z
J a
14.642 С x
14.643 Г ?¦*
csch2 аж da; = - x coth ax + Л In sinh aa;
a
,„ _ _1 аж 7(aa;K
ax 6 + 1080 +
Г
J 9 + p
14.644
14.645 Г csch"
2(-i)nB2n-i _
Bn + 1)!
csch aa; q Ч J p + q sinh aa;
,™ j™ — ~ cschnaa; coth аж
Bn - 1)Bте)!
[See 14.553]
— 1)
_ n^2 Г
n — 1 J
cschn-2a;,
14.646
14.647
14.648
14.649
14.650
14.651
14.652
14.653
14.654
14.655
14.656
14.657
14.658
INTEGRALS INVOLVING INVERSE HYPERBOLIC FUNCTIONS
Г sinh-1-da; = a; sinh - - V^+a2
J a a
a: sinh — dx =
a
J
I x2 si
J
— ) 81ПП — —
4/ а
x хл/х2 + а2
J
¦ v. i « j
sinh-1 - dte
a
sinn (a;/a)
x
x3 . , , x Ba2 - a;2) V^Ta2
= ^smha + 9^
x _ (x/aK 1 • 3(ж/аM _ 1 - 3 - 5(a;/aO ,
а 2-3-3 2-4-б-5 2-4«6-7'7
In2 Bж/а) _ (a/a;J 1 • 3(a/a;L _ 1 • 3 • 5(а/жN ....
2 2-2-2 2-4-4-4 2-4-6-6-6
In2 (-2ж/а) (a/a;J _ 1 • 3(a/a;L 1 - 3 • 5(g/a;)8 _ ...
2 + 2-2-2 2-4-4»4 2-4-6-6-6
Г sinh-1 (x/a) dx _ sinh-i (ж/а) _ I ln
J ж2 a; a
2-2-2 2-4-4»4
а + Va;2 + a2
a;
J
r
I x
cosh-eta =
a
x
cosh — dx =
a
a;2 cosh-1 - dx -
a
a; cosh (x/a) — Va;2 — a2 , cosh (a;/a) > 0
x cosh (x/a) + Va;2 - a2 , cosh-1 (ж/о) < О
[ iBx2 ~ a2) cosh (x/a) - ?a;Vz2 - a2 , cosh (ж/а) > О
[ lBx2 - a2) cosh-i (x/a) + \x^Jx2-a2, cosh (x/a) < 0
Г ia;3 cosh-1 (x/a) - Ш2 + 2a2) Vz2 - a2 , cosh (ж/а) > О
\
[ \x* cosh (x/a) + 1(ж2 + 2a2) Va;2 - a2, cosh (x/a) < 0
+ if cosh (ж/а) > 0, - if cosh (ж/а) < 0
|«| <a
x > a
x < —a
cosh-Hx/a) _ 1
1,
a
Г cosh (x/a) ^x _ _ cosh
Г tanh-i ? dx = x tanh-1 - + § In (a2 - a;2)
J a a 2
Г x tanh-1 - dx = ^ + А(ж2 - a2) tanh~^
j a ? " a
[- if cosh (a;/a) > 0,
+ if cosh (x/a) < 0]
Г x2 tanh-1 ^ dx = ^+ ^tanh-1- + f ln (a2 - a;2)
J a 6 3 а 6

INDEFINITE INTEGRALS
93
14.659 Г tanh"J
14.660 ftanh-4«/a) da.
J a;2
14.661 fcoth-^da; =
52
.
14.662 J
14.663
14.664
14.665 Г
a
a
(ж/а)
_ tanh
x coth x
f + ?¦
+
ж2-
(ж/а)
21П
1
2а
(ж2-а
2)
ж
a
a;2
coth-Чж/»)
6 3 a ¦ 6 "
'а (а/жK , (а/жM
coth'1 (ж/а) X1 / ж2
2a
- а2
14.666
a; sech - dx
a
14.667 Г
14.668 fsech Чж/а)
J ж
14.669 fcsch^-aa;
J а
14.670 ГжсэсЬ-!^
J а
ж sech-Чж/а) + а sin~Чж/а), sech-Чя/а) > О
ж sech-Чж/а) — а sin-1 (x/a), sech-Чж/а) < О
[ -|ж2 sech-Чж/а) + ±а\/а2 - х2, sech'1 (ж/а) < О
Г-i In (а/ж) In Dа/ж) - ^jf^ - о .* ^JaT ~ ' ' ' sech (ж/а) > О
i In (a/x) In Dа/ж) + (ж/аJ + ^ ' f(^/а)^ + • • •, sech (ж/а) < О
x csch - ± a sinh -
da
[+ if ж > 0, - if ж < 0]
Y csch-i | ± а^х2 + а2 [+ if ж > 0, - if ж < 0]
In (ж/а) In Dа/ж) + „ — -—. . h • ¦ • 0 < ж < a
14.671
Г csch (ж/а)
J a;
in (-,/.) in (-/4a) -
• 3(а/ж)«
_
ж 2-3-3 2-4-5-5
—а<х<0
\х\ > а
14.672 ГсспвШ^^ж =
J a
sinh
14.673 Г жт cosh- da; =
J a
a m + l
ж
•m + l
/a;2 + а2
da;
C°Sh а ~ ^TIJ
/Ж2-(
га + 1
а т
14.674
14.675 J
da; =
д.т + 1 х „ Г д.т + 1
= tanh-1 —-— I —
т+1 а то + 1 J а2 — ж
п+1 у „ С ™т+1
-— coth-1 *— -I
+ 1 а га + 1 J a2 - ж
- a2
da;
da; cosh (a;/a) < 0
J
14.676 I жmsech-1-dж =
m+l
x
sech-1 - +
а т+1
.2
жmdж
dж
а2 — ж2
14.677 f »m csch -1 - da; =
га
_ X
1 х а С х
ISeC а ~ m + lj ^
+1J ,/:
.m+l , , Ж
—- csch — ± —
ra + 1 am
'a;2 + а2
sech (x/a) > 0
seen (ж/а) < О
[+ if x > 0, - if x < 0]

DEFINITION OF A DEFINITE INTEGRAL
Let f(x) be denned in an interval a S x S b. Divide the interval into n equal parts of length Ax ~
(b — a)/n. Then the definite integral of f(x) between x = a and x = b is defined as
15.1 I f(x)dx = lim {f(a)Ax + f(a + Ax)Ax + f(a + 2Ax)Ax + ••• + f(a+(n~l)Ax)Ax}
The limit will certainly exist if f(x) is piecewise continuous.
If f(x) = -j-g(x), then by the f
can be evaluated by using the result
If f(x) = -j-g(x), then by the fundamental theorem of the integral calculus the above definite integral
15.2 j f(x)dx = J J-x9(-x)dx = 9{x)
b
= 9(b) - g(a)
If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral
is called an improper integral and can be defined by using appropriate limiting procedures. For example,
15.3 Г f(x)dx = lim Г f(x)dx
15.4 Г f(x) dx = lim С f(x)dx
f(x) dx = lim I f(x) dx if 6 is a singular point
15.6 I f(x)dx — lim I f(x)dx if a is a singular point
GENERAL FORMULAS INVOLVING DEFINITE INTEGRALS
J*b y-»b s*b f*b
{f(x) ± g(x) ± k(x) ± ¦ ¦ ¦} dx = I f(x)dx ± I g(x) dx ± I h(x) dx ± ¦ ¦ ¦
a J a Ja Ja
15.8 I cf(x)dx = с I f(x)dx where с is any constant
Л •'о
15.9 С f(x)dx = 0
15.10 Г f(x) dx = - fa/(a;)da;
15.11 Г /(ж) dx = Cf(x)dx+ Г /(ж) dx
15.12 I f(x) dx = (b — a) f(c) where с is between a and b
This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in
a S x S b.
94

DEFINITE INTEGRALS 95
f(x) g(x) dx = f(c) I g(x) dx where с is between a and b
a Ja
This is a generalization of 15.12 and is valid if f(x) and g(x) are continuous in а Ш х Ш b
and g(x) a 0.
LEIBNITZ'S RULE FOR DIFFERENTIATION OF INTEGRALS
15.14 -j- I F(x, a) da; = I 'д~^а; + F(<f>2>a) ~ F{<t>u°)
APPROXIMATE FORMULAS FOR DEFINITE INTEGRALS
In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x0,
xu x2, ..., xn_u xn = b and we let j/o = f(xo)> Vi = /(«i). Vz = f{x2), ¦¦¦,Уп = /(«»), h = (b - a)ln.
Rectangular formula
cb
15.15 I f(x)dx ~ h(y0 + 2/x + y2 +••¦+ yn-i)
Trapezoidal formula
f(x) dx = т;(Уо + 2j/i + 2j/2 + ' ' ' + 2i/tl_1 + 2/n)
Simpson's formula (or parabolic formula) for n even
15.1/ I /(a;) da; ~ — \j/q + 42/j + Zj/2 + ^1/3 + • • • + Zj/n_2 + 42/n_i + 2/n)
"'a J
DEFINITE INTEGRALS INVOLVING RATIONAL OR IRRATIONAL EXPRESSIONS
15.18 I
,0
dx __ jr_
+ a? 2a
0 < m + 1 < m
15.19 Г x" lda: = —^—, 0<p<l
Jo 1 + x sm p?r
15.20 С xmdx = 7r"m + 1~n , ,
JQ xn + an n sin [(m + 1)тг/п]
<e .. /*°° xm dx tt sinm
¦ л ^^^
15.22 Г da;
15.23 Г
/n] Г(р
, ^ , 2x cos /? + x2 sin OTjr sin ,
/а*-х2 2
/a2 — x2 dx = ——
4
15.24
o nr[{m + l)/n + p + 1)
/•« xmdx (-l)r~ 17гат + 1-пгГ[(т + 1)/и]
15.25 J (a.» + an)r = и sin {(m + 1)тг/и](г - 1)! T[(m + l)/n - r + 1] ' 0 < m + 1 < mr

96
DEFINITE INTEGRALS
DEFINITE INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS
All letters are considered positive unless otherwise indicated.
JO m, n integers and m Ф n
sin mx sin nx dx = ¦s
о I я-/2 m, n integers and m = n
fir
15.26 I si
15.27 Г
0 m, n integers and m Ф n
cos mx cos nx dx = <
о I Я-/2 m, n integers and m = n
15.28 I sin mx cos nx dx =
0
m,n integers and m + n odd
2ml(m2 — n2) m,n integers and m + n even
15.29 Г si
sin2 x dx = I cos2 x dx = —
J 4
15.30 I sin2 x da; = I
¦>m j 1 • 3 • 5 • • • 2m — 1 я- , o
cos2m * da; = — ¦ — , m = 1, 2, .. .
2462 2
XTT/Z
bill Л
X-/2
sin^-ia;,
ir/2
2•4•6•• ¦ 2m 2
__ 2 • 4 • 6 • • • 2w
1•3•5 • • • 2m +
. m = 1,2, ...
q)
15.33
psinj
irl2 p > 0
da; = «j 0 p — 0
-7T-/2 p < 0
15.34 1 sin px cos ga;da,
a;
15.35 f"s
sin px sin qx
a;2
15.36 f"»
sin2 pa;
тр
2
0 p > q > 0
! tr/2 0 < p < q
| ir/4 p = <jr > 0
я-р/2 0 < p ё q
n-q/2 p g q > 0
.5.4, f
¦°° a; sin ma:
: + a2
dx =
Ee-ma
2
15.37
= ??
15.38 Г
•'о
15.39 С
15.40 Г
cos pa; — cos qx
cos Vх — cos 9^
a;2
cosmx , _ jr „_
a;2+ a2
dx = -f-.
2a
15.42 l -Щ^Лх
15.43
ln dx
2тг
15.44 I
Jo a +
a + b sin x
l7r da;
cos а;
Х7Г/2
-
da;
+ b cos л;
cos (b/a)
y/a2 - 62

15.46 С2" —
DEFINITE INTEGRALS 97
dx 27га
+ Ъ sin a;J JQ (а + Ь cos a;J (а2 - Ь2K'2
0<a<l
J-»2tt 7
Ё* = 2^ ,
0 1 — 2а cos a; + а2 1 — а2
1548 р ж sin х dx = Г Ог/а) In A + а) |а] < 1
Jo 1 - 2а cos х + а2 | ^ ь A + 1/а) ,а| > х
15.49 С cosmxdx = „а™ а2 < х m = о, 1, 2,
Jo 1 - 2а cos х + а2 1 - а2
15.50 I sin aa;2 dx = I cos ах2 da; = "s Л/тг~
Л Л *
15.51 Г sin ах" da; = —— ГA/и) sin -f-, п>1
Jo иа1/и 2п
cosaa;nda; = —i— rfl/m) cos —-, п>1
щ"» 2m
R1T1 X ¦, I COS DC t
^^^ dx = I —— da; =
о Va; ^o V*
15.54 r^dx = 2r(p)sl;(pW2). o<P.
cos a;
1С EC ^ CU»X 7 7Г n
15.55 I da; = _ . , ;—j^rr , 0
Jo xp 2Г(р) cos (ртг/2)
X00 1 t / k2 7^2
sin aa;2 cos 26a; da; = тгл/тгЧсов sin—
2 \ 2a V а а
Г i / т 2 1.2
cos aa;2 cos 2ba; da; = - \l ~- ( cos h sin —
15.58 Г ™
15.59 f" ™
3 a; , _ 3^
3 8
ia; я-
-da; = 7Г
15.60 Г ^"J
Jo a;
15.61 f'\
15.62 Г" « ¦ - ' l l ¦ X X
da;
,0 ^ + tanma;
0 sin x 112 32 T 52 72
15.64 i 2i^—iEda; = ^ In 2
..64 f ™^2
15.65 Г' X ~ cos x dx - f"cosx&
Л x Л x
15.66 I -;—: 5 — COS X = 7
Jo \1 + a;2 /a;
itA7 Г tan px — tan да; j _
1Э.О/ I ax - 2 ...
= ?ln?

98 DEFINITE INTEGRALS
DEFINITE INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS
15.68 I e~°* cos bxdx = 2 a
15.69 Г e-« sin bx dx = *
Jo a* + №
15.70 Ce~a*
Л
15.71 Г c~aI
15.72
dx =
15.73 Г e~a
15.74
x2cosbxdx = iA[e
2 V a
2 V a
и
n
where erf с (p) = -7= ( e~x% dx
V - .Sp
15.75 С е-<ах2 + Ьх+с) da. = JTe(b2-4ac)/4a
15.76 Гх-е-«& - HlL+li
ш/q a
15.77 Г"—-»2- - r[(wl+1>/2l
15.78 Г
ж"
'0
15.80
"'0
For even n this can be summed in terms of Bernoulli numbers [see pages 108-109 and 114-115].
xdx _ 1 1,1 1 , _ T2
For some positive integer values of n the series can be summed [see pages 108-109 and 114-115].
15.83 Г fnmx dx = Icoth^-J-
J e2™ - 1 4 2 2m
15.84 f"f-i—«-'^ = у
15.85 f г*-*''*. = Ь
f
15.86
х " 1 x

DEFINITE INTEGRALS 99
1 г at I С ах — С bx 1
15.87 I dx = -
J x sec pa; 2
15.88 Ce~ax-e-bxdx = u...
J x esc px V V
~ax _ e-bx i Ь . .a
dx = tan tan —
15>89 Г- e-«*a-cosx) dx = cot-ie-?ln(a*+l)
Л x l
DEFINITE INTEGRALS INVOLVING LOGARITHMIC FUNCTIONS
15.90 Г
J n
m>-1, n = 0,l,2, ...
If пФ 0,1,2,... replace n! by Г(п-
15-91 • TT~x = 2
15.92
{ T^~xdX
15.93 С ln A + x)
15.94 Г1 Mk=j*)
Л
15.95 I 1пж1пA + ж)
ln x ln A - ж) da; = 2 - —
15.97 f°° xP-1 ln ж dx = - я-2 esc ртг cot ря- 0 < p < 1
Л X + x
15.98 f1
12
_e!
6
dж = 2 -
тл ~ хП dx = ln2L±I
/0 ln ж и + 1
= -у
e-l2lna;da; = -^—(у + 2 1п2)
о 4
15.101 Г ъ (*+!)& = d
Jo ^ - lyl 4
J. тг/2 f тг/2
In sin ж dx = I ln cos x dx = — - ln 2
о "'o z
X^/2 /. 7Г/2
(In sin жJ da; = I (In cos жJ da; = f(ln!
Л
15.104 Г жlnsinxdж = -^1п2
J-.U/2
sin x ln sin ж dж = In 2 — 1
о
15.106 \ In (a + 6 sin ж) da; = I In (a + 6 cos ж) da; = 2я- In (a + Va2 ~ '

100 DEFINITE INTEGRALS
15.107 Г
J n
In (a + b cos x)dx = ж In
a + Уд2 - 62
2
Г 2тг In a, a S ft > о
15.108 Г In (a2 - 2ab cos x + b2) dx = <|
^o \ 2ж In 6, b й а > О
Г4
15.109 I ln(l + tanx)dx - § In 2
Л 8
J-7T/2 / \
sec * In ( ] + " cos X dx = A{(cos-i aJ - (cos bJ}
0 yl + a cos x) z
15.111 J In 2s,
• ж\ j /sin a sin 2a sin 3a.
s,n- dx = _/ + +
о v 2/ V I2
See also 15.102.
15.112
DEFINITE INTEGRALS INVOLVING HYPERBOLIC FUNCTIONS
CK _sir
Л sin
4^dx = ^tanhf
o sinh bx 26 2b
cos aa; , _ ж -.аж
- dx тгг sec*177Г
« 26 26
15.114 Г
sinh ax 4a2
If n is an odd positive integer, the series can be summed [see page 108].
MISCELLANEOUS DEFINITE INTEGRALS
15118 Г f(ax) - f(bx) dx = {/(o)-/(-)} 1„ А
Л
This is called Frullani's integral. It holds if f'(x) is continuous and I '^x' ~ ^°°^ da; converges.
J, x
15.120 Г

DEFINITION OF THE GAMMA FUNCTION Г(те) FOR n > 0
16.1
Г(т) = I tn
m>0
16.2
16.3
RECURSION FORMULA
r(m + 1) = n r(m)
r(m+l) = n\ if n = 0,1,2, ... where 0!
THE GAMMA FUNCTION FOR «<0
For n < 0 the gamma function can be defined by using 16.2, i.e.
16.4 r(m) = — -
n
!
GRAPH OF THE GAMMA FUNCTION
Tin)
16.5
16.6
16.7
Fig.16-1
SPECIAL VALUES FOR THE GAMMA FUNCTION
Щ) =
Bm - 1)
m- 1,2,3,
101

102 THE GAMMA FUNCTION
RELATIONSHIPS AMONG GAMMA FUNCTIONS
16.8 r(p)r(l-p) = .v -
16.9 22*
This is called the duplication formula.
16.10 Т(х)т(х+1У\т(х+^)---г(х+^^
\ mj \_ mj у m
For m = 2 this reduces to 16.9.
OTHER DEFINITIONS OF THE GAMMA FUNCTION
1611 Г(* + 1> = Hm l^
(ж + l)(x + 2) ¦ • ¦ (x + k)
16.12 — = хеУ* П
This is an infinite product representation for the gamma function where у is Euler's constant.
DERIVATIVES OF THE GAMMA FUNCTION
16.13 Г'A) = Г e-*lnxdx = -y
16.,4
2 a; + 1
ASYMPTOTIC EXPANSIONS FOR THE GAMMA FUNCTION
16.15 r(x + l) =
This is called Stirling's asymptotic series.
If we let x = n a positive integer in 16.15, then a useful approximation for n! where n is large
[e.g. n > 10] is given by Stirling's formula
16.16 n! ~ ¦\/2тгпппе~п
where ~ is used to indicate that the ratio of the terms on each side approaches 1 as n -» °°.
MISCELLANEOUS RESULTS
i
16.17 |r(ix)|2 =
x sinh жх

17.1
DEFINITION OF THE BETA FUNCTION B{m,n)
B(m,n) = Г t™-1^-*)»-! dt m > 0, n>0
RELATIONSHIP OF BETA FUNCTION TO GAMMA FUNCTION
17.2
B(m,n) =
T(m + n)
Extensions of B(m, n) to m < 0, n < 0 is provided by using 16.4, page 101.
17.3
17.4
17.5
17.6
SOME IMPORTANT RESULTS
B(m, n) = B(n, m)
B(m,n) = 2 I sin2*? cos2"9
, и) = Г
fm-1
dt
, n)
J.i
-
103

18
BASIC DIFFERENTIAL EQUATIONS
and SOLUTIONS
DIFFERENTIAL EQUATION
18.1 Separation of variables
/i(») 9\{У) dx + f№) 92{У) dy - 0
18.2 Linear first order equation
g + P(x)y = Q(x)
18.3 Bernoulli's equation
g + P(x)y = Q(x)yn
18.4 Exact equation
M(x, y) dx + N(x, y)dy = 0
where дМ/ду = BN/dx.
18.5 Homogeneous equation
dy ip{y\
ax V2'/
SOLUTION
1 7-7-;:dx + 1 —гтаУ - c
J /2(ж) J 9\(У)
yeSPdx - J QeSpdxdx + с
ve(l-n)Jp<ta = A-й) jQe»-n) /Pd:cda; + с
where v = y1~n. If n = 1, the solution is
In 2/ = Г (Q ~ P) dx + С
i Мдх + (lN -4~ \Mdx)dy = с
J J \ «у J J
where 3a; indicates that the integration is to be performed
with respect to * keeping у constant.
J F(v) - v
where v = y/x. If F(v) = v, the solution is у — ex.
104

BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
105
DIFFERENTIAL EQUATION
18.6
yF{xy)dx + xG(xy)dy = 0
107 Linear, homogeneous
second order equation
dP-y dy ,
To + a~f- + by = 0
dx2 dx
a, b are real constants.
18 8 Linear, nonhomogeneous
second order equation
3 + -S + * - E«
a, b are real constants.
18.9 Euler or Cauchy equation
X2fl + axf- +by = S(X)
dx1 dx
SOLUTION
, Г G(v)dv .
where v = xy. If G(v) = F{v), the solution is xy = c.
Let m1, m2 be the roots of m2 + am + b = 0. Then
there are 3 cases.
Case 1. m1, m2 real and distinct:
у = ciemix + c2emzx
Case 2. mx, ra2 real and equal:
у = Olemi* + c2xemix
Case 3. m^ = p + qi, m2 — p — qi:
у = epx(cl cos qx + c2 sin qx)
where p = — ct/2, q = т/Ь — аУА .
There are 3 cases corresponding to those of entry 18.7
above.
Case 1.
у = c1emix + c2em2x
+ e™lX Cc~™ixR(x)dx
W2 - «1 J
Case 2.
1/ = cxem\x + сгхе!1
+ же!1 I e~mixR(x)dx
— emix I xe~mtx R(x) dx
Case 3.
у = еР*^! cos 9a; + c2 sin q«)
+ eP*sing* fe-wfi(x)cos9x«fa
- eP*cosgx Гс-««й(а,)81пд!8Лв
Putting x — ef, the equation becomes
and can then be solved as in entries 18.7 and 18.8 above.

106
BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
18.10
х d
18.11
Х2<^ +
Х dx2 +
18.12
A-х2
Bessel's equation
% + х% + №х2-пг)у = °
Transformed Bessel's equation
Bр + 1)х~У; + (a2x2r + /32)y = 0
Legendre's equation
)g_2xg+m(m + 1J/ = о
See pages
У —
where q -
See pages
У =
136-137.
x-"|Cl
= Vp2~^J*
У -
146-148.
ClJn(\x) + c2YJx)
ClPn(x) + c2Qn(x)

ARITHMETIC SERIES
19.1 a+ (a + d) + (a + 2d) + •¦• + {a
where I = a + (n — l)d is the last term.
Some special cases are
19.2
19.3
1 + 2 + 3+
1 + 3 + 5 +
= \n{2a
+ n = \n(n + 1)
+ Bи - 1) = n2
- -Ln(a +
GEOMETRIC SERIES
19.4 a + ar + ar2 + or3 + ¦•• +
where I — arn~1 is the last term and г Ф 1.
g(l - г") _ a - rl
1-r ~ 1-r
If -1 < r < 1, then
19.5
a + ar + ar2 + ar3 +
1-r
ARITHMETIC-GEOMETRIC SERIES
19.6 a + (a + d)r
where г Ф 1.
(a
If -1 < r < 1, then
19.7 a + (a + d)r
1-r
a rd
1-r ' A-rJ
A
SUMS OF POWERS OF POSITIVE INTEGERS
19.8
+ 2" + 3p + • • • +
P + l
2!
4!
where the series terminates at n2 or n according as p is odd or even, and Bk are the Bernoulli
numbers [see page 114].
107

108 SERIES OF CONSTANTS
Some special cases are
19.9 i + 2 + 3+---+« = и(те +1)
19.10 12 + 22 + 32 + ... + m2 = n(n + l)Bw + 1)
6
19.11 is + 23 + з3 + ••• + n3 = n4n +1J = A + 2 + 3+¦••+nJ
4
19.12 I4 + 24 + 34 + • • • + n4 =
+ n
30
If Sk = lk + 2k + 3fc + • • • + nk where к and n are positive integers, then
19.13
SERIES INVOLVING RECIPROCALS OF POWERS OF POSITIVE INTEGERS
,9.14 i_! + !_! + !_... = m2 .j.t
1916 1-1 + 7-Й + Й-- = ^ + lln2 0,
1917 i.l + '-^i.^-.. —
1У-17 Х б + 9 13 + 17
,9.,8 I_| + i_l+_l_... =
,9.19 J_+J_+J^ + J^.+ ... = 5l
,9 20 1 + 1 + 1+1+... = ?i
ly.ZU 14 + 24 + g4 + 44 + 90
19.21 ^+1 + 1 + ^+... =-^^,0173430^
19.22 12-1 + 1-1+ ... = g
19 53 1 1 + 1 1+... — """
¦¦»•-«•» 14 24 T 34 44 "•" 720
,««>.! 1.1 1 ,
3
"'" 16 26 n 36 46 ' 30,240
19.25 ^ + 1 + 1 + 1+... =^ [9.33700.^
19.26 ^ + 1 + 1 + 14+... =
19-27 й + ^ + ^ + fe+ ¦•• = U 1.0СИЧ7О77
,9.28 - -^ + ^з - ^з + " • • =
?5+33~p~Y3+""" - 128
19.30 ГТз + з75+бГ7+7Т9+"" = \
,9.31 ^-z + ^-l+J-l+J-&+... =|
19 30 _!_ + 1 + 1 + 1 + • • • = *-2~8 П
12 • 32 32 . 52 ' 52 . 72 ~ 72 . 92 ~ Jg (У ¦

^4-^4
SERIES OF CONSTANTS
Ю9
19 33 1,1,1,
I2 • 22 • 32 22 • 32 • 42 32 • 42 • 52
1
19 34 I _ 1 i 1
a a + d a + 2d a + 3d
19.35 J-+—+—+ — +••• =
12p 22p 32p 42p
19.36 1-+-1-+-1-+A-+... =
19.37 -L _ /
19.38 -i-
32р
52P+1 72P+1
4y2 - 39
16
1 м"^
1 + ud
2Bр)!
22p + 2Bp)!
MISCELLANEOUS SERIES
19.39 i + cosa + cos2a + ••• + cosma = ^— f—
2 2 sin (a/2)
._ ^_ sin fl(m + l)la sin ina
19.40 sin a + sin 2a + sin 3a + ••¦ + sin ma = -^ ^— ==—
sin (a/2)
19.41 1 + rcosa + r2cos2a +
19.42 rsina + r2 sin 2a + т-з sin 3a +
19.43 1 + r cos a + rz cos 2a + • • • +
~ Г C0S a
-—„ ,
1 — 2r cos a + rz
sm a
rn cos na =
/ -0 ,
1 — 2r cos a + r2
+ 2 COS Па —
r\ < 1
1 COS (
— Г COSa
л л ¦ i ч ¦ n i i r, ¦ т sin a — rn + 1 sin (n + l)a + rn + 2 sin ma
.44 r sm a + r2 sin 2a + ¦ • • + rn sin na = — ^—J '—5
1 — 2r cos a + r2
(n + l)a +
1 — 2r cos a + r2
THE EULER-MACLAURIN SUMMATION FORMULA
19.45 2 F(k) = Г
fc=x Jo
30,240
1,209,600
19.46
THE POISSON SUMMATION FORMULA
2
к — — <
i. IX
J
eMm*F{x)dx

TAYLOR SERIES FOR FUNCTIONS OF ONE VARIABLE
20.1
/(ж) = f(a) + f'(a)(x-a) +
f"(a)(x-aJ
2! (n - 1)!
where Rn, the remainder after n terms, is given by either of the following forms:
20.2 Lagrange's form Rn = ¦¦
n\
20.3 Cauchy's form
Rn =
The value ?, which may be different in the two forms, lies between a and x. The result holds if f(x) has
continuous derivatives of order n at least.
If lim Rn = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If
а = 0 the series is often called a Maclaurin series. These series, often called power series, generally
converge for all values of x in some interval called the interval of convergence and diverge for all x outside
this interval.
BINOMIAL SERIES
(а + ж)" = а" + та"'1* +
20.4
= an
Special cases are
20.5 (a + xJ = a2 + 2aa; + a;2
20.6 (a + a;K = a3 + 3a2a; + Заа;2 + a;3
20.7 (a + x)* = a4 + 4a3a; + 6o.2a;2 + 4аж3 + a;4
20.8 (l + ж)-1 = 1 - x + x2 - x3 + a;4 - •••
20.9 A + a;)-2 = 1 — 2x + 3x2 - 4xs + бж4 - • •
+ x)~3 = 1 - Зж + 6a;2 - 10a;3 + 1бж4 -
ап-2х2 +
20.10
20.11
20.12
20.13
20.14
.x)V2 = 1 +_,.
1-3
1-3-5
2-4-6;
+
2-4'
+
= ^b + w
xI'3 = 1
3-6"
2-4-6
_ 1-4-7,
3-6-91
3-6-9X
2^2!—1KZL—?! ап-з
-1 < х < 1
-1 < х < 1
-1 < х < 1
-1 < х S 1
-К ж S 1
-1 < ж S 1
-К х S 1
110

TAYLOR SERIES
111
SERIES FOR EXPONENTIAL AND LOGARITHMIC FUNCTIONS
20.15 e* =
20.16 ax =
20.17 In A4
20.18 ±l
1 + x + й + It + '''
«X In a = 1 + ,lna + {xlnaJ + (Ж1П^3 +
2!
3!
= x - ? + ^ -^
20.19 In, =
-1 < X < 1
5 
SERIES FOR TRIGONOMETRIC FUNCTIONS
20.21 sin ж
20.22 cos x
20.23 tan,
20.24 cot,
20.25 sec x
20.26 esc x
20.27 sin i
-co <d CC <C °°
22иB2"-1)ВГ1,2и-1
1 _ ?. _ e! _ ^x5 _
ж 3 45 ~~ 945 ~
ж2 5,* 61,6
1 + Y + ~2Л + 720
1 ж 7^ 31,5
ж 6 360 15,120
Bn)l
Bn)!
1 ,3 , 1 • 3 ,5 , 1-3-5 x7
20.28 cos a; = ^— sin a;
1 х3 1 • 3 ,5
2 \х + 2 3 ^ 2-4 5
5
20.29 tan~!x =
,3 , ,5 а;7
-I + -i ^
2 х З,3 5,5
20.30 cot, = ^-
20.31 sec
20.32 csc-i, = sin-i(l/*) =
x3 , ,5
1 1
2 \ х 2
2 • 4
ж
+
1«3
2 • 4 • 5,5
«I < g
0 < 1,1 <
+ ¦¦¦ 0 < |x| < ж
x < 1
< 1
[+ if Ж & 1, - if х =? -1]
< 1
-ig - ¦•¦ [p = о if , > 1, p = 1 if ж < -1]
1,1 > 1
1,1 > 1

112
TAYLOR SERIES
SERIES FOR HYPERBOLIC FUNCTIONS
20.33 sinh * =
20.34 cosh * = l + ^+^+^
20.35 tanhx = , - g + gg %g
d 15 315
20.36 coth* = - + f-J+gL
20.37 sech* = 1-Т + -24- -W
20.38 cschx = -_i + — -
-°°<z<«
(-D-1^- 1)B.»»-'
Bm)!
1 • 3a;5 1 • 3 • 5a;7
20.39 sinh-l x =
'Ы \2x\ + jj-1
20.40 cosh x - ±. \ In Bж) -
"? 5 7
20.42 coth-i* = 1 + J- + J-+-L
я 3a;3 5a;5 7x7
2-3 2-4-5 2-4-6-7
1-3 , 1-3-5
x2 2-4-4x4 2-4-6-6x6
-и 1*3 I 1-3*5 . _
I ^* * л J I -^ j ^ ^ ?* Г
+
x\ < 1.
Г+ifa;^! Л
L- if ж == -ij
+ if cosh~l a; > 0, a;
— if cosh~la; < 0, x
< 1
ж|
MISCELLANEOUS SERIES
20.43 в»»-* =
20.44 e^s^ =
20.45 etani =
20.46 ex sin x
20.47 ex cos x —
20.48 In |sin x\ =
20.49 In |cos x\ =
20.50 ln|tan«| =
20.51 1Э-1
* + Ж + 3 30 90
ill
m 1*1 - -g- - 180 - 2835
ж* _ xf _ 17a;8
12 45 2520
90 2835
, 2"/2 sin (тиг/4) а;" ,
m!
mBn)!
mBm)!
mBm)!
— eo < a; < «>
— °° < a; < eo
—» < ж < во
О < |ж| < я-
0<И<|

TAYLOR SERIES ЦЗ
REVERSION OF POWER SERIES
If
20.52 у = сгх + c2x2 + c3a;3 + c4a;4 + csxs + c6x6
then
20.53 x = ClV + C2y2
where
20.54 clC1 = 1
20.55 e\C2 = -c2
20.56 c\C3 = tc\ - cxc
20.57 c\C± - Ъсгс2сг - Ьс\ -
20.58 с\Сь = 6^c2c4 + 3c*4 - c\c5 + 14c| - 21
20.59 c^Cg = 7cfc2c5 + 84Clc|c3 + 7C3c3c4 - 28c2c2c32 - cjc6
TAYLOR SERIES FOR FUNCTIONS OF TWO VARIABLES
20.60 /(*,*) = f(a,b) + (x-a)fx(a,b) + (y-b)fy(a,b)
+ Yi((x~ «J/xx(«, b) + 2(x - a)(y - b)fxy(a, b) + („ - Ь)«/И(а, Ь)} +
where /х(а, Ь), /н(а, Ь), ... denote partial derivatives with respect to x,y, .. . evaluated at x = а, у — b.

DEFINITION OF BERNOULLI NUMBERS
The Bernoulli numbers Bu B2, B3, . . . are defined by the series
;tx2 B2x* B3x6
21.1
21.2 1-fcotf = 2|
X BlX
2 + 2!
4!
6!
x\<2v
4!
6!
DEFINITION OF EULER NUMBERS
The Euler numbers Elt E2, Es, ... are defined by the series
1 Esx*
21.3 secha; = 1 -
21.4 sec a; = 1
2!
+
1
4!
2! 4!
1
6!
6!
+
TABLE OF FIRST FEW BERNOULLI AND EULER NUMBERS
B2
B3
в*
B5
B6
B7
Bs
B9
Bx
Bx
Вг
Bernoulli numbers
= 1/6
= 1/30
= 1/42
= 1/30
= 5/66
= 691/2730
= 7/6
= 3617/510
- 43,867/798
о = 174,611/330
! = 854,513/138
2 = 236,364,091/2730
Ег
E2
Es
#4
E5
E6
E7
E8
Es
#10
#11
#12
Euler numbers
= 1
= 5
= 61
= 1385
= 50,521
= 2,702,765
= 199,360,981
= 19,391,512,145
= 2,404,879,675,441
= 370,371,188,237,525
= 69,348,874,393,137,901
= 15,514,534,163,557,086,905
114

BERNOULLI AND EULER NUMBERS
115
RELATIONSHIPS OF BERNOULLI AND EULER NUMBERS
21.5
/2и +
V 2
2n+ 1
4
2и + 1
6
21.6
6/
21.7
SERIES INVOLVING BERNOULLI AND EULER NUMBERS
21.10 В„ =
B2п-1-1)я.2п
Л/
M.» I
21.11
32n+l
I
ASYMPTOTIC FORMULA FOR BERNOULLI NUMBERS
21.12

22
FORMULAS from
VECTOR ANALYSIS
VICTORS AND SCALARS
Various quantities in physics such as temperature, volume and speed can be specified by a real number.
Such quantities are called scalars.
Other quantities such as force, velocity and momentum require for their specification a direction as
well as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed
line segment indicating direction. The magnitude of the vector is determined by the length of the arrow,
using an appropriate unit.
NOTATION FOR VECTORS
A vector is denoted by a bold faced letter such as A [Fig. 22-1]. The magnitude is denoted by |A| or
A. The tail end of the arrow is called the initial point while the head is called the terminal point.
FUNDAMENTAL DEFINITIONS
1.
2.
3.
Equality of vectors. Two vectors are equal if they have the same
magnitude and direction. Thus A = В in Fig. 22-1.
Multiplication of a vector by a scalar. If m is any real number
(scalar), then wiA is a vector whose magnitude is \m\ times the
magnitude of A and whose direction is the same as or opposite
to A according as m > 0 or m < 0. If m = 0, then tttA = 0 is
called the zero or null vector.
Fig. 22-1
Sums of vectors. The sum or resultant of A and В is a vector С = A + В formed by placing the
initial point of В on the terminal point of A and joining the initial point of A to the terminal point
of В [Fig. 22-2F)]. This definition is equivalent to the parallelogram law for vector addition as in-
indicated in Fig. 22-2(c). The vector A - В is defined as A + (-B).
С = A + B
(b)
Fig. 22-2
116

FORMULAS FROM VECTOR ANALYSIS
117
Extensions to sums of more than two vectors are immediate. Thus Fig. 22-3 shows how to obtain
the sum E of the vectors А, В, С and D.
О
(a)
Fig. 22-3
4. Unit vectors. A unit vector is a vector with unit magnitude. If A is a vector, then a unit vector in
the direction of A is a = A/A where A > 0.
LAWS OF VECTOR ALGEBRA
If А, В, С are vectors and m, n are scalars, then
22.1 A + В = В + A Commutative law for addition
22.2 A + (B + C) = (A + В) + С
22.3 m(nA) = (mn)A = n(mA)
22.4 (m + n)A — жА + пА
22.5 m(A + В) = mA + mB
Associative law for addition
Associative law for scalar multiplication
Distributive law
Distributive law
COMPONENTS OF A VECTOR
A vector A can be represented with initial point at the
origin of a rectangular coordinate system. If i, j,k are unit
vectors in the directions of the positive x, y, z axes, then
22.6
A = Ati + AJ + A3k
where Ati, A^j, A3k are called component vectors of A in the
i, j,k directions and А^,АЪА^ are called the components of A.
DOT OR SCALAR PRODUCT
Л3к
Fig. 22-4
22.7
A • В = AB cos 9 0 ё 9 ^
where в is the angle between A and B.

118
FORMULAS FROM VECTOR ANALYSIS
Fundamental results are
22.8
22.9
22.10
where A = Asi +A2j + A3k, В =
A-B = B'A
A • (B + C) = A • В + A • С
А-В = AiBj + A2B2 + A3B3
,i + B2j + B3k.
Commutative law
Distributive law
CROSS OR VECTOR PRODUCT
22.11 AXB = ABsintfu OSes,
where в is the angle between A and В and u is a
unit vector perpendicular to the plane of A and В
such that A, B, u form a right-handed system [i.e. a
right-threaded screw rotated through an angle less
than 180° from A to В will advance in the direction
of u as in Fig. 22-5].
Fundamental results are
22.12 AxB =
Al A2 AS
B, Bo B»
= (A2B3-A3B2)i + (AgBj-A^j + (A^
22.13 AXB = -BXA
22.14 AX(B+C) = AXB + AXC
22.15 | A X B| = area of parallelogram having sides A and В
Fig. 22-5
MISCELLANEOUS FORMULAS INVOLVING DOT AND CROSS PRODUCTS
22.16 A'(BXC) =
A1 A2 As
A2B3C1
22.17 | A • (B X C)| = volume of parallelepiped with sides А, В, С
22.18 AX(BXC) = B(A'C) - C(A«B)
22.19 (AXB)XC = B(A • C) - A(B • C)
22.20 (AXB)-(CXD) = (A'C)(B-D) - (A-D)(B-C)
22.21 (AXB)X(CXD) = C{A • (B X D)} - D{A • (B X C)}
= B{A • (C X D)} - A{B • (C X D)}
- A3B2C1 -

FORMULAS FROM VECTOR ANALYSIS Ц9
DERIVATIVES OF VECTORS
The derivative of a vector function A(u) = Ax(u)i + A2(u)j + A3(u)k of the scalar variable и is
given by
оооо d\ A(m + Дм) - А(м) dAi dA2- dA3
22.22 -=— = lim = —=—l H—3—j H — к
ещ ди-»о Дм ам ещ ам
Partial derivatives of a vector function A(xry,z) are similarly defined. We assume that all derivatives
exist unless otherwise specified.
FORMULAS INVOLVING DERIVATIVES
22.23 #(A-B) = A.^ + ^-B
du du du
22.24 A(AXB) = Axf + f хв
22.25 ^{A.(BXC)} = f.(Bxc) + A.(f xc)+A.(Bxf
22.26 a-^ = A^
du du
22.27 A • ~ = 0 if IA is a constant
THE DEL OPERATOR
The operator del is defined by
22.28 V = i— + i4- + k-2-
5a; З2/ dz
In the results below we assume that U = U(x,y,z), V = V(x,y,z), A = А(ж,г/, z) and В = B(x,y,z)
have partial derivatives.
THE GRADIENT
22.29 Gradient of U = grad U = VU = fiA+jA+ki_\f/ = ?Е{ + ^j + ^
e \ dx By dz I dx By dz
THE DIVERGENCE
22.30 Divergence of A = div A = V • A = ( i — + j ~ + k^- ) • (Aji + A2j + A3k)
\ ox oy dz
dA1 dA2 3A3
dx dy dz

120
FORMULAS FROM VECTOR ANALYSIS
22.31 Curl of A = curl A = V X A
i j к
A A JL
дх ду dz
ду
A2 As
dA.
Hz
i +
THE CURL
3z
dx
dA2
22.32 Laplacian of U =
22.33 Laplacian of A =
THE LAPLACIAN
= V(V?7) = Щ. + Щ + Щ
дх2 ду2 dz2
THE BIHARMONIC OPERATOR
22.34 Biharmonic operator on U = V4J = V2(V2I7)
За;4
ду*
dz*
дх2ду2
, d*U
1 ду2 dz2
d*U
дх2 dz2
MISCELLANEOUS FORMULAS INVOLVING V
22.35 V(U + V) = VU + VV
22.36 V (A + B) = V'A+V-B
22.37 VX(A+B) = VXA+VXB
22.38 VA7A) = (V17)«A + ?7(V-A)
22.39 VX(PA) = (VI/) X A + I7(VXA)
22.40 V • (A X В) = В • (V X A) - A • (V X B)
22.41 V X (A X B) = (B • V)A - B( V 'A) - (A • V)B + A( V • B)
22.42 V(A-B) = (B- V)A + (A» V)B + В X (V X A) + A X (V X B)
22.43 V X (VU) = 0, i.e. the curl of the gradient of U is zero.
22.44 V • (V X A) = 0, i.e. the divergence of the curl of A is zero.
22.45 V X (V X A) = V(V • A) - V2A

FORMULAS FROM VECTOR ANALYSIS
121
INTEGRALS INVOLVING VECTORS
If A(m) = — B(m), then the indefinite integral of А(м) is
du
22.46
J A(«)
dit = В(м) + с с = constant vector
The definite integral of A(«) from и = a to м = Ь in this case is given by
22.47 I a(m) du = BF) - B(a)
The definite integral can be denned as on page 94.
LINE INTEGRALS
Consider a space curve С joining two points Pj(a.b a2, as) and
Р2Ф1, b2, b3) as in Fig. 22-6. Divide the curve into n parts by points
of subdivision (xuyltzt), .. ., (xn-1,yn^1,zn^1). Then the line integral
of a vector A(x, y, z) along С is defined as
22.48
i
A-dr =
A-dr - lim 2 А(жр, ур, zp) • Дгр
where Дгр = Axv i + Ayv j + &.zv к, Джр = xv + 1 — жр, Дз/р = yp + i — yp,
Дгр = гр +! — гр and where it is assumed that asn-*« the largest
of the magnitudes |Дгр| approaches zero. The result 22.48 is a gen-
generalization of the ordinary definite integral [page 94].
The line integral 22.48 can also be written
22.49
Г A-dr = Г (А1
dx + A,z dy + A3 dz)
using A = .Aji + A2j + A3k and dr = dx\ + dyj + dzk.
PROPERTIES OF LINE INTEGRALS
(*p. V».
Fig. 22-6
22.50
22.51
A'dr
A-dr = I A'dr+I A-
INDEPENDENCE OF THE PATH
In general a line integral has a value which depends on the particular path С joining points Pt and P2
in a region %. However, in case A = V0 or V X A = 0 where </> and its partial derivatives are con-
continuous in %, the line integral I A • dv is independent of the path. In such case
22.52 Г A-dr = Г 2 A-dr = ф(Р2) - ф(Р{)
''с Jp,

122
FORMULAS FROM VECTOR ANALYSIS
where ф(Рг) and ф(Р2) denote the values of d> at Рг and P2 respectively. In particular if С is a closed curve,
22.53
XA>dr = i
A • dr = ф A • dr = 0
where the circle on the integral sign is used to emphasize that С is closed.
MULTIPLE INTEGRALS
Let F(x,y) be a function defined in a region % of the
xy plane as in Fig. 22-7. Subdivide the region into n parts
by lines parallel to the x and у axes as indicated. Let дАр =
Axp Аур denote an area of one of these parts. Then the in-
integral of F(x, y) over % is defined as
22.54
Г F(x,y)dA = lim 2 F(xp, yv) дАр
•^en n-*« J>=1
provided this limit exists.
In such case the integral can also be written as
22.55
I F(x, y) dy dx
x = a ' m = /j(x)
= I i I F{x,y)dy\dx
ux = a l*/v = f,lx) J
Fig. 22-7
where у — ft(x) and j/ = /2(x) are the equations of curves PHQ and PGQ respectively and a and b are
the x coordinates of points P and Q. The result can also be written as
22.56
I F(x,y)dxdy = I
у = с^х = д1(уУ Jy =
dx V dy
J
where a; = gi(y), x = g2(y) are the equations of curves HPG and HQG respectively and с and d are the j/
coordinates of Я and G.
These are called double integrals or area integrals. The ideas can be similarly extended to triple or
volume integrals or to higher multiple integrals.
SURFACE INTEGRALS
Subdivide the surface S [see Fig. 22-8] into n elements of
area ASp, p = 1,2,... ,n. Let A(a;p, yv, zp) = Ap where (xp, yp, zp)
is a point P in ASp. Let Np be a unit normal to ASp at P. Then
the surface integral of the normal component of A over S is
defined as
22.57
X
A • N dS = lim
2^
p=i
Np
/x
Fig. 22-8

FORMULAS FROM VECTOR ANALYSIS
123
RELATION BETWEEN SURFACE AND DOUBLE INTEGRALS
If % is the projection of S on the xy plane, then [see Fig. 22-8
ХХ.ЭО I A • IN uo —
THE DIVERGENCE THEOREM
Let S be a closed surface bounding a region of volume V; then if N is the positive (outward drawn)
normal and dS = N dS, we have [see Fig. 22-9]
22.59
Г V • A dV = Г
A-dS
The result is also called Gauss' theorem or Green's theorem.
Fig. 22-9
Fig. 22-10
STOKE'S THEOREM
Let S be an open two-sided surface bounded by a closed non-intersecting curve С [simple closed curve]
as in Fig. 22-10. Then
22.60 ? A • dr = Г (V x A) • dS
where the circle on the integral is used to emphasize that С is closed.
GREEN'S THEOREM IN THE PLANE
? (Pdx + Qdy) = Г (<>Q-i!E\dxdy
X JK \dx dy
Tc JR \dx dy)
where R is the area bounded by the closed curve С This result is a special case of the divergence theorem
or Stoke's theorem.

124
FORMULAS FROM VECTOR ANALYSIS
22.62
GREEN'S FIRST IDENTITY
J {ф Vfy + (V*) • (V*)} dV = j
where 96 and ф are scalar functions.
GREEN'S SECOND IDENTITY
22.63
J& Vfy - Ф VV) dV = Г
dS
MISCELLANEOUS INTEGRAL THEOREMS
22.64 i VXAdV = feJSXA 22.65 Г ф dr = I dSXJ7ф
CURVILINEAR COORDINATES
M3 curve
L ul curve
«2 curve
A point P in space [see Fig. 22-11] can be lo-
located by rectangular coordinates (x, y, z) or curvi-
curvilinear coordinates (ult щ, и3) where the transforma-
transformation equations from one set of coordinates to the
other are given by
22.66 x - x(ultu2,u3)
у = у(щ,и2>щ)
z = z(uvu2,u3)
If u2 and u3 are constant, then as щ varies, the
position vector г = xi + y] + zk of P describes a
curve called the щ coordinate curve. Similarly we
define the u2 and щ coordinate curves through P. The
vectors dr/dui, дг/дщ, дг/ди3 represent tangent vec-
vectors to the uv u2, u3 coordinate curves. Letting
ei>e2>e3 be unit tangent vectors to these curves, we
have
22.67 il
where
22.68 hx =
are called scale factors. If e1( e2, e3 are mutually perpendicular, the curvilinear coordinate system is
called orthogonal.
Fig. 22-11
or
1. _
hlii1' 3«2 "*"*
dr
dr
3m2
Jl =
duz
, h3 =
i a
г3е3
dr
du3

FORMULAS FROM VECTOR ANALYSIS
125
FORMULAS INVOLVING ORTHOGONAL CURVILINEAR COORDINATES
22.69
22.70
——
дЩ
dr = —— dut + — du2 + — du3 =
д 0U2 OU3
ex + h2 du2 e2 + h3 du3 e3
ds2 = dr-dr = h\du\ + h\du\ + fcfditf
where ds is the element of arc length.
If dV is the element of volume, then
22.71 dV = \(h1e1du1)-(h2e2du2)X (h3e3du3)\ — h1h2h3duidu2du3
dr # 3r 3r
du3
du2 du$ =
d(x, y, z)
д(щ, u2, u3)
du2 du3
where
22.72
is called the Jacobian of the transformation.
S(x, y, z) _
, u2, u3)
дх/dUi дх/ди2 дх/ди3
ду/дщ ду/ди2 ду/ди3
dz/дщ dz/du2 dz/дщ
TRANSFORMATION OF MULTIPLE INTEGRALS
The result 22.72 can be used to transform multiple integrals from rectangular to curvilinear
coordinates. For example, we have
2273 fffF(x,y,z)dxdydz = /#«(«1.%.*з)|г?^|^^^з
where %' is the region into which % is mapped by the transformation and G(w1(m2, u3) is the value of
F(x, y, z) corresponding to the transformation.
GRADIENT, DIVERGENCE, CURL AND LAPLACIAN
In the following, Ф is a scalar function and A = Atet + A2e2 + A3e3 a vector function of orthogonal
curvilinear coordinates щ, и2, u3.
22.74 Gradient of Ф = grad Ф =
П
22.75 Divergence of A = div A = V-A =
П^ ditj Л2 au2 h3 du3
+
22.76 Curl of A = curl A = V X A =
h2e2 h3e3
д
дЩ
ди2
д
ди3
h2A2 h3A3
+
h3hj дф \ в /hth2 эф
h2 du2j du3 \ h3 дщ
22.77 Laplacian of Ф =
Note that the biharmonic operator У4Ф = У2(У2Ф) can be obtained from 22.77.

126
FORMULAS FROM VECTOR ANALYSIS
SPECIAL ORTHOGONAL COORDINATE SYSTEMS
Cylindrical Coordinates (r, 9, z) [See Fig. 22-12]
22.78
22.79
x = r cos в, у — r sin в, z = z
h\ = 1, hi - r2, hi = 1
22.80
еч 1зф ±эч
дг2 г дг г2 дв2
Fig. 22-12. Cylindrical coordinates.
Fig. 22-13. Spherical coordinates.
Spherical Coordinates (г, в, ф) [See Fig. 22-13]
22.81
22.82
22.83
x = r sin в cos Ф, у = r sin в sin ф, z = r cos в
hi = 1, hi = г2,
v2 -1A/ 2f*
r2 dr\r 3r
r2sins
= r2 sin2 9
дв)
дф2
Parabolic Cylindrical Coordinates (u, v, z)
22.84
22.85
22.86
= -|(w
-|(w2--y2), i/ = uv,
г = z
h\ = hi = u2
v2, hz3 = i
и2 + v2 \ди2 dv2 т dz2
The traces of the coordinate surfaces on the xy
plane are shown in Fig. 22-14. They are confocal
parabolas with a common axis.
Fig. 22-14

FORMULAS FROM VECTOR ANALYSIS
127
Paraboloidal Coordinates (u, v, ф)
22.87
where
22.88
22.89
x = uv cos Ф, у — uv sin ф, z — i(n2 — v2)
и g 0, v § 0, 0 ё ф < 2тт
hf - h\ = v? + v2, hi = A2
.-Ч„**) +
w(m2+v2) du\ dul v(u2 + v2) Sv \ dvl u2v'
Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 22-14 about the
x axis which is then relabeled the z axis.
Elliptic Cylindrical Coordinates (u, v, z)
22.90
where
22.91
22.92
x = a cosh и cos v, у — a sinh и sin v, z = z
м s o, 0 s v < 2л-, -«о < z < oo
-i = h2 — a2(sinh2tt. + &m2v), h3 =
+
a2(sinh2 и + sin2 v) \du2 Bv2
+
The traces of the coordinate surfaces on the xy plane are shown in Fig. 22-15. They are con-
focal ellipses and hyperbolas.
Fig. 22-15. Elliptic cylindrical coordinates.

128 FORMULAS FROM VECTOR ANALYSIS
Prolate Spheroidal Coordinates (?, -q, ф)
22.93 x = a sinh ? sin 4 cos ф, у — a sinh ? sin -q sin <f>, z — a cosh ? cos 4
where ? = О, Оёт/ёгг, 0 s 0 < 2тг
22.94 h\ = h\ = a2(sinh2? + sin2,), Л3 = a2 sinh2 ? sin2 v
22.95 V2<I> =
a2(sinh2 ? + sin2 ч) sinh ? d? V d?
. 1 д ( . дФ
+ -9/,:_1_9 » i _;__9 4 -¦•_ 7- Sin 7) -—
a2(sinh2 ? + sin2 y) sin ч аД 5т; j a2 sinh2 ? sin2 , дф2
Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 22-15 about the
x axis which is relabeled the z axis. The third set of coordinate surfaces consists of planes passing
through this axis.
Oblate Spheroidal Coordinates (?, rj, ф)
22.96 x = a cosh ? cos 4 cos ф, у = a cosh ? cos у sin ф, z = a sinh ? sin -q
where 1 = 0, -тг/2 ^ ч = т/2, 0 S 96 < 2тг
22.97 h\ = hi - a2(sinh2 ? + sin2 4), A§ = a2 cosh2 ? cos2 ч
22.98 V4> =
a2(sinh2 ? + sin2 4) cosh ? dt
. 1 о/ дф \ 1 1 а^Ф
a2(sinh2? + sin2^) cos?; ду у Ч Эт/у/ a2 cosh2? cos2); Э02
Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 22-15 about the у
axis which is relabeled the z axis. The third set of coordinate surfaces are planes passing through
this axis.
Bipolar Coordinates (u,v,z)
22.99 x = "sinhv , у = asinM , z = z
cosh v — cos и cosh v — cos и
where 0 = и < 2тг, —« < -у < «, — °° < z < <*>
or
22.100 x2 + (y - a cot uJ = a2csc2w, (x — a coth vJ + y2 = a2 csch2 v, z = z
22.101 hi = fc22 = . . a2 r,, hi = 1
(cosh v — cos uJ
tt 1П9 V72* — (cosh v — cos иJ /д2Ф , д2Ф\ , д2Ф
VZ V Ф - -2 ^du2+J^J + Sz2
The traces of the coordinate surfaces on the xy plane are shown in Fig. 22-16 below.

FORMULAS FROM VECTOR ANALYSIS
129
Fig. 22-16. Bipolar coordinates.
Toroidal Coordinates (u, v,
22.103
_ a sinh v cos Ф
cosh v — cos и '
_ a sinh v sin <f>
a smu
cosh v — cos и ' cosh v — cos и
22.104
,2 _ ,2 _
hl - k2 -
(cosh v- cos u)*
.2 _
' 3 ~
a2 sinh2 t;
(cosh t) — cos mJ
22 105 V2<f> = (cosh ^ ~ cos иK э
ди \cosh v — cos и ди j
, (cosh v — cos u)s _д_ / sinh у дф\ , (cosh v — cos иJ д2Ф
п% ritiVi 41 Л/11 \ рпоЬ 1» — <^r»c! it Лч1 / /тг.2 einV>2 /л Л^,2
a2 sinhv Si; I cosh v — cos и dv
a2 sinh2/y
The coordinate surfaces are obtained by revolving the curves of Fig. 22-16 about the у axis
which is relabeled the z axis.
Conical Coordinates (A, p, v)
22.106
X/it» ^ j (^ — o^){v2 — a2) X /
62 - a2
22.107

130
FORMULAS FROM VECTOR ANALYSIS
22.108
or
22.109
22.110
22.111
or
22.112
22.113
X2
«2-
X2
a2-
X2
__ a2-
X2 "
y2 =
z2 -
h2
r x2
a2-
a2-
{ a2-
T2
У2 --
z -
h1
hi
hi
Confocal Ellipsoidal
У2 z2
X b2 - X c2 - X
, У2 | z2
Ii b2 - ii c2 - ii
, У2 , z2
v b2 — v C2 — v
. (a2 - X)(a2 - д)(а2 -
(a2 - b2)(a2 _ C2)
F2 - a2)F2 - c2)
(c2 - a2)(c2 - b2)
(/i - X)(v - X)
4(a2-X)F2-X)(c2-
4(ct — /i)(o — fi){c ~
(X - У)(д - r)
4(a2 — v)(b2 — v)(c2 -
- 1
- 1
v)
v)
-x)
Confocal Paraboloidal
, У2
X 62 - X
ц ' b2 - n
, У2
_ (a2 - X)(a2 - rf(a2 -
62-a2
_ (Ь2-Х)(Ь2-д)(Ь2-
a2- b2
= X + ii + v - a2 -
4(a2-X)(b2-X)
_ (v — /i)(X — д)
4(a2 — /j)F2 — л)
16(a2-r)(b2-^)
X -«
д 62 <
^ a2 <
-v)
¦v)
- b2
Coordinates (A, /a, v)
X < c2 < b2 < a2
c2< ц<Ъ2<а2
c2 < b2 < v < a2
Coordinates (A, p, v)
< X < 62
/t < a2
V < °°

DEFINITION OF A FOURIER SERIES
The Fourier series corresponding to a function f(x) defined in the interval с SI x S с + 2L where e
and L > 0 are constants, is defined as
23.1
where
23.2
a0
n=l
V + 2 ( «n cos -|— + bn sin -g-I
1 I ,, . ПтгХ
an = 7 I /(*) cos -j-
t /*c + 27j
1 I ./ \ ПтгХ
Y I /(x) sin —=-
dx
n =
If /(ж) and f'(x) are piecewise continuous and f(x) is defined by periodic extension of period 2L, i.e.
f(x + 2L) = f(x), then the series converges to f(x) if x is a point of continuity and to ^{/(x + 0) + f(x — 0)}
if x is a point of discontinuity.
COMPLEX FORM OF FOURIER SERIES
Assuming that the series 23.1 converges to f(x), we have
23.3 f(x) = J_ cne
where
23.4 cn = 1
|АК-гЬ„)
n > 0
и < О
и = О
PARSEVAL'S IDENTITY
23.5
i f
23.6
GENERALIZED PARSEVAL IDENTITY
aoca »
/(x) er(*) dx = — + 2 («A + bndn)
* n=l
where ал, Ь„ and cn, dn are the Fourier coefficients corresponding to f(x) and g(x) respectively.
131

132
FOURIER SERIES
SPECIAL FOURIER SERIES AND THEIR GRAPHS
23.7
23.8
ТГ
2
23.9
23.10
23.11
2
7Г
fix) =
4 / sin ж ,
4 /cos ж
/(х) = х,
„ /sin ж
Ч 1
/(х) = а;
„ /sin х
Ч 1
Г 1 0 < X < 77
[-1 -77 < X < 0
sin Зж , sin 5х . \
3 ' Б ' /
Г X 0 < X < 77
|ж| = «
[-Х -77 < Ж < 0
, cos За; , cos 5х , \
' 32 ' 52 ' 7
—77 < X < ТГ
sin 2x , sin Зх ^
2 ' 3 )
, 0 < ж < 2,7
, sin 2а; , sin Зж , \
2 3 /
/(ж) = |sin х|, —тг < х < тг
4 /cos 2a;
77^ 1-3
cos 4x cos 6x \
3«5 5*7 у
!
| —2л- ' —77
~2тг —ТТ
/
YV
-4тг -2v
— 27Г Я"
Fig.
\
Fig.
Л
Fig.
/
/
Fig.
Fig.
0
23-1
fix)
/
0
23-2
fix)
/
0
23-3
fix)
- 2-
/
0
23-4
fix)
r
0
23-5
1
1
177
1
1
77
' 1
1
1 _
I/
2л-
V
77
1
1
12„ Х
1
1
/
/С x
/1
^:
2»

FOURIER SERIES
133
23.12 /(x) =
sin x 0 < x < w
0 TT < X < 277
1.1.
+ m
2 /cos 2x , cos 4x . cos 6x ,
5Г \ 1 • a A' 5 5*7
Fig.23-6
23.13 /(x) =
COS X 0 < X < 77
— COS X —7Г < Ж < 0
8^ /sin 2x 2 sin 4x 3 sin 6a;
я- \l-3 3-5 5*7
Fig. 23-7
23.14 /(ж) = x2, -я- < x < я-
я^ _ /cos x _ cos2x cos3x
3 4l 12 22 + 32
Fig. 23-8
23.15 /(ж) = х(тг - x), 0 < ж < я-
?^ _ /c°s 2a; cos 4a; cos 6x
"б" ~ \ I2 22 32
-2тг
Fig.23-9
23.16 f(x) = х(тг - x)(v + x), -т
sin x _ sin2a; sin Sx
13 23 33
Fig. 23-10

134
FOURIER SERIES
23.17 f(x) =
a_ _ 2_ /sing cos a; _ sin 2a cos 2a;
тг тг \ 1 2
, sin 3a cos 3a; _
¦•- 2a — -•¦ 2a —
! ' ' !
1 ! !
1 ' 1
1 1 1
1 1
1 1 1
II II
-Зл- -2л- -тг
fix)
0
2a
tr
2тг
2a »
Зл-
Fig. 23-11
23.18 /(*) =
х(тг — x) 0 < x < ir
—x(w — x) — v < x < 0
sin x sin 3x sin 5a; ,
Д 1» + ~~33 ' 53~
Fig. 23-12
MISCELLANEOUS FOURIER SERIES
23.19
23.20
23.21
23.22
23.23
/(*)
fix)
fix)
= sin fix, —1г<х<тт, ii?= integer
2 sin hit I sin x 2 sin 2a; , 3 sin За; ^
* V12-
= COS liX, —JT < X < IT,
2ц sin цтг ( 1 .
тг \2r*
= tan [(a sina;)/(l — a
a sin x +
= In A - 2a cos x + a2),
—2 la cosx
= |tan-i[Basina;)/(l-
a sin x +
H2 22 - u? ' 32 - ifi 1
/i ?= integer
cos x cos 2a; cos За; \
lz — /iz 22 — д2 3Z — /л1' у
cos a;)], — ж < x < тг, a\ < 1
-r- sin 2a; + -r- sin 3a; + • • •
и о
—тг < x < тг, \а\ < 1
a2 a3 \
+ -r- cos 2x + -5- cos За; + • • • I
2 3 у
a2)], -тг < x < тг, |a| < 1
a3 a5
-5- sin За; + — sin 5a; + • • •
a 0

FOURIER SERIES 135
23.24 f(x) = \ tan-I[Bacos«)/(l-a2)], -тг < x < тг, \a\ < 1
Ld
a3 „ , w>
a cos x — — cos ox + -=- cos ox — ¦ • •
о О
23.25 f(x) - e»x, -тг < x < тг
2 sinh fiTr /J_ , -S (—l)n(/j cos wa; — n sin nx)
23.26 fix) — sinh цх, —тг < х < тг
2 sinh цтг I sin a; _ 2 sin 2a; 3 sin 3x
~тг I12 + ^ 22 + ,j.2 32 + M2
23.27 /(a;) = cosh цх, -тг < x <
2ц sinh /iTT f _1 cos a; cos2» _ cos 3x ,
~^ 12 + Д2 + 22 + Д2 ~ 32 + Д2 +
23.28 f(x) = In |sin |ж|, 0 < x < тг
23.29 /(a;) = ln|cosia;|, -тг < x < тг
cos x cos 2x cos Зх ,
1 ' 2 3 +
23.30 f{x) = U
cos x cos 2a; cos 3a; ,
12 ' 22 32
23.31 f(x) =
sin a; sin2x sin 3a; .
13 ' 23 33
23.32 /(a;) = ^774 -^j
cos x cos 2a; cos 3x ,
14 ' 24 34

BESSEL'S DIFFERENTIAL EQUATION
24.1 x2y" + xy' + (x2-n2)y = 0 n is 0
Solutions of this equation are called Bessel functions of order n.
BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n
n\xl —
Г(и
2 • 4Bn + 2)Bи + 4)
24.3
24.4
- n) \ 2B - 2«) 2 • 4B - 2n)D -
J_n(x) = (-l)"Jn(x) n = 0,1,2, ...
24.5
24.6
24.7
If n?= 0,1,2, ..., Jn(x) and J-n(x) are linearly independent.
If n?= 0,1,2, ..., Jn(x) is bounded at x = 0 while J-n(x) is unbounded.
For n — 0,1 we have
/y.2 л.4 т€
42 ~ 22 • 42 • 62
22 . 42 . e ~ 22 • 42 • 62 • 8
BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n
sin-гея-
24.8
^n (*) =
hm
„^0,1,2,
n = 0,1,2,
sin pir
This is also called Weber's function or Neumann's function [also denoted by Л^п(
136

BESSEL FUNCTIONS 137
For n = 0,1,2, ..., L'Hospital's rule yields
24.9 Yn(x) = - {In (x/2) + y}Jn(x) - ^ V (w~?!~1)!(»/2)»-»
(>
"¦ fc=o A:! (n +
where у = .5772156. . . is Euler's constant [page 1] and
24.10 ф(р) = 1 + 1 + 1+ ... +I( Ф@) = о
For n = 0,
24.11 Г0(х) = |{1
24.12 Y-n(x) = (-l)»Yn(x) №=0,1,2,...
For any value n & 0, Jn(x) is bounded at x = 0 while Yn(x) is unbounded.
GENERAL SOLUTION OF BESSEL'S DIFFERENTIAL EQUATION
24.13 xt = AJ (x) -\- BJ (x) 7i ?= 0 1 2
24.14 у - AJn(x) + BYn(x) all»
24.15 2/ = AJn(x) + BJn(x) jxj2(x) alln
where A and В are arbitrary constants.
GENERATING FUNCTION FOR Jn{x)
24.16 ex(t-l/»/2 = ^
П= —o
RECURRENCE FORMULAS FOR BESSEL FUNCTIONS
24.17 J»+i(*) =
24.18 ^(ж) =
24.19 ж^(ж) = ajJn_!(«) - nJn(x)
24.20 ж^(ж) = nJn(x) - xJn+1(x)
24.21 ^-{^^(ж)} = xnjn_1(a;)
24.22 ^{a;-«Jn(*)} = -x-"Jn + l(x)
The functions Fn (ж) satisfy identical relations.

138 BESSEL FUNCTIONS
BESSEL FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER
In this case the functions are expressible in terms of sines and cosines.
«I ¦<« т , \ „ / 2 i>. «I . . , . / 2 /cosx ,
24.23 Ji/2(a;) = Д| — sin» 24.26 J_3/2(a;) = Д| — (-^— + si
sin a;
24.24 J.,,^) = Д/^cosz 24.27 J^fc) = \\-^\(±-
24.25 J3/2(*) = *Tl/'^*_ cos*") 24.28 J_tn(x) = yj-^-Цвтх + (^-l) cosx
V wX \^ X J У irX \X \X~ J
For further results use the recurrence formula. Results for Y1/2(x), Y3/2(x), ... are obtained from 24.8.
HANKEL FUNCTIONS OF FIRST AND SECOND KINDS OF ORDER n
24.29 Н<»(х) = Jn(x) + iYn(x) 24.30 H™(x) = Jn(x) - i Yn(x)
BESSEL'S MODIFIED DIFFERENTIAL EQUATION
24.31 x2y" + xy' - (x2 + n*)y = 0 иёО
Solutions of this equation are called modified Bessel functions of order n.
MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n
1Л 32 J (v\ — i~" 7 liv\ — i> — mri/2 1 (i~\
""" 1n\x/ ~ г Jn\lx> ~~ e Jn\lX)
X» f X2 X* +...}= V ^/2^ + 2*
2» r(n +1) 1 2Bn-LO^ o.jra»±o\/i)«j_ii ' r .«.
24.33
— П\Л' *~* * ^ — П I1*/ — e
ж-» Г a;2 ж4 4. ... 1 = V (*/2Jk-n
2B-2и) 2 • 4B - 2и)D - 2и) J k=o ft! r(fc + 1 -1
24.34 /_n(a;) = In(x) n = 0,1,2,..
If n ?= 0,1,2, ..., then /„(a;) and /_n(o;) are linearly independent.
For и = 0,1, we have
24.35 I0(x) = 1 + |^ + 22^p + xl—- + • • •
24.36
24.37 /i(«) = I^x)

BESSEL FUNCTIONS 139
MODIFIED BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n
{I-n(x)-In(x)} n ?> 0,1,2, ...
2 sin Птг
.24.38 Kn(x)
lim —4- U-P(x) - Ip(x)} n = 0,1,2,...
For n = 0,1, 2, . . ., L'Hospital's rule yields
1 n-i
24.39 Kn(x) = (-l)»+i{ln(o;/2) + y}In(x) + ~ 2 (-l)k(n-fc-1)! («/2J*-»
* fc=o
where Ф(р) is given by 24.10.
For n = 0,
24.40 K0(x) = -{In (x/2) + y}I0(x) + g-
24.41 K_n(x) = Кп(х) w = 0,l,2,...
GENERAL SOLUTION OF BESSEL'S MODIFIED EQUATION
24.42 у = AIn(x) + BI_n(x) n ?-0,1,2,..
24.43 у = Aln(x) + BKn(x) all»
24.44 у - AI Ax) + BIn(x) f df all и
J xll(x)
where A and В are arbitrary constants.
GENERATING FUNCTION FOR h(x)
24.45 eict+i/»/2 = 2
П =
RECURRENCE FORMULAS FOR MODIFIED BESSEL FUNCTIONS
24.46 In + 1(x) = /„_!(*) - ^In(x) 24.52 Kn+1(x) = K^x) + ^^»(«)
24.47 /;(*) = |{7n_1(x) + 7n+1(x)} 24.53 K'n(x) = tfK^x) + Kn+1(x)}
24.48 ж/;(ж) = xln-iix) - nln(x) 24.54 xK'jx) = -xK^^x) - nKn(x)
24.49 xl'n(x) = х/„+1(х) + nln(x) 24.55 i^W = rf,(a;) - x
24.50 А{жп/п(а;)} = «»/„_!(*) 24.56 ^{Л„(!()} = -«»«„_
24.51 ^{*-»/я(*)} = *-»7в+1(«) 24.57 J-{*-«#»} = -*-»

140 BESSEL FUNCTIONS
MODIFIED BESSEL FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER
In this case the functions are expressible in terms of hyperbolic sines and cosines.
24.58 Im(x) = лГ^впЛж 24.61 I-a/2(x) = л Г^ ( sinha _
24.59 1-V2(x) = J— coshx 24.62 I5/2(x) = J~ \ D + 1 ) sinha; - - cosh x
\ WX \ irX l\X2 j X
24.60 Is/2(x) = л[^(со5Ъх-^^-) 24.63 7_B/2(«) = -M-jf4+1^sh* i
Y п"Ж \ ^ J v ^"-^ (^ x*^
For further results use the recurrence formula 24.46. Results for K1/2(x), Ks/2(x), . .. are obtained
from 24.38.
Ber AND Bei FUNCTIONS
The real and imaginary parts of Jn (xe3iri/i) are denoted by Bern (x) and Bein (x) where
64 Ber«(a;) = 3o'klT(n + k + l)cos 1
24.66 W ' Ber(*) = l-
24.67 Bd(.) = feW
» +
Ker AND Kei FUNCTIONS
The real and imaginary parts of e~n77i/2Xn(a;e77i/4) are denoted by Kern (x) and Kein (a;) where
24.68 Kern (x) = -{In (ж/2) + у} Bern (ж) + ?г Bein (ж)
2 Jn к\
24.69 Kein (ж) = -{In (ж/2) + у} Bein (ж) - \* Вег„ (ж)
"^ in - к - 1)! (ж/2р-» . (Зи + 2fc)ff
i Г] sin 2
fc=0 Л! 4
1 » (a;/2)" + 2fc Cn
+ 2 Д fc!(n + fc)! {Ф№) + *(И + Л)} Sln
and Ф is given by 24.10, page 137.
If n = 0,
24.70 Ker (x) = -{ln(*/2) + y}
24.71 Kei (ж) = -{In (ж/2) + у} Bei (x) - J Ber (ж) + («/2J -

BESSEL FUNCTIONS
141
DIFFERENTIAL EQUATION FOR Ber, Bei, Ker, Kei FUNCTIONS
24.72
«V + xy' -
= 0
The general solution of this equation is
24.73 у = А{Вег„ (x) + г Bein (x)} + B{Kern (x) + i Kein (x)}
GRAPHS OF BESSEL FUNCTIONS
Fig. 24-1
Fig. 24-3
Bei x
Fig.24-2
Fig.24-4
Fig. 24-5
Fig. 24-6

142
BESSEL FUNCTIONS
INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
24.74 CxJ0(x)dx = xJ,(x)
24.75 j x2J0(x)dx = x2J1(x) + xJ0(x) — I J0(x) dx
24.76 Cx™J0(x)dx = xmJ1(x) + (m-l)x'"-1J0(x) - (m-1J fx">-2J0(x)
dx
24.77
24.78
dx =
= Ji(x) ~ -
Ji(x)
/ M
x) dx
(m - lJxm
24.79 Cj1(x)dx = -J0(x)
24.80
J/лЛ -| f* J
0 V*/ J- I 
~2 (m — l)xm~1 (m — IJ »/ xn
m-2
Г xJ^x) da; = -xJ0W + Г Л (ж) «to
24.81 CxmJ1(x)dx = -xmJ0(x) + m \ xm~iJo(x)dx
24.82 C^p-dx = -Ji(x) + fjo(x)dx
dx
24.84 j a;" Jn_x(a;) da; = a;"Jn(a;)
24.85 JV«Jn+1(a;)da; = -x-"Jn(x)
С xmJn(x)dx = ~xmJn-1(x) + (m + n-1) j xm~1Jn-l(x) dx
24.86
x{a Jn(j3x) J'n(ax) - /3 JJ
/32- „2
j x Jn(ax)Jn(px) dx =
j X Jl {ax) dx = ^ {J'n (ax)}2 + у (l - ^) Vn M>
24.87
24.88
The above results also hold if we replace Jn(x) by Yn(x) or, more generally, A Jn(x) + В Yn(x) where
A and В are constants.
DEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
24.89 Г e-<**J0(bx)dx = -
•Л)
24.90 Г" e~"*Jn(bx) dx = '
24.91 j cos ax J0(bx) dx -
a2 + 62
a2 + 62 - a)"
n > -1
a > 6
a< 6

BESSEL FUNCTIONS 143
24.92 f °° Jn(bx) dx = | и > -1
С Jn(bx)
А
24.93 I -±—dx = j- n = 1,2,3,...
n
24.94 Г e-"*J0(b\fx~) dx = e
Jn a,
24.95 J xJn{ax)Jn(px)dx = — y,_^a
24.96
24.97
о
INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS
24.98 J0(x) = -\ cos (x sin e) de
24.99 Jn(x) = — I cos <ne — x sin в) de, n = integer
24.100 Jn(x) = ——— Г cos (x sin e) cos2n 6 de, n > -|
24.101 Y0(x) = --J cos (x cosh u) du
1 Cw 1 C2v
24.102 I0(x) - -I cosh (x sin e) de = -=- I e*staede
ASYMPTOTIC EXPANSIONS
24.103 Jn(x) - Л/ — cos ( x - ~ - ^ ) where a; is large
24.104 Yn(x) ~ \[^-sm(x-^-^) where a; is large
\ ttX у 2. 4 у
1 / \n
24.105 Jn(x) ~ -^(|?) where и is large
У2п \2riy
24.106 Уп(ж) ~ -\\(i?) where re is large
24.107 /„ (ж) ~ where a; is large
V2a;
24.108 if,, (ж) ~ where x is large

144
BESSEL FUNCTIONS
ORTHOGONAL SERIES OF BESSEL FUNCTIONS
Let Xj, X2, X3, ... be the positive roots of RJn(x) + SxJ'n(x) = 0, n > —1. Then the following series
expansions hold under the conditions indicated.
s =
24.109
where
24.110
In particular
24.111
where
24.112
24.113
where
24.114
In particular
24.115
where
24.116
24.117
where
24.118
In particular
24.119
where
24.120
= 0
if
if
if
/(*) =
n — 0,
/(*) =
/W =
Ak =
n = 0,
/(*) =
A,
U«, =
i.e.
AX
Afc
A,
Ak
A,
J2{>
A,
С
—
2 (re
re = 0 so that
—
i:
Ai, A2, A3,
/.<M> +
2
/»(Xi«) +
2
JnM +
2
fc)-^n l(
Jo(Xi«) +
2
4(x,) +
¦
A
4)
A
)
'S
A
A
J
R/S
Aox» + A]
Г1
+ 1I «"
2
Xk) - >/„_!
Д = 0 [i.e
Л Л- А
Л-0 • Л1
° = 'X
2
j
h
X
h
1
л
. are positive roots of Jn(%) = 0
2Jn(X2a;)
+ A3Jn(X3a;) + •••
- Г x f(x) JJ\kx) dx
Jo
2 J0(X2a;)
J* */(*)
> —те
2Jn(X2a;)
Jn + 1(Xfc)
г^о^г^)
Г1
P^k) Л>
= -те
„(Xja;) +
/Co-1 da-
)Jn+li4
1> ^2> X3, .
(Xxa;) + /
fix) dx
Jo
+ Л3 J0(X3a;) + • • •
Joi\kx) dx
+ A3Jni\3x) + •••
J ж/(жO„(Хкж) da;
+ A3 J0i\3x) + •¦¦
A2Jn(k2x) + •¦¦
С x J x
) Jo k
. . are the positive roots of Jiix) — 0],
i.2Joi\2x) + ¦ ¦¦

BESSEL FUNCTIONS 145
In this case
and we have
24.121
where
24.122
there are two
/(*) =
Uo -
[Ak =
pure
Aol
4(x
R/S
imaginary
n(\ox) + A
2
o) + /„_i(X0
2
< -n
roots —гЛ0
1Jn(X1a;) +
)W*o) .
as
A
Г
С
well as
2 Jn(X2a;)
*/<*)/»
x f(x) Jn
the positive roots X1; X2, X3, . ..
+ •••
(\ox) dx
(\kx) dx
MISCELLANEOUS RESULTS
24.123 cos (a; sin») = J0(x) + 2 J2 (x) cos 26 + 2J4(x) cosie + ¦¦¦
24.124 sin (x sin в) = 2Jx(a;)sini9 + 2 J3 (x) sin 30 + 2 J5 (a;) sin Ъб + •••
24.125 ./„(а-+ г/) = _2юЛ(ж)^п-ьB/) и = 0,±1,±2, ...
This is called the addition formula for Bessel functions.
24.126 1 = J0(x) + 2J2(x) + ••• + 2J2n(a;) + •••
24.127 x = 2{Jt(x) + 3J3(x) + 5J5(x) + ¦¦¦ + Bn + 1) J2n + 1(x) + •••}
24.128 x2 = 2{4J2(a;) + 16 J4(x) + 36J6(a;) + ••• + BnJJ2n(a;) + •••}
x Ji (x)
24.129 —-— = J2(a;) - 2J4(x) + 3J6(x) - •••
24.130 1 = J\(x) + 2j\(x) + 2J\(x) + 2Js(a;) + •••
ЛД 1^1 Т1Г/пл\ 1/7 /™\ О 7 /пл\ _l_ 7 /~,\\
24.132 J';'(x) = ±{Jn-3(x) - 3Jn_1(a;) + 3Jn+1(x) - Jn + 3(x)}
Formulas 24.131 and 24.132 can be generalized.
24.133 J^(x)J_n(x) - JLnJn(x) =
24.134 J,(x)J_n + Ax) + J-,(x)J,_,i
24.135 Jn + 1(x)Yn(x) - Jn(x)Yn+1(x) = Jn(x)Y'n(x) - J'n(x)Yn(x) = ^
24.136 sina; = 2{Jx(a;) - J3(x) + J5(x) - •••}
24.137 cos a; = J0(x) - 2J2(a;) + 2 J4(«) - •••
24.138 sinh ж = 2{/i(a;) + Is{x) + I5(x) + • ¦ ¦}
24.139 cosh x = Jo(«) + 2{72(a;) + 74(a;) + Ie(x) + ¦ ¦ •}

LEGENDRE'S DIFFERENTIAL EQUATION
25.1 {I~x2)y" - 2xy' + n(n+l)y = 0
Solutions of this equation are called Legendre functions of order n.
LEGENDRE POLYNOMIALS
If n = 0,1,2, ..., solutions of 25.1 are Legendre polynomials Pn{x) given by Rodrigue's formula
SPECIAL LEGENDRE POLYNOMIALS
25.3 P0(x) = 1 25.7 P4(a-) = iC5a-4 - ЗОж2 + 3)
25.4 Pi (ж) = x 25.8 P5{x) = iF3x5 - 70x3 •
25.5 P2(x) = ^(Зж2-1) 25.9 Р6(ж) = ^B31жв - 315ж4 + 105а;2 - 5)
25.6 Р3(х) = J-Ex3-3a;) 25.10 Р7(х) = ^D29ж7 - 693ж5 + 315ж3 - 35ж)
LEGENDRE POLYNOMIALS IN TERMS OF $ WHERE z=cosd
25.11 P0(cos9) = 1 25.14 P3(cos9) = ^C cos 9 + 5 cos 3e)
25.12 P^cos») = cose 25.15 P4(cosi?) = -6^(9 + 20 cos2e + 35 cos49)
25.13 P2(cos0) = A(l + 3cos29) 25.16 P5(cos0) = ^ C0 cos в + 35 cos 39 + 63 cos 5»)
25.17 P6(cos9) = 5^E0 + 105 cos 29 + 126 cos40 + 231 cos 69)
25.18 P7(cos«) = 1щA7б cos в + 189 cos 3» + 231 cos 59 + 429 cos 70)
GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
25.19 х = 2
Vl - 2tx + t2 «=o
146

LEGENDRE FUNCTIONS 147
RECURRENCE FORMULAS FOR LEGENDRE POLYNOMIALS
25.20 (n+l)Pn + 1{x) - Bn + l)xPn(x) + пРп_г(х) = О
25.21 P» + i(*) - xP'nix) = (n + l)Pn(x)
25.22 xP'n(x) - P'n.x(x) = nPn(x)
25.23 P» + i(*) " P'n-iix) = Bn+l)Pn(x)
25.24 {x*-l)P'n{x) = nxP.W - яР„.,И
ORTHOGONALITY OF LEGENDRE POLYNOMIALS
25.25
Г Pm(x)Pn(x)dx = 0
Г (pn(*)
^-1
25.26 | {PJxWdx = o , 1
J_l пч 2w + 1
Because of 25.25, Pm(x) and Pn(#) are called orthogonal in —1 = a; S 1.
ORTHOGONAL SERIES OF LEGENDRE POLYNOMIALS
25.27 f(x) = A0P0(a;) + A^^x) + A2P2(a;) + •••
where
25.28 А„ = -
i z^1
J
SPECIAL RESULTS INVOLVING LEGENDRE POLYNOMIALS
25.29 Pn(l) = 1 25.30 Pn(-1) = (-1)" 25.31 Pn(-x) = (-1)«P,(«)
О и odd
wl.,.5-(»D
(-l)n/2 —г—-.—j—i L n even
v ' 2'4•6 • ••n
25.33 Р_(ж) = -Г (х + y/x2 - 1 cos 0)"
25.34 I P. (ж) da; =
2n
25.35 |РЯ(*)| ё 1
25.36 m() ^j
where С is a simple closed curve having x as interior point.

148 LEGENDRE FUNCTIONS
GENERAL SOLUTION OF LEGENDRE'S EQUATION
The general solution of Legendre's equation is
25.37 у = AUn(x) + BVn(x)
where
,(M „ , , . n(n+l) n(n-2)(n+l)(n
25.38 Un(x) = 1 gj x2 + ^
04 4Q T/ / ч (n - l)(n + 2) (rt-l)(n-3)(n + 2)(n
25.39 Уп(ж) = a; — x3 + ^
These series converge for — 1 < x < 1.
LEGENDRE FUNCTIONS OF THE SECOND KIND
If n = 0,1,2, . . . one of the series 25.38, 25.39 terminates. In such cases,
[Un(x)IVn{l) n = 0,2,4,...
25.40 Pn(x) = \
[Vn(x)/Vn(l) n = 1,3,5, ...
where
25.41 I7n(l) = (-1)«/2 2«Г(У)!ТД! « = 0,2,4,...
25.42 У„A) = (-^("-"/гг"-! f^Y^j! /и! п = 1,3,5,...
The nonterminating series in such case with a suitable multiplicative constant is denoted by Qn(x) and
is called Legendre's function of the second kind of order n. We define
Г Г7пA)У„(а;) и = 0,2,4,...
25.43 Qn(x) = ^
|-У„A)У„(*) 71 = 1,3,5,...
SPECIAL LEGENDRE FUNCTIONS OF THE SECOND KIND
25.44 Qa(x) = fl
25.45 QAx) = ? 1„ i^JE -i
3x
,..7 - , ч 5a;3 - 3x . (\ + x\ 5a;2 . 2
25.47 Qa(x) = —jp- In (^rTr^J - -2~ + з
The functions Qn(x) satisfy recurrence formulas exactly analogous to 25.20 through 25.24.
Using these, the general solution of Legendre's equation can also be written
25.48 у = APn(x) + BQn(x)

LEGENDRE'S ASSOCIATED DIFFERENTIAL EQUATION
26.1 (l-x2)y" - 2xy' + \n(n+l) - ——\y = О
I 1-х2)
Solutions of this equation are called associated Legendre functions. We restrict ourselves to the im-
important case where m, n are nonnegative integers.
ASSOCIATED LEGENDRE FUNCTIONS OF THE FIRST KIND
26.2
n \ / \ / ^д-m n \ / 2"и' da;"
where Pn(x) are Legendre polynomials [page 146]. We have
26.3 P°n(x) = Pn{x)
26.4 P™(x) = 0 if m > n
SPECIAL ASSOCIATED LEGENDRE FUNCTIONS OF THE FIRST KIND
26.5 P{(x) = A-х2I'2 26.8 P\(x) -
26.6 P\(x) - Зл;A - я2I/2 26.9 р|(ж) =
26.7 p\(x) = m-x2) 26.10 Р»(ж) =
GENERATING FUNCTION FOR
26.11 2"w!a
RECURRENCE FORMULAS
26.13 P" + 2(s) - A_ 2-^-P™ + 1(a;) + (n - m)(n + m + 1) P™(x) = 0
149

150 ASSOCIATED LEGENDRE FUNCTIONS
ORTHOGONALITY OF Р™(ж)
26.14 Г P™(x)P?(x)dx = 0 if n?-l
J-l
f1 m( \M ,1 - 2 (n + m)!
26.15
26.18
ORTHOGONAL SERIES
26.16 /(ж) = АтР™(ж)
where
26.17 At = -
WTm)l J.
ASSOCIATED LEGENDRE FUNCTIONS OF THE SECOND KIND
where Qn(%) are Legendre functions of the second kind [page 148].
These functions are unbounded at x = ±1, whereas P™(x) are bounded at x = ±1.
The functions Q™(«) satisfy the same recurrence relations as P™(x) [see 26.12 and 26.13].
GENERAL SOLUTION OF LEGENDRE'S ASSOCIATED EQUATION
26.19 у = AP™(x) + BQ™(x)

27.1
HERMITE'S DIFFERENTIAL EQUATION
y" — 2xy' + 2ny = 0
If и = 0,1,2,...
Rodrigue's formula
27.2
HERMITE POLYNOMIALS
then solutions of Hermite's equation are Hermite polynomials Hn(x) given by
Hn(x) = (-!)«„*• |l(
SPECIAL HERMITE POLYNOMIALS
27.3
27.4
27.5
27.6
H0(x)
Hl{x)
H2(x)
HJx)
= 1
= 2a-
- 4ж2 -
= 8a;3-
2
12a;
27.7
27.8
27.9
27.10
Щ(Х)
H5(x)
He(x)
H7(x)
= 32a-5 -
= 64a;6 -
= 128x7 -
48a-2 + 12
160a-3 + 120a;
480a-4 + 720a-2 - 120
- 1344ж5 + ЗЗбОж3 - 1680a-
27.11
GENERATING FUNCTION
e2'*-' = 2
Hn(x) t"
27.12
27.13
RECURRENCE FORMULAS
х) = 2xHn(x) -
H'n{x) = 2nHn_1(x)
151

152 HERMITE POLYNOMIALS
ORTHOGONALITY OF HERMITE POLYNOMIALS
27.14 Г e~x2 Hm(x) Hn(x) dx - 0 m?=n
27.15 С е-*' {Hn(x)Y dx = 2«nlyfc
ORTHOGONAL SERIES
27.16 f(x) = АоЯо(я0 + А1Н1(ж) + А2Я2(а;) +
where
27.17 A» = Ц= Г е-*2 f(x) Hk(x) dx
SPECIAL RESULTS
27.18 Н„(ж) = Bа;)" - ^-pJ Ba;)
27.19 Нв(-я,) = (-1)»НЯ(«) 27.20 Я2п_!@) = О
27.21 Я2п@) = (-1)»2» • 1 • 3 • б • • • Bи - 1)
27.22 I HJt)dt =
2(п + :
27.23 ?{<Г*2Я„(*)} = -е-12Я„ + 1И
27.24 Гг'!йл(()й = Я„_г@) - в"»1 »„_!(*)
•'о
27.25 Г t"e~e Hn(xt) dt = yf*n\Pn(x)
27.26 Я„<
This is called the addition formula for Hermite polynomials.
« Як(а;) ЯьB/) Я„ + ,(*) Hn(y) - Нп(х) Я„
07

LAGUERRE POLYNOMIALS
28.1
LAGUERRE'S DIFFERENTIAL EQUATION
xy" + A — x)y' + ny = 0
LAGUERRE POLYNOMIALS
If n — 0,1,2, ... then solutions of Laguerre's equation are Laguerre polynomials Ln(x) and are given
by Rodrigue's formula.
4) Q *S T / \ -Г/М -r\
SPECIAL LAGUERRE POLYNOMIALS
28.3 L0(x) = 1
28.4 L^x) = -x + 1
28.5 L2(x) = a;2 - 4a; + 2
28.6 L3(x) = -ж3 + 9a-2 - 18a- + 6
28.7 L4(«) = ж4 - 16а-з + 72a;2 - 96a- + 24
28.8 L5(x) = -a;8 + 25a-4 - 200a;3 + 600a;2 - 600a; + 120
28.9 Le(x) = a;6 - 36a;5 + 450x4 - 2400JC3 + 5400ж2 - 4320a; + 720
28.10 L7(x) = -a;7 + 49a;6 - 882ж5 + 7350x4 - 29,400a-3 + 52,920a;2 - 35,280a; + 5040
28.11
GENERATING FUNCTION
e-xtn-t _ « Ln(x) t*
1 - t ~ „=n «I
RECURRENCE FORMULAS
28.12
28.13
28.14
(x) - Bn+l- x) Ln(x) + n2Ln_1 (x) = 0
b'n(x) - nb'n-iix) + kL4_,(x) = 0
xb'n(x) = nLn(x) -
153

154 LAGUERRE POLYNOMIALS
ORTHOGONALITY OF LAGUERRE POLYNOMIALS
28.15 ( e~* Lm(x) Ln(x) dx = 0 тФп
Л
28.16 Г е-* {Ln(x)}2 dx = (%!J
ORTHOGONAL SERIES
28.17 f(x) = A0L0(x) + AtL^x) + A2L2(x) + •••
where
28.18 Ak = -^ J" e-' f(x) Lk(x) dx
SPECIAL RESULTS
28.19 L.@) = n! 28.20 f Ln(t) dt = Ln(x) - ^~
28.21 LM = (-1)" {*¦ - ^f + ^-^-' - - (
0 if p < n
Г
xPe-*Ln(x)dx =
[(-l)"(n!J if p =
28>23 Д
28.24 2 №,У = etj 0BVxl)
28.25
X™

LAGUERRE'S ASSOCIATED DIFFERENTIAL EQUATION
29.1
xy" + (m + 1 — x)y' + (n — m)y = 0
ASSOCIATED LAGUERRE POLYNOMIALS
Solutions of 29.1 for nonnegative integers m and n are given by the associated Laguer,re polynomials
29.2
where Ln(x) are Laguerre polynomials [see page 153].
29.3
29.4
Ll(x) = Ln(x)
L™(x) = 0 if m > n
SPECIAL ASSOCIATED LAGUERRE POLYNOMIALS
29.5
29.6
29.7
29.8
29.9
L\{x)
Ы(х)
h\{x)
LjW
Ll(x)
= -1
= 2x-
= 2
= -3a;2
= -6a;
4
+
+
18a; - 18
18
29.10 lA(x)
29.11 L\{x)
29.12 b\(x)
29.13 L\(x)
29.14 bt(x)
- -6
4cc3 - 48ж2 + 144ж - 96
12ж2 - 96a; + 144
24cc - 96
24
GENERATING FUNCTION FOR
29.15
n=m n-
155

156 ASSOCIATED LAGUERRE POLYNOMIALS
29.18
RECURRENCE FORMULAS
29.16 ^ n±±Ln+i(x) + {x-\ m~2n-l)Ln(x) + ФLn-ф) = 0
и + 1
29.17 ^{L"{*» = L«+1H
29.19 xj~{Ln(x)} = (x-m)Ln(x) + (m-n- 1) bZ~\x)
ORTHOGONALITY
~ со
29 20 I r™e~*Tm(r)Lm(x) dr - 0
29.21
?xme-*{L™(x)}2 dx =
- ..
(w!K
(n — m)!
ORTHOGONAL SERIES
29.22 f(x) = AmLm(x)
where
29.23 Ak -
SPECIAL RESULTS
29.24
и! Г и(и-т) _ , . и(и-1)(и-т)(и-т-1)
^"~ - -Ц; ia:"'m Л хп~т 2
1! 2!

30.1
CHEBYSHEV'S DIFFERENTIAL EQUATION
A - x2)y" - xy' + n2y - 0 n = 0,1,2, . ..
CHEBYSHEV POLYNOMIALS OF THE FIRST KIND
Solutions of 30.1 are given by
30.2 Tn(x) = cos (n cos z) = x" -
SPECIAL CHEBYSHEV POLYNOMIALS OF THE FIRST KIND
30.3
30.4
30.5
30.6
T0(x)
Тх(х)
г,<«>
Ых)
= 1
= X
= 2х2
= 4*з
- 1
-Зх
30.7
30.8
30.9
30.10
Т4(х)
г.(«)
г.(.)
Г7(х)
= 8x4-
= 16ж5
= 32сс6-
= 64ж7¦
¦ 8ж2 + 1
- 20x3 +
- 48сс4 +
- 112*5 .
5х
18ж2
30.11
GENERATING FUNCTION FOR Tn(x)
1 - tx
l-2tx
75 = 2 Тп(х) t»
30.12 Tn(-x) =
30.13 Tji) = i
SPECIAL VALUES
30.14 rn(-i) = (-1)»
30.15 Г2„@) = (-1)"
157
30.16 Г2п + 1@) = О

158
CHEBYSHEV POLYNOMIALS
RECURSION FORMULA FOR Tn(x)
30.17
Tn + 1(x) - 2xTn(x) + rn_i(ae) = 0
ORTHOGONALITY
30.18
30.19
Tm(x)Tn{x)
dx = 0 m Ф n
J-i VT^
dx =
it if «. = 0
tt/2 if и = 1, 2, . .
30.20
where
30.21
ORTHOGONAL SERIES
,4 T ^'уЛ -4- A T (w\ -4- A T (v\
2 f1 f(x)Tk(x)
Ak —
2 f1 f(x)Tk
— I —_
vJ-x VT^
dx
CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
30.22
Un(x) =
_ sin {(и + 1) cos x}
sin (cos x)
SPECIAL CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
30.23
30.24
30.25
30.26
U0(x)
Ux(x)
U2(x)
Us(x)
= 1
= 2x
= 4«.-l
= 8x3 — 4x
30.27
30.28
30.29
30.30
и,(х)
U5(x)
U6(x)
U7(x)
= 16a;4 -
= 32x5 _
= 64xe -
= 128x7
- 12x2 + 1
32x3 + 6x
• 80x4 + 24x2 -
- 192x5 + 80x3
GENERATING FUNCTION FOR Un(x)
30.31
1 - 2tx +
— 2 и (х)
n=0

30.32 Un(-x) = (-1)» !/„(*)
30.33 Un(l) - n + 1
CHEBYSHEV POLYNOMIALS
SPECIAL VALUES
30.34 Un(-1) = (-l)"(n+l)
30.35 ?/2и@) = (-1)"
159
30.36 t/2n+1@) = 0
30.37
RECURSION FORMULA FOR Un(x)
- 2xUn(x)
= 0
30.38
30.39
ORTHOGONALITY
-l
dx = 0 то
J
30.40
where
30.41
ORTHOGONAL SERIES
f(x) = Ao U0(x)
A2 U2(x)
Ak - — Г \/l-x2f(x) Uk(x) dx
77 J-i
30.42
30.43
30.44
30.45
RELATIONSHIPS BETWEEN Tn(x) AND Un(x)
ТП(Х) = Un(x) - ![/,_,(!)
A-x^Un-^x) = xTn{x) - Tn+1(x)
30.46
GENERAL SOLUTION OF CHEBYSHEV'S DIFFERENTIAL EQUATION
\a + В sin ж
if n = 1, 2, 3,
if n- 0

31.1
HYPERGEOMETRIC DIFFERENTIAL EQUATION
x(l - x)y" + {e - (a + b + l)x}y' - aby - 0
HYPERGEOMETRIC FUNCTIONS
A solution of 31.1 is given by
319 sv h ^ 14-а-6 , a(a+1NF + 1) „ , a(a+ l)(g + 2NF +1)F + 2) , ,
31.2 F(a,b;c;x) = 1 + j^x + 1. 2 • c(C + 1) ** + 1 • 2 • 3 • c(c + l)(e + 2) *3 +
If a, 6, с are real, then the series converges for —1 < x < 1 provided that с — (a+ 6) > —1.
SPECIAL CASES
31.3
31.4
31.5
31.6
31.7
F(-p,
F(i,l;
lim F
Щ.-
ЩЛ
1;
2;
41
¦?
;1
l;-*) =
;-*) = [1
,n; 1; x/n)
; ^; sin2 x)
; sin2 x) =
A + ae)p
!n(l + x)]/x
= COS X
sec x
31.8
31.9
31.10
31.11
31.12
щ>
Щ,
F(l,
F(n
F(n,
J.-3
2'2
+ 1,
—и;
;ж2) =
;-*2)
;*) =
-и; 1;
i;d-
(sin x)/x
= (tan-1»)/:
1/A - *)
(l-«)/2) =
-*)/2) = Г,
ж
(X)
GENERAL SOLUTION OF THE HYPERGEOMETRIC EQUATION
If c, a — b and с — a — b are all nonintegers, the general solution valid for \x\ < 1 is
31.13 у = AF(a, b;c;x) + Bx^-c F{a-c+1, Ь - e+1; 2-с; ж)
31.14
31.15
31.16
31.17
MISCELLANEOUS PROPERTIES
F(a, 6; c; 1) =
F(c) F(c — a — b)
Г(с — а) Г(с — 6)
~F(a,b;c;x) = —F(a + 1, b + 1; e+1; x)
doc с
F(a,b;c;x) =
F(a, b;c;x) = A-x)c-a~bF(c-а, с-Ь; с; x)
160

DEFINITION OF THE LAPLACE TRANSFORM OF F(t)
32.1
<{F(t)} = Г e-stF(t)dt = /(e)
In general f(s) will exist for s > a where a is some constant. ^ is called the Laplace transform
operator.
DEFINITION OF THE INVERSE LAPLACE TRANSFORM OF /(s)
If J?{F(t)} = f(s), then we say that F(t) = <~1{/(s)} is the inverse Laplace transform of f(s).
1 is called the inverse Laplace transform operator.
COMPLEX INVERSION FORMULA
The inverse Laplace transform of f(s) can be found directly by methods of complex variable theory.
The result is
32.2
F(t) =
estf(s)ds
i rc+"
es« f(s) ds
where с is chosen so that all the singular points of f(s) lie to the left of the line Re {s} = с in the complex
s plane.
161

162
LAPLACE TRANSFORMS
TABLE OF GENERAL PROPERTIES OF LAPLACE TRANSFORMS
32.3
32.4
32.5
32.6
32.7
32.8
32.9
32.10
32.11
32.12
32.13
32.14
32.15
a/i(s) + b/2(s)
f(sta)
f(s-a)
e~as f(s)
sf(s)-F@)
s*f(s) -sF@) -F'@)
s"f(s) - s«-iF@) - s"-2F'@) - ... - f(n-i) (o)
/'(«)
/"(s)
/Cn)(s)
s
м
sn
/(s) g(s)
F(t)
aFx(t) + bF2(t)
aF(at)
e<*F(t)
(F(t-a) t> a
\0 t<a
F'(t)
F"(t)
F<»>(t)
-tF(t)
t2F(t)
(-l)nt»F(t)
j F(u) du
С1 Сг Гг <t-v)n-i
Jo ¦¦¦HFMdun-{ («-I)! ^dM
Г F(m) G(« - u) du
Л

LAPLACE TRANSFORMS
163
32.16
32.17
32.18
32.19
32.20
32.21
32.22
32.23
32.24
/* CO
) /(и) du
1 Гт
l-e-.J-J e suF№du
f№)
s
f(s + 1/s)
s2 + 1
2yW0
/(In s)
s Ins
P(s)
Q(s)
P(s) = polynomial of degree less than n,
Q(s) = (s - ttl)(s - a2) ¦ ¦ ¦ (s - an)
where ax, a2, .. .,an are all distinct.
F(t)
F(t)
t
F(t) F(t + T)
—— С e-u2/4tF{u)du
VirtJ0
1 J0BVui)F(u)du
/•»
tn/2 1 u-~n/2JnBVut)F(u)du
Jo
ft .
1 Jo Bл/м(« - м)) F(m) dM
Jo
Г t»F(u)
Jo T(u+l)du

164
LAPLACE TRANSFORMS
TABLE OF SPECIAL LAPLACE TRANSFORMS
32.25
32.26
32.27
32.28
32.29
32.30
32.31
32.32
32.33
32.34
32.35
32.36
32.37
32.38
1
s
1
s2
Д n-1,2,3,...
1
s — a
-—^-— к = 1,2,3,..,
(s - a)"
Ц- «> 0
(s — a)"
1
s2 + a2
s
s2 + a2
1
(s - 6J + a2
8- Ь
(S - бJ + a2
1
S2-a2
s
s2-a2
1
(s - 6J - a2
F(t)
1
t
(n-l
Г(и)
eot
fn"leaf, 0!-l
(n-l)!'
fn-l eat
Г(и)
sin at
a
cos at
ebt sin at
a
ebt cos at
sinh at
a
cosh at
ebt sinh at
a

LAPLACE TRANSFORMS
165
32.39
32.40
32.41
32.42
32.43
32.44
32.45
32.46
32.47
32.48
32.49
32.50
32.51
32.52
32.53
32.54
32.55
s- b
(s - 6J - a2
(s - a)(s - 6) аФЬ
7 w ГГ аФ b
(s — a)(s — b)
1
(S2 + a2J
S
(«2 + a2J
S2
(S2 + a2J
S3
(s2 + a2J
S2-a2
(S2 + a2J
1
(S2 - №2J
S
(S2 - a2J
S2
(S2 _ a2J
83
(s2 - a2J
s2 + a2
(S2 - №2J
1
(S2 + №2K
s
(S2 + a2K
S2
(«2 + Cl2K
S3
(S2 + a2K
F(t)
ebt cosh af
ebt _ eat
6 - a
6ebt — aeat
b — a
sin at — a.? cos at
2a3
t sin at
2a
sin at + at cos at
la
cos at — ^at sin at
t cos at
at cosh at — sinh at
га?
t sinh at
2a
sinh at + at cosh at
2a
cosh at + ^at sinh at
t cosh at
C — a2t2) sin at — 3at cos at
8a5
* sin at — at2 cos at
8a3
A + аЧ2) sin at — at cos at
8a3
St sin at + at2 cos at
8a

166
LAPLACE TRANSFORMS
32.56
32.57
32.58
32.59
32.60
32.61
32.62
32.63
32.64
32.65
32.66
32.67
32.68
32.69
32.70
32.71
32.72
s4
(s2 + a2K
sS
(s2 + a2K
3s2 - a2
(s2 4 a2K
s3 - 3a2s
(s2 4 a2K
s4 - 6a2s2 4 a4
(s2 4 a2L
s3 — <z2s
(s2 4 a2)*
1
(s2 - a2K
8
(s2 - a2K
82
(S2 - a2K
S3
(s2 - a2K
s4
(g2 _ a2K
85
(s2 - a2K
3s2 4 a2
(S2 _ a2K
s3 4 3a2s
(s2 - a2K
s4 4 6a2s2 4 a4
(S2 _ a2L
s3 4 a2s
(S2 _ a2L
1
s3 4 a3
F(t)
C — аЧ2) sin at 4 5at cos at
8o
(8 — аЧ2) cos at — lat sin at
8
t2 sin at
2a
¦Jt2 cos at
^t3 cos at
t3 sin at
24a
C 4 a2;2) sinh at - 3at cosh at
8a5
at2 cosh at — t sinh at
8a3
at cosh at 4 (a2t2 — 1) sinh at
8a3
3t sinh at 4 at2 cosh at
8a
C 4 a2t2) sinh at 4 5at cosh at
8a
(8 4 аЧ2) cosh at 4 lat sinh at
8
t2 sinh at
2a
|t2 cosh at
it3 cosh at
t3 sinh at
24a
eot/2 Г /г . V^at \/3at J
l^jVSsm 2 cos 2 4 e-satrtj

LAPLACE TRANSFORMS
167
32.73
32.74
32.75
32.76
32.77
32.78
32.79
32.80
32.81
32.82
32.83
32.84
32.85
32.86
32.87
32.88
32.89
/(8)
S
s3 + a3
s2
s3 + a3
1
s3 - a3
s
s3 — a3
s3 — a3
1
s4 + 4a4
s
s4 + 4a4
s4 + 4a4
s3
s4 + 4a4
1
s4- a4
s
s*-a*
s4-^
1
y"s + a + y"s + b
1
sy's + a
1
¦\fs (s — a)
1
ys — a + b
F(t)
gat/2 [ y^at ^ . yUat , ,„,1
-s— i cos—s Ь V3 sin — e iatn >
3a [ 2 2 J
о 2 1 ^ COS ~ \ о Sill q г
o_ 1 V d Sin i-uS „ + g-at/2V.
oa t z &
1/ V3 aA
3 ( e°t + 2e-»*/2 cos v__ 1
—j (sin at cosh at — cos at sinh at)
sin at sinh at
2a2
— (sin at cosh at + cos at sinh at)
cos at cosh at
^—j (sinh at — sin at)
1
¦5—^(cosh at — cos at)
7j- (sinh at + sin at)
¦J(cosh at + cos at)
g —bt — g —at
2F - a) y^ffi
erf Vat
eat erf Vat
¦\fa
eat j-J ъ ebh erfc (bVO I
[yVt J

168
LAPLACE TRANSFORMS
32.90
32.91
32.92
32.93
32.94
32.95
32.96
32.97
32.98
32.99
32.100
32.101
32.102
32.103
32.104
32.105
№
l
7s2 + a2
1
Vs2 + a2
(s - Vs2 - a2)"
Vs2 - a2
ebts- vV + o*)
v>+^
1
(s2 + а2K^2
s
(s2 + a2K'2
s2
(s2 + a2K'2
1
(S2 _ a2K/2
S
(S2 _ a2K/2
S2
(S2 _ a2K/2
1
s(es-l) s(l-e-s)
See also entry 32.165.
1
s(es — r) s(l—re~s)
es - 1 1-е-.
s(es — r) s(l — re~s)
See also entry 32.167.
g-a/s
Vi
jo'
j
F(t) -n, n
where [f
F(t) = r", n
tJ^at)
a
t Jo (at)
r0(at) - atJ
tl^at)
a
tlo(at)
S(<»+1,
Ш
F(t) - 2
k=l
= greatest
< t < w + 1,
cos2V^
v^
« < 6
iM
(.0
% - 0,1,2, ...
integer = t
n- 0,1, 2, ...

LAPLACE TRANSFORMS
169
32.106
32.107
32.108
32.109
32.110
32.111
32.112
32.113
32.114
32.115
32.116
32.117
32.118
32.119
32.120
32.121
/«
e-a/s
s3/2
e-a/s
S
s
e-aJs
Vs(Vs+ b)
, f' + A
ln r~T
In[(s2 + a2)/a2]
2s
ln [(s + a)la]
s
(y + ln s)
s
у = Euler's constant = .5772156...
6S + s
у = Euler's constant = .5772156...
ln s
s
ln2s
s
F(t)
sin 2\/ot
/ \n/2
(-) JnByfa~t)
\a/
a 2/
erf (a/2yfi)
eric(a/2y/i)
\ 2VT/
д — bt g—at
t
Ci (at)
Ei(at)
Ш
2 (cos at — cos bt)
t
- (ln t + y)
у = Euler's constant = .5772156...
у = Euler's constant = .5772156...

170
LAPLACE TRANSFORMS
32.122
32.123
32.124
32.125
32.126
32.127
32.128
32.129
32.130
32.131
32.132
32.133
32.134
32.135
32.136
32.137
32.138
/W
V'(n + 1) - Г(и + 1) In s
gn+l n -" 1
tan л (als)
tan {als)
s
f,a/s
—— erfc (yals )
es a erfc (s/2a)
/*/4a* erfc (s/2a)
s
eas erfc Vas
V~s
eas Ei(as)
- cos asi-— Si (as) 5- — sin as Ci (as)
a I [2 J J
sin as < — — Si (as) у + cos as Ci (as)
I2 J
cos asi— — Si (as) у — sin as Ci (as)
s
sin as 4 — — Si (as) 1 + cos as Ci (as)
I2 J
s
Г "I2
|- Si (as) + Ci2(as)
0
1
e-as
e-os
s
See also entry 32.163.
F(t)
t" Int
sin at
t
Si (at)
g-2^t
V7t
erf (at)
1
y^+a~)
1
t + a
1
«2 + a2
t
4-2 _j_ /»2
tan-1 (t/a)
1 /t2 + a2\
2 4 »2 J
1 A2 + a^\
tll\ a* )
°K(t) = null function
S(t) = delta function
8(t-a)
4i(t - a)

LAPLACE TRANSFORMS
171
32.139
32.140
32.141
32.142
32.143
32.144
32.145
32.146
32.147
32.148
32.149
32.150
32.151
32.152
32.153
32.154
32.155
sinh sx
s sinh sa
sinh sx
s cosh sa
cosh sx
s sinh as
cosh sx
s cosh sa
sinh sx
s2 sinh sa
sinh sx
s2 cosh sa
cosh sx
s2 sinh sa
cosh sx
s2 cosh sa
cosh sx
s3 cosh sa
sinh ^V^
sinh aVs"
cosh xyfs
cosh aVs"
sinh x^fs
т/s cosh a\^s
cosh xyfs
*\fs sinh a\As
sinh xyfs
s sinh aVs"
cosh xyfs
s cosh aVs"
sinh x*\fs
s2 sinh aVlT
cosh x~\fs
s2 cosh aVs
F(i)
X , 2 S (-1)" . ИггХ №«
- + 2 sin cos
a я- и = 1 и a a
4 ^ (-1)" . Bn-l)irx . Bn-l)irt
2* n * sm _ sin
¦a „T^ 2% - 1 2a 2a
« , 2 ^ (-1)" rnrx . v-t
1 > cos sm
а ж n~i n a a
ч , 4 ? (-1)" Bn-l)wx Bn-l)vt
14 2 iT 4 cos S—— cos S— —
г- „=! 2m — 1 2a 2a
xt , 2a ^ (-1)" . ип-а; . итт«
1 5- .> -1—^sin sm
a я-2 n=j и2 а а
, 8a ^, (-1)" . Bn-lWx Bw-l)jrt
ж H 2 2 7S T4i sm 5 cos n— —
я-2 „ = i Bn — IJ 2a 2a
t2 , 2a ? (-1)" шх(л nvt\
„ + , i —5—cos 1 1 cos 1
2a я-2 n'T1 n2 a \ a )
x , 8a ^, (-1)" Bи-1Ь-а; . Bn-l)vt
t + \r* 3iVn-lJC°S 2a Sm 2a
4i2 + x2 a') 16a2 i ()П со3Bи-1O7Хсо3B?г-1М
^(t +x a) ^ ДBи-1K^00 2a C°° 2a
a2 „^ ( X) )lC °Ш a
"¦„ 2 ( l)»-42n l)e-<2»-i)V./4a« CosB№):7^
» n=i ^a
2 2 ( l)«-ie-<2«-i)VW BinBn71)>rae
a n=1 ^Ja
X + 2 2 ( 1)-в--1Л'»1СО8№ГЯ!
Ж , 2 ^ (-1)" „2„2., 2 . П7ГХ
1 2 e n7rt/a sm
a жn=j и a
1+- 2 9(-1)П1в-B--1)'Л,4а.С08B«-1)^
я- „-г1! 2и - 1 2a
-* + 2f 2 (~fd e-«v./-')8in^
a И n = j и3 a
4x^ „,¦>) | f 16«2 V (-1)" e (?.« nVt/VC0SB»-lK*
z(X a) ' * ^ Д Bn-lK" ""° 2a

172
LAPLACE TRANSFORMS
32.156
s J0(iw\fs)
1-2 2
where \lt X2> • • • are the positive roots of
32.157
Jo (i
s2 J0(i
I(x2-a2) + t + 2a2 2
where Xlt\2> ¦ ¦ ¦ are the positive roots of Jo(\) = 0
32.158
as
Triangular wave function
1-
2a
4a
6a
Fig. 32-1
32.159
Square wave function
F(t
1-
)
i a
——
2a
i
i
|3a
i
4a i5a
Fig. 32-2
32.160
a2s2 + я-2
Rectified sine wave function
F(t)
1-
2a
Fig. 32-3
3a
32.161
тгЧ
n2)(l-e~as)
Half rectified sine wave function
1-
2a 3a
Fig. 32-4
32.162
as2 s(l-
Saw tooth wave function
Fig. 32-5

LAPLACE TRANSFORMS
173
F(t)
32.163
See also entry 32.138.
Heaviside's unit function li(t — a)
F(t)
1-
Fig. 32-6
32.164
Pulse function
F(t)
Fig.32-7
32.165
See also entry 32.102.
Step function
F(t)
3-
2-
1
2a
Fig.32-8
—i—
3a
—I
4a
32.166
e~s -I- e-2s
F(t) = и2, n^ t <n+l, n -0,1,2,
4-
3-
2-
1-
1
2 3
Fig.32-9
32.167
s(l-re-s)
See also entry 32.104.
F(t) = rn, n ? t < n + 1, n = 0,1,2,
1 2 3
Fig.32-10
32.168
тгаA + e~gs)
sin (irtla)
0
t> a
1-
Fig. 32-11

FOURIER'S INTEGRAL THEOREM
33.1 f(x) = I {A (a) COS ax + B{a) sin ax} da
•'o
where
1 C"
(A(a) = - I /(x)coSaxdx
i Г"
B(a) = — I /(x) sin ax dx
' — CO
Sufficient conditions under which this theorem holds are:
(i) f(x) and f'{x) are piecewise continuous in every finite interval — L < x < L;
(ii) I |/(a;)|dx converges;
(iii) /(x) is replaced by ?{/(x + 0) + f(x — 0)} if x is a point of discontinuity.
EQUIVALENT FORMS OF FOURIER'S INTEGRAL THEOREM
33.3 f(x) = ^ \ \ f(u) cos a(x-u) du da
a — — x it = — »
33.4 /(x) = J- Г eia* da Г /(м) e-ia" d
= ^Г Г f(u) eb*ix-u) du da
33.5 /(ж) = — I sin «a; da I f(u) sin aw d
where f(x) is an odd function [/(—x) = — /(»)].
33.6 /(x) = — I cos ax da I f(u) cos км d
17 Л Л
where f{x) is an even function [/(—x) = /(x)].
174

FOURIER TRANSFORMS 175
FOURIER TRANSFORMS
The Fourier transform of f(x) is denned as
33.7 f{f(x)} - F(a) =
Then from 33.7 the inverse Fourier transform of F(a) is
33.8 T-4F(a)} = f(x) = -~
We call f(x) and F(a) Fourier transform pairs.
CONVOLUTION THEOREM FOR FOURIER TRANSFORMS
If F(a) = Hf(x)} and G(a) = ТЫх)}, then
33.9 ¦?- I F(a) G(a) eia* da = I f(u) g(x -u)du = f*g
where f*g is called the convolution of / and fir. Thus
33.10 f{f*g) = f{f}f{g}
PARSEVAL'S IDENTITY
If F(a) = f{f(x)}, then
33.11 С \!(Х)\ЫХ = ^J" \F(a)\2da
More generally if F(a) = f{f(x)} and G(a) = f{g(x)}, then
33.12 Г f(x)gjx~)dx = ^- Г F(a)G~W)da
where the bar denotes complex conjugate.
FOURIER SINE TRANSFORMS
The Fourier sine transform of f(x) is defined as
33.13 Fs(a) = fs{f(x)} = Г f(x) sin ax dx
Then from 33.13 the inverse Fourier sine transform of Fs(a) is
33.14 f(x) = f-HFsW = -Г Fs(a) sin ax da

176 FOURIER TRANSFORMS
FOURIER COSINE TRANSFORMS
The Fourier cosine transform of f(x) is defined as
33.15 Fc(a) = fc{f(x)} = Г f(x)cosaxdx
Then from 33.15 the inverse Fourier cosine transform of Fc(a) is
33.16
2 C
f(x) = f-HFc(a)} = ~ I Fc(a) cosaxda
SPECIAL FOURIER TRANSFORM PAIRS
33.17
33.18
33.19
33.20
33.21
33.22
A \x
t° \x
< 6
> 6
1
X2 + 62
X
«2+ 62
F(a)
2 sin ba
a
we~ba
b
6e
inanF(a)
in
d«"
x , A
6 V 6 у

FOURIER TRANSFORMS
177
SPECIAL FOURIER SINE TRANSFORMS
33.23
33.24
33.25
33.26
33.27
33.28
33.29
33.30
33.31
33.32
33.33
33.34
33.35
33.36
[1 0 < x < b
\o x > b
X
X2 + 62
g-bx
xn 1 g bx
хе-ь**
X-n
sin 6a;
X
sin 6a;
cos 6x
X
tan-Чж/Ь)
esc ba;
1
e2x_l
Fc(«)
1 — cos ba
a
77
2
2e
o:
«2+ 62
Г(и) sin (n tan alb)
(a2 + 62)«/2
^ _aV4b
7ra" csc (nir/2) 0 •-' n •-' 2
2Г(п)
1 (a+b\
2Ы\а-ь)
jva/2 a< b
\nb/2 a> b
ГО а < b
J тг/4 a- b
[тг/2 a > 6
26tEnh26
/ \ !
4 C° yT/ 2a

178
FOURIER TRANSFORMS
SPECIAL FOURIER COSINE TRANSFORMS
33.37
33.38
33.39
33.40
33.41
33.42
33.43
33.44
33.45
33.46
33.47
33.48
33.49
33.50
fl 0 < x < 6
to x > 6
1
X2 + 62
g-bl
%n 1 g bx
е~ьх>
Х-
X "
1.,-, 1 1
\ 2 _i_ 2 /
sin 6x
X
sin 6a;2
cos 6x2
sech 6x
cosh(V^x/2)
cosh (\/?r x)
V5
sin 6 a
26
6
«2 + 62
Г(п) cos (n tan a/6)
(a2 + 62)"/2
Hi---
*«*-*«* (nr/2)
2 T(n)
g-ca _ g-ba
Гтг/2 a < b
J tt/4 a - 6
[ 0 a > 6
ПГ/ «2_ _ ^a^X
Д/ 86 у 46 46/
Y °6 у 46 46y
\ sech ^
26 26
p cosh (Vira/2)
-\/|-{cosB6V^) - sinB6\/a)}

INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
34.1
where
= F(k,<p) = I
de
Г
dv
'о Vl — к2 sin2 e
am и is called the amplitude of и and x = sin 0, and where here and below 0 < к < 1.
COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
34.2
К
= F(k,ir/2) = Г"
•'n
0 Vl - *:2 sin2 0
INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND
34.3
Е{к,ф) = I Vl — &2 sir
. -1>2
COMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND
34.4
E = E(k, Я-/2) = J Vl - k2 sin2 e de = J
= if" I
1.3V fc*
2-4/ 3 \2-4«6/ 5
INCOMPLETE ELLIPTIC INTEGRAL OF THE THIRD KIND
34.5 П №.».
0 A + w sin2 (?) Vl - fe2 sin2 в Jo A + w2) V(l -
-Г
179

180
ELLIPTIC FUNCTIONS
34.6
П№,П,7г/2) = Г
COMPLETE ELLIPTIC INTEGRAL OF THE THIRD KIND
de _ Г1 dv
о A + п sin2 в) л/l - к2 sin2 в Л A + nv2) л/A - v2)(l - кЧ2)
LANDEN'S TRANSFORMATION
34.7
This yields
34.8
tan ф =
sin 2<f>1
к + cos 2
or к sin 0 = sin B0! — <
F(k, Ф)
= С
•Л) Л/1 — I
л/1 — fe2 sin2 в,
1
where fcj = 2\/fc/(l + fc). By successive applications, sequences klt k2, ks, . . . and <f>v <j>2, Фз, ¦ ¦ ¦ are obtained
such that к < fcj < k2 < k3 < ¦ • • < 1 where lim kn = 1. It follows that
34.9
where
34.10
- л 7^ J . = J t;
\ fc ^0 л/l - sin2 о \ "
-я Ф
In tan 2 +f
к, =
and Ф = lim
The result is used in the approximate evaluation of F(k, ф).
JACOBI'S ELLIPTIC FUNCTIONS
From 34.1 we define the following elliptic functions.
34.11 x = sin (amm) = snu
34.12 л/l — x2 = cos (amм) — спи
34.13
— л/l ~ k2 sn2 м =
We can also define the inverse functions sn x, en x, dn x and the following
34.14
34.15
34.16
nsM.
пси
ndn
snit
1
enw
1
34.17
34.18
34.19
SCM =
sdw. =
edit =
dnw
snu
en и
sun
dnM
спи
dn и
34.20 cs ь
34.21 dcM =
34.22 dsM =
snw.
en и
dntt
snit
34.23
34.24
34.25
sn (u + v)
en (u + v)
dn (u + v)
ADDITION FORMULAS
snM.cnt>dnt> + cnitsntidmt
1 - k2 sn2 и sn2 v
спи cnt) — snM snt> dnM dnt)
1 — k2 sn2 и sn2 v
dnM dnt> — k2 snu snt> спи спг>
1 - к2 sn2 м sn2 v

ELLIPTIC FUNCTIONS
181
DERIVATIVES
34.26 — sn и = en и dn и
du
34.27 -— en и = — sn и dn и
du
34.28 —— dn и = — к2 sn и en и
du
34.29 -y-scu — dcuncu
du
SERIES EXPANSIONS
34.30
34.31
34.32
snu = и -
16fe4) |!
dn и - 1 - /с2 |у + fc2D + fc2) j^ - Щ16 + 44fc2 + k4) gy
CATALAN'S CONSTANT
34.33
\$
dk = I
./.:
de dk
- fc2 sin2 s
l2 32 52
= .915965594.
Let
34.34
Then
34.35
34.36
34.37
К =
PERIODS OF ELLIPTIC FUNCTIONS
¦nil ,
о Vl — к2 sin2 i
т* i2 do
о Vl — k'2 sin2 <
sn и has periods 4X and 2iK'
en м has periods 4K and 2X + 2Ж'
dn и has periods 2J5T and 4iK'
where k' = vl — k2
IDENTITIES INVOLVING ELLIPTIC FUNCTIONS
34.38 sn2M + сп2м — 1
34.40 dn2м - fc2 en2и = к'2 where к' = VT^
dn 2m + en 2m
34.42 сп2 и =
34.44
1 + dn 2г«
— сп 2м sn и dn м
+ сп 2м
34.39 dn2 и + к2 sn2 м = 1
1 — en 2m
сп и
34.41 Sn2M =
34.43 dn2 м =
34.45
- dn2M
1 + dn 2м
1 - к2 + dn 2м + fc2 en м
1 + dn 2м
к sn м сп м
+ dn2M
dnM

182
ELLIPTIC FUNCTIONS
SPECIAL VALUES
34.46 snO = 0 34.47 cnO = 1 34.48 dnO = 1 34.49 scO = 0 34.50 amO = 0
INTEGRALS
34.51 I sn и du
34.52 I en и du
34.53 Г dn и
34.54 I scu du
du =
-г In (dnи — к спи)
-г cos (dnu)
sin (sn«t)
1
In (dc и + Vl — k2 nc u)
34.55 I cs и du = In (ns и — ds u)
34.56 I cd и du
-r In (nd и + к sd и)
34
du = In (nc и + scu)
-1
.57 I dew
34.58 Г sd и du
34.59 I ds и du — In (ns и — cs u)
34.60 I ns и du = In (ds и — cs u)
34.61 I ncu du
34.62 Г nd и du
sin (fc edit)
1 . /, , scm
:^=rr In ( dc и +
cos (cdw)
34.63
where
34.64
34.65
E =
E' =
Г
LEGENDRE'S RELATION
EK' + E'K - KK' = jt/2
- fe2 sin2 *
/2
К =
К' =
J.

ERROR FUNCTION erf (ж) = -fr i e~u* du
35.1 erf(x) = ~= x
X° X'
35.2 erf (x) - 1 -
1 1-3 l-3'5
Bx2J Bx2K ^
35.3 erf (-ж) = -erf(x), erf @) = 0, erf(») = 1
COMPLEMENTARY ERROR FUNCTION erfc (ж) = 1 - erf (ж) =
VttJx
du
35.4 erfc (x) = 1 --^(x-
3-1! 5*2! 7-3!
1
35.5 erfc (ж) ~ —
35.6 erfc@) = 1, erfc(») = 0
1 . 1'3 1-3-5
Ba;2
EXPONENTIAL INTEGRAL Ei{x) = Г
35,
35
35
35
35
35
35
.7
.8
.9
.10
.11
.12
.13
ЕЫ =
Ei(x) -
Ei(x) ~
Si(x) -
Si(x) ~
Si(-x) =
У
у ¦
0
X
1-1!
ж
2
- lna; + Г
- lna; + (-
^1- —+ -
SINE
X3
! 3-3! '
sin x /l ;
X \X ;
-Si(x), SilO) =
1-е-»
x a;2
•1! 2-2!
+
)
INTEGRAL
Xs X
5-5! 7-
3! , 5!
Xs X5
-- 0, Si(«>)
7
7!
л
=
a;3
3-3!
Si(x) =
+ ...
, COS X 1
* x 1
: jr/2
•Л
z*1 sin и
"Jo и
f\ 2! + 4!
\
du
183

184 MISCELLANEOUS SPECIAL FUNCTIONS
COSINE INTEGRAL Ci(x) - Г
Jx
35.14 Ciix) = -y-lnx+ C^-^
Jo u
35.15 СЦх) = -у -
35.16 Сг(^) ~ ^^fl-3! + 5!_...\sm?/ 2! 4!
ж \x 3 5 у а; \ 2 4
\ x3 я5 у а; \ х2 х4
35.17 Сг(<=°) = О
FRESNEL SINE INTEGRAL S(x) ~ л/- \ sinU2 du
Y 7Г /О
»= ^/ix*
35.18 S(x) =
35.,,
! 7-3!
35.20 S(-x) = -S(«), S@) = 0, S(») = |
FRESNEL COSINE INTEGRAL C(x) = J- Г соаи2 du
35.22
35.23 С(-ж) = -C(x), C@) = 0, C(») = I
Li
RIEMANN ZETA FUNCTION C(x) = Jx + ^x +
35.24 rt.) = -J-JJ^-idu, X>X
35.25 f(l~^) = 21~хтг-хТ(х) cos(ttx/2)$(x) [extension to other values]
35.26 JBfc) = ^j Л = 1,2,3,...

TRIANGLE INEQUALITY
36.1
36.2
Ы - Ы =
Ы + Ы
\-an\ S
CAUCHY-SCHWARZ INEQUALITY
36.3 |в1Ь, + a2b2 +•••+ anbn\
The equality holds if and only if a1/b1 = a2/b2 = • • • = «„/&„.
INEQUALITIES INVOLVING ARITHMETIC, GEOMETRIC AND HARMONIC MEANS
then
36.4
where
36.5
If A, G and H are the arithmetic, geometric and harmonic means of the positive numbers av a2, ..., an,
я g G ё А
36.6
36.7
Я
( +
и У»! a2
an
The equality holds if and only if aj = a2 = "-" — an-
36.8 !«!&! + a2b2 + ¦ • • + anbn\
where
36.9
The equality holds if and only if
duces to 36.3.
HOLDER'S INEQUALITY
Np+
+
= |an|p-V|bn|. For p = q = 2 it re-
re185

186 INEQUALITIES
CHEBYSHEV'S INEQUALITY
If at & a2 & • • • s an and b1 ? Ь2 ? • • • = 6n, then
+ b2 + • • • + K\ ai6i + «2^2 + • • • + anbn
,, , - /«i + ffl2 + + an\ fb\ + b2 + + K\ ^ ai6i + «2^2 + + anbn
36.10 ()(?
or
36.11 («i + a2+ ••• +«„)(&!+ 62+ ••• +ЬП) й и(а1Ь1 + а2Ь2 + ¦ • • + anbn)
MINKOWSKI'S INEQUALITY
If ац, a2, . . .,an, bj, 62, . . . bn are all positive and p > 1, then
36.12 {(а1 + Ь1)р + (а2+Ь2)Р+ ••• +(an+6n)P}i/P S (< + «1+ • • • + О1'" + F
The equality holds if and only if a^&j = a2lb2 = ¦ ¦ ¦ = ajbn.
CAUCHY-SCHWARZ INEQUALITY FOR INTEGRALS
36.13 IJ f(x) g(x) dx
IJ
The equality holds if and only if f(x)/g(x) is a constant.
HOLDER'S INEQUALITY FOR INTEGRALS
\f(x)g(x)\dx ё -М \f(x)\>dxl jj
36.14
where 1/p + 1/q = 1, p > 1, q > 1. If p = q = 2, this reduces to 36.13.
The equality holds if and only if |/(«)|p~V|fi'(a!)| is a constant.
MINKOWSKI'S INEQUALITY FOR INTEGRALS
If v > 1,
[ Cb ~\ i/p ( rb -j i/p г ^ь ч l/p
36.15 JJ |/(*) +jr(aO|p<fcj- S jj l/t*)!'^ jj ^
The equality holds if and only if f(x)/g(x) is a constant.

37.1 cota; = - + 2x\-^—,+ + - +
X
37.2 esc a; =
а; I «2 — ^-2
37.3
sec x =
37.4 tan, =
37.5
_i_ + —:_
37.6 csc2a; = —г
1 , 1
(x + я-)
,
37.7 cotha; = -
37.8 cscha; = - - 2а;
х
37.9 sech a; =
9тг2 + 4а;2
37.10 tanh* = 8х^-^1^+.о| ,,„-¦ !
4x2
187

38.1 sin a;
38.2 cos* = 1-^1-^
4х2
38.3 sinhz = x
x*\ * , x"
38.4 cosh x =
7Г2
See also 16.12, page 102.
where \lt X2, X3, ... are the positive roots of Jo (x) = 0.
38.7 J,(x) = x 1-3 1-Г5 1-72
where \lt X2, X3, ... are the positive roots of Jx (x) = 0.
38.8
cos — cos — cos — cos —
38.9
2 2 4 4 6 6
• — • — • — • — •
13 3 5 5 7
This is called Wallis' product.
188

BINOMIAL DISTRIBUTION
39.1
¦(*) = Д Г )?««"-* Р>0, Q>0,
POISSON DISTRIBUTION
39.2
«•<*) = 2
t<x
x > о
HYPERGEOMETRIC DISTRIBUTION
39.3
•И*) = 2
\n-t
tgl /Г + 8
п
NORMAL DISTRIBUTION
39.4
Ф(х) =
= —^= С e-f
-f2 dt
STUDENT'S t DISTRIBUTION
39.5
Ф(х)
и + 1
2
Г(и72)
,/ZZ Г(п/2) J
dt
39.6
CHI SQUARE DISTRIBUTION
1 /"*
v^^ 2"/2Г(и/2) J
F DISTRIBUTION
39.7
*(*) =
Г(их/2) Г(и2/2) ^ 0
189
Г «"

40
SPECIAL MOMENTS
OF INERTIA
The table below shows the moments of inertia of various rigid bodies of mass M. In all cases it is
assumed the body has uniform [i.e. constant] density.
TYPE OF RIGID BODY
40.1 Thin rod of length a
(a) about axis perpendicular to the rod through the center of
mass,
F) about axis perpendicular to the rod through one end.
40.2 Rectangular parallelepiped with sides a, b, с
(a) about axis parallel to с and through center of face ab,
(b) about axis through center of face be and parallel to c.
40.3 Thin rectangular plate with sides a, b
(a) about axis perpendicular to the plate through center,
F) about axis parallel to side 6 through center.
40.4 Circular cylinder of radius a and height h
(a) about axis of cylinder,
F) about axis through center of mass and perpendicular to
cylindrical axis,
(c) about axis coinciding with diameter at one end.
.- с Hollow circular cylinder of outer radius a,
inner radius b and height h
(a) about axis of cylinder,
F) about axis through center of mass and perpendicular to
cylindrical axis,
(c) about axis coinciding with diameter at one end.
MOMENT OF INERTIA э
JgM(a2 + &2)
JLMDa2 + 62)
JLM(a2 + 62)
T2Ma2
-^M(h? + 3«2)
^МD^2 + За2)
¦|М(а2 + б2)
З^МСЗа2 + ЗЬ2 + щ
JjMCa2 + ЗЬ2 + 4Л2)
190

SPECIAL MOMENTS OF INERTIA
191
40.6 Circular plate of radius a
(a) about axis perpendicular to plate through center,
F) about axis coinciding with a diameter.
._ — Hollow circular plate or ring with outer radius a
and inner radius 6
(a) about axis perpendicular to plane of plate through center,
F) about axis coinciding with a diameter.
40.8 Thin circular ring of radius a
(a) about axis perpendicular to plane of ring through center,
F) about axis coinciding with diameter.
40.9 Sphere of radius a
(a) about axis coinciding with a diameter,
F) about axis tangent to the surface.
40.10 Hollow sphere of outer radius a and inner radius 6
(a) about axis coinciding with a diameter,
F) about axis tangent to the surface.
40.11 Hollow spherical shell of radius a
(a) about axis coinciding with a diameter,
F) about axis tangent to the surface.
40.12 Ellipsoid with semi-axes a,b,c
(a) about axis coinciding with semi-axis c,
F) about axis tangent to surface, parallel to semi-axis с and
at distance a from center.
40.13 Circular cone of radius a and height h
(a) about axis of cone,
F) about axis through vertex and perpendicular to axis,
(c) about axis through center of mass and perpendicular to axis.
40.14 Torus with outer radius a and inner radius b
(a) about axis through center of mass and perpendicular to
plane of torus,
F) about axis through center of mass and in the plane of the
torus.
\Ma?
\Ma?
\M(a? + 62)
\M(a? + б2)
Ma?
\Ma?
\Ma?
\Ma?
Щ(аь - 65)/(a3 - 63)
|M(«5 - Ь5)/(аз _ 63) + Ma?
Ma?
2Ma?
|M(tt2 + 62)
\M Fa2 + 62)
~M(a* + Ah?)
|Щ4а2+Л2)
iMGa2 - 6ab + 3b2)
\M(9a2 - 10a6 + 562)

Length
Area
Volume
Mass
Speed
Density
Force
Energy
Power
Pressure
1 kilometer (km)
1 meter (m)
1 centimeter (cm)
1 millimeter (mm)
1 micron (/j.)
1 millimicron (m/j.)
1 angstrom (A)
= 1000 meters (m)
= 100 centimeters (cm)
= Ю-2 m
= 10-3m
= 10~6 m
= Ю-9 m
= 10-Wm
1 mch (in.)
1 foot (ft)
1 mile (mi)
1 mil
1 centimeter
1 meter
1 kilometer
= 2.540 cm
= 30.48 cm
= 1.609 km
= 10-3 jn.
= 0.3937 in.
= 39.37 in.
= 0.6214 mile
1 square meter (m2) = 10.76 ft2
1 square foot (ft2) = 929 cm2
1 square mile (mi2) = 640 acres
1 acre = 43,560 ft2
1 liter (I) = 1000 cm3 = 1.057 quart (qt) = 61.02 in3 = 0.03532 ft3
1 cubic meter (m3) = 1000 I = 35.32 ft3
1 cubic foot (ft3) = 7.481 U.S. gal = 0.02832 m3 = 28.32 I
1 U.S. gallon (gal) = 231 in3 = 3.785 I; 1 British gallon = 1.201 U.S. gallon = 277.4 in3
1 kilogram (kg) = 2.2046 pounds (lb) = 0.06852 slug; 1 lb - 453.6 gm = 0.03108 slug
1 slug = 32.174 lb = 14.59 kg
1 km/hr = 0.2778 m/sec = 0.6214 mi/hr = 0.9113 ft/sec
1 mi/hr = 1.467 ft/sec - 1.609 km/hr = 0.4470 m/sec
1 gm/cm3 = 103 kg/m3 = 62.43 lb/ft3 - 1.940 slug/ft3
1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft3 = 0.5154 gm/cm3
1 newton (nt) = 105 dynes = 0.1020 kgwt = 0.2248 lbwt
1 pound weight (lbwt) — 4.448 nt = 0.4536 kgwt = 32.17 poundals
1 kilogram weight (kgwt) = 2.205 lbwt = 9.807 nt
1 U.S. short ton = 2000 lbwt; 1 long ton = 2240 lbwt; 1 metric ton = 2205 lbwt
1 joule = lntm = 107 ergs - 0.7376 ft lbwt = 0.2389 cal = 9.481 X 10-* Btu
1 ft lbwt = 1.356 joules = 0.3239 cal = 1.285 X lO Btu
1 calorie (cal) = 4.186 joules - 3.087 ft lbwt = 3.968 X 10~3 Btu
1 Btu (British thermal unit) - 778 ft lbwt = 1055 joules = 0.293 watt hr
1 kilowatt hour (kw hr) = 3.60 X 106 joules = 860.0 kcal = 3413 Btu
1 electron volt (ev) = 1.602 X 109 joule
1 watt = 1 joule/sec — 10? ergs/sec = 0.2389 cal/sec
1 horsepower (hp) = 550 ft lbwt/sec = 33,000 ft lbwt/min = 745.7 watts
1 kilowatt (kw) = 1.341 hp = 737.6 ft lbwt/sec = 0.9483 Btu/sec
1 nt/m2 = 10 dynes/cm2 = 9.869 X 10 atmosphere = 2.089 X 10 lbwt/ft2
1 lbwt/in2 = 6895 nt/m2 = 5.171 cm mercury = 27.68 in. water
1 atmosphere (atm) = 1.013 X 105 nt/m2 = 1.013 X 10» dynes/cm2 = 14.70 lbwt/in2
= 76 cm mercury = 406.8 in. water
192

Part II
TABLES

SAMPLE PROBLEMS
ILLUSTRATING USE OF THE TABLES
COMMON LOGARITHMS
1. Find log 2.36.
We must find the number p such that 10» = 2.36 = N. Since 10° = 1 and 101 = 10, p lies
between 0 and 1 and can be found from the tables of common logarithms on page 202.
Thus to find log 2.36 we glance down the left column headed N until we come to the first two digits,
23. Then we proceed right to the column headed 6. We find the entry 3729. Thus log 2.36 = 0.3729,
i.e. 2.36 = W°-3729.
2. Find (a) log 23.6, F) log 236, (c) log 2360.
From Problem 1, 2.36 = 100-3729. Then multiplying successively by 10 we have
23.6 = 101-3729, 236 — Ю2-3729, 2360 = Ю3-3729
Thus
(a) log 23.6 = 1.3729
(b) log 236 = 2.3729
(c) log 2360 - 3.3729.
The number .3729 obtained from the table is called the mantissa of the logarithm. The number
before the decimal point is called the characteristic. Thus in (b) the characteristic is 2.
The following rule is easily demonstrated.
Rule 1. For a number greater than 1, the characteristic is one less than the number of digits before
the decimal point. For example since 2360 has four digits before the decimal point, the
characteristic is 4 — 1=3.
3. Find (a) log .236, F) log .0236, (c) log .00236.
From Problem 1, 2.36 = 10°-3729. Then dividing successively by 10 we have
.236 = Ю0.3729-1 = Ю9.3729-10 — Ю~-6271
.0236 = Ю0-3729 = 1O«-3729-10 = lO-6271
.00236 = lO0-3729 = Ю7-37290 = Ю-2-627!
Then
(а) log .236 = 9.3729 - 10 - -.6271
(б) log .0236 = 8.3729 - 10 = -1.6271
(c) log .00236 = 7.3729 - 10 = -2.6271.
The number .3729 is the mantissa of the logarithm. The number apart from the mantissa [for
example 9 — 10, 8 — 10 or 7 — 10] is the characteristic.
The following rule is easily demonstrated.
Rule 2. For a positive number less than 1, the characteristic is negative and numerically one more
than the number of zeros immediately following the decimal point. For example since .00236
has two zeros immediately following the decimal point, the characteristic is —3 or 7 — 10.
194

SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 195
4. Verify each of the following logarithms.
(а) log 87.2. Mantissa = .9405, characteristic = 1; then log 87.2 = 1.9405.
(б) log 395,000 = 5.5966.
(c) log .0482. Mantissa = .6830, characteristic = 8-10; then log10 .0482 = 8.6830 - 10.
(d) log .000827 = 6.9175 - 10.
5. Find log 4.638.
Since the number has four digits, we must use interpolation to find the mantissa. The mantissa
of log 4638 is .8 of the way between the mantissas of log 4630 and log 4640.
Mantissa of log 4640 = .6665 Mantissa of log 4.638 = .6656 + (.8)(.0009)
Mantissa of log 4630 = .6656 = .6663 to four digits
Tabular difference = .0009 Then log 4.638 = 0.6663
If desired the proportional parts table on page 202 can be used to give the mantissa directly
F656 + 7).
6. Verify each of the following logarithms.
(а) log 183.2 = 2.2630 B625 + 5)
(б) log 87,640 = 4.9427 (9425 + 2)
(c) log .2548 = 9.4062-10 D048 + 14)
(d) log .009848 = 7.9933 - 10 (9930 + 3)
COMMON ANTILOGARITHMS
7. Find (a) antilog 1.7530, F) antilog G.7530 - 10).
(a) We must find the value of 101-7530. Since the mantissa is .7530 we glance down the left column
headed p in the table on page 205 until we come to the first two digits 75. Then we proceed right
to the column headed 3. We find the entry 5662. Since the characteristic is 1, there are two
digits before the decimal point. Then the required number is 56.62.
(b) As in part (a.) we find the entry 5662 corresponding to the mantissa .7530. Then since the char-
characteristic is 7 —10, the number must have two zeros immediately following the decimal point.
Thus the required number is .005662.
8. Find antilog (9.3842 - 10).
The mantissa .3842 lies between .3840 and .3850 and we must use interpolation. From the table
on page 204 we have
Number corresponding to .3850 = 2427 Given mantissa = .3842
Number corresponding to .3840 = 2421 Next smaller mantissa = .3840
Tabular difference = 6 Difference - .0002
Then 2421 + ^B427 - 2421) = 2422 to four digits, and the required number is 0.2422.
The proportional parts table on page 204 can also be used.
9. Verify each of the following antilogarithm.
(а) antilog 2.6715 = 469.3
(б) antilog 9.6089 - 10 = .4063
(c) antilog 4.2023 = 15,930

196 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
COMPUTATIONS USING LOGARITHMS
10. P = G84268)B°431) ¦ logP = log 784.6 +log .0431 -log 28.23.
log 784.6 = 2.8947
(+) log .0431 = 8.6345 - 10
11.5292-10
(-) log 28.23 = 1.4507
logP = 10.0785-10 = .0785. Then P = 1.198.
Note the exponential significance of the computation, i.e.
04311 no2-8947WlO8-6345-i°1
.ии!) _ y±v )\±v j 1П2.8947 + 8.6345-10-1.4507 — 100785 — 1
23 ~ W145 ~
loo
28.23 ~ W1-45
11. P = E.395)8. logP = 8 log 5.395 = 8@.7320) = 5.8560, and P = 717,800.
12. P = V387.2 = C87.2I'2. logP = i log 387.2 = 1B.5879) = 1.2940 and P = 19.68.
13. P = ^.08317 = (.08317I/5. logP = 1 log .08317 = 1(8.9200-10) = 1D8.9200-50) = 9.7840-10
and P = .6081.
14. P = — 4 ' -. logP = 1 log .003654 + 3 log 18.37 - D log 8.724 + 1 log 743.8)
(8.724L V743.8
Numerator N Denominator D
1 log .003654 = 1G.5628 - 10) 4 log 8.724 = 4@.9407) = 3.7628
= 1A7.5628 - 20) = 8.7814 -10 1 log 743.6 = ^7B.8714) = 0.7178
3 log 18.37 = 3A.2641) = 3.7923 Add: logD = 4.4806
Add: logAT = 12.5737-10
logAT = 12.5737-10 '•
(-) log D = 4.4806
logP = 8.0931-10. Then P = .01239
NATURAL OR NAPIERIAN LOGARITHMS
15. Find (a) In 7.236, (b) In 836.2, (c) In .002548.
(a) Use the table on page 225.
In 7.240 = 1.97962
In 7.230 = 1.97824
Tabular difference = .00138
Then In 7.236 = 1.97824 + J^(.OO138) = 1.97907
In terms of exponentials this means that e1-97907 = 7.236.
(b) As in part (a) we find
In 8.362 = 2.12346 + ^B.12465 - 2.12346) = 2.12370
Then
In 836.2 = In (8.362 X 102) = log 8.362+ 2 In 10 = 2.12370 + 4.60517 = 6.72887
In terms of exponentials this means that e6-72887 = 836.2.
(c) As in part (a) we find
In 2.548 = 0.93216 + ^@.93609-0.93216) = 0.93530
Then
In .002548 = In B.548X10-3) = In 2.548 - 3 In 10 = 0.93530-6.90776 = -5.97246
In terms of exponentials this means that e'5-97246 = .002548.

SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 197
TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)
16. Find (a) sin 74°23', F) cos 35°42\ (c) tan 82°56'.
(a) Refer to the table on page 206.
sin74°30' = .9636
sin74°20' = .9628
Tabular difference = .0008
Then sin 74°23' = .9628 + ^(.0008) = .9630
F) Refer to the table on page 207.
cos35°40' = .8124
cos35°50' = .8107
Tabular difference = .0017
Then cos 35°42' = .8124 - .^(.0017) = .8121
or cos35°42' = .8107 + j%(.0017) = .8121
(c) Refer to the table on page 208.
tan82°60' = tan83°0' = 8.1443
tan82c50' = 7.9530
Tabular difference = .1913
Then tan82°56' = 7.9530 + ^(.1913) = 8.0678
17. Find (a) cot45°16', (b) sec 73°48\ (c) csc28°33'.
(a) Refer to the table on page 209.
cot45°10' = .9942
cot45°20' = .9884
Tabular difference = .0058
Then cot45°16' = .9942 - ^(.0058) - .9907
or cot45°16' = .9884 +-^(.0058) - .9907
F) Refer to the table on page 210.
sec73°50' = 3.592
sec73°40' = 3.556
Tabular difference = .036
Then sec 73°48' = 3.556 + ^(.036) = 3.585
(c) Refer to the table on page 211.
esc 28°30' = 2.096
csc28°40' = 2.085
Tabular difference = .011
Then csc28°33' = 2.096 - ^(.011) = 2.093
or csc28°33' = 2.085 +-^j(.Oll) = 2.093

198 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
INVERSE TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)
18. Find (a) sin (.2143), F) cos (.5412), (c) tan A.1536).
(a) Refer to the table on page 206.
sinl2°30' = .2164
sinl2°20' = .2136
Tabular difference = .0028
oi до oi ос
Since .2143 is . ' = J of the way between .2136 and .2164, the required angle is
.0028
12°20' + 1A0') = 12°22.5'.
F) Refer to the table on page 207.
cos57°10' = .5422
cos57°20' = .5398
Tabular difference = .0024
Then cos (.5412) = 57°20/ - 412 ~ 398 A0') = 57°14.2'
v ' .0024 v '
or cos-1 (.5412) = 57°10' + '5422 ~ 412 A0') = 57°14.2'
.0024
(c) Refer to the table on page 208.
tan49°10' = 1.1571
tan49°0' = 1.1504
Tabular difference = .0067
Then tan-1 A.1536) = 49°0' + 1-1536 ~ i-1504 A0>) = 49043/
.0067
Other inverse trigonometric functions can be obtained similarly.
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS (RADIANS)
19. Find (a) sin (.627), F) cos A.056), (c) tan (.153).
(a) Refer to the table on page 213.
sin (.630) = .58914
sin (.620) = .58104
Tabular difference = .00810
Then sin (.627) = .58104 + JL(.00810) = .58671
(b) Refer to the table on page 214.
cos A.050) = .49757
cos A.060) = .48887
Tabular difference = .00870
Then cos A.056) = .49757 - ^(.00870) = .49235
or cos A.056) = .48887 + -^(.00870) = .49235
(c) Refer to the table on page 212.
tan (.160) = .16138
tan (.150) = .15114
Tabular difference = .01024
Then tan (.153) = .15114 + ^(.01024) = .15421
Similarly other trigonometric functions are obtained.

SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 199
20. Find sin (-512,) in radians.
Refer to the table on page 213.
sin (.540) = .51414
sin (.530) = .50553
Tabular difference = .00861
Then sin'1 (.512) - .530 + 12 ~ -50553 (,qi) = .5375 radians
Similarly the other inverse trigonometric functions are obtained.
COMMON LOGARITHMS OF TRIGONOMETRIC FUNCTIONS
21. Find (a) log sin 63°17', F) log cos 48°44'.
(a) Refer to the table on page 217.
log sin 63°20' = 9.9512 - 10
Iogsin63°10' = 9.9505 - 10
Tabular difference = .0007
Then log sin 63°17' = 9.9505 - 10 + J6(.O007) = 9.9510 - 10
(b) Refer to the table on page 219.
log cos48°40' = 9.8198 - 10
log cos48°50' = 9.8184 - 10
Tabular difference = .0014
Then Iogcos48°44' = 9.8198 - 10 - y^(.0014) - 9.8192-10
or Iogcos48°44' = 9.8184 - 10 + ^(.0014) = 9.8192-10
Similarly we can find logarithms of other trigonometric functions. Note that log sec x = —log cos x,
log cot x = —log tan x, log esc x = —log sin x.
22. If log tan ж = 9.6845-10, find ж.
Refer to the table on page 220.
log tan25°50' = 9.6850 - 10
Iogtan25°40' = 9.6817-10
Tabular difference = .0033
Then x = 25°40' + 9-6845 ~ 9.6817 AQ,) _ 25°48.5'
.0033
CONVERSION OF DEGREES, MINUTES AND SECONDS TO RADIANS
23. Find 75° 28' 47" in radians.
Refer to the table on page 223.
70° = 1.221730 radians
5° = .087267
20' = .005818
8' = .002327
40" = .000194
7" = .000034
Adding, 75° 28' 47" = 1.317370 radians

200 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
CONVERSION OF RADIANS TO DEGREES, MINUTES AND SECONDS
24. Find 2.547 radians in degrees, minutes and seconds.
Refer to the table on page 222.
2 radians = 114° 35' 29.6"
.5 = 28° 38' 52.4"
.04 = 2° 17' 30.6"
.007 = 0° 24' 3.9"
Adding, 2.547 radians = 144° 114' 116.5" = 145° 55' 56.5"
CONVERSION OF RADIANS TO FRACTIONS OF A DEGREE
25. Find 1.382 radians in terms of degrees.
Refer to the table on page 222.
1 radian = 57.2958°
.3 = 17.1887°
.08 = 4.5837°
.002 = .1146°
Adding, 1.382 radians = 79.1828°
EXPONENTIAL AND HYPERBOLIC FUNCTIONS
26. Find (a) e5-2*, (b) e"-158.
(a) Refer to the table on page 226.
е5.зо = 200.34
e5.2o = 181.27
Tabular difference = 19.07
Then e5-24 - 181.27 + -^A9.07) = 188.90
F) Refer to the table on page 227.
e-.i5o - .86071
e-.ieo = .85214
Tabular difference = .00857
Then e-158 = .86071 - ^(.00857) = .85385
or e--iss = .85214 + yU.00857) = .85385
27. Find (a) sinh D.846), F) sech (.163).
(a) Refer to the table on page 229.
sinh D.850) = 63.866
sinh D.840) = 63.231
Tabular difference = .635
Then sinh D.846) = 63.231 + ^(.635) = 63.612
(b) Refer to the table on page 230.
cosh (.170) = 1.0145
cosh (.160) = 1.0128
Tabular difference = .0017
Then cosh (.163) = 1.0128 + ^(.0017) = 1.0133
and so sech (.163) = L—- = , * = .98687
cosh (.163) 1.0133

SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 201
28. Find tanh (.71423).
Refer to the table on page 232.
tanh (.900) = .71630
tanh (.890) = .71139
Tabular difference = .00491
Then tanh-i (-71423) = .890 + -71423 ~ -71139 (i0) = .8958
INTEREST AND ANNUITIES
29. A man deposits $2800 in a bank which pays 57c compounded quarterly. What will the deposit
amount to in 8 years?
There are n — 8 • 4 = 32 payment periods at interest rate r = .05/4 = .0125 per period. Then the
amount is
A - $2800A + .0125K2 = $2800A.4881) = $4166.68
using the table on page 240.
30. A man expects to receive $12,000 in 10 years. How much is that money worth now, considering interest
at 6% compounded semi-annually?
We are asked for the present value P which will amount to A = $12,000 in 10 years. Since there
are n = 10 • 2 = 20 payment periods at interest rate r = .06/2 = .03 per period, the present value is
P = $12,000A+ .03)-2<> = $12,000(.55368) = $6644.16
using the table on page 241.
31. An investor has an annuity in which a payment of $500 is made at the end of each year. If interest
is 4% compounded annually, what is the amount of the annuity after 20 years?
Here r = .04, n = 20 and the amount is [see table on page 242],
$500 Г1 + •04J° ~ 11 = $500B9.7781) = $14,889.05
L -04 J
32. What is the present value of an annuity of $120 at the end of each 3 months for 12 years at 6%
compounded quarterly?
Here n = 4 • 12 = 48 payment periods, r = .06/4 = .015 and the present value is
$120 Г1 ~ <1-015)^! | = $120C4.0426) = $4085.11
.015
using the table on page 243.

TABLE
1
FOUR PLACE COMMON LOGARITHMS
log10 N or log iV
N
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
N
0
0000
0414
0792
1139
1461
1761
2041
2304
2553
2788
3010
3222
3424
3617
3802
3979
4150
4314
4472
4624
4771
4914
5051
5185
5315
5441
5563
5682
5798
5911
6021
6128
6232
6335
6435
6532
6628
6721
6812
6902
6990
7076
7160
7243
7324
0
1
0043
0453
0828
1173
1492
1790
2068
2330
2577
2810
3032
3243
3444
3636
3820
3997
4166
4330
4487
4639
4786
4928
5065
5198
5328
5453
5575
5694
5809
5922
6031
6138
6243
6345
6444
6542
6637
6730
6821
6911
6998
7084
7168
7251
7332
1
2
0086
0492
0864
1206
1523
1818
2095
2355
2601
2833
3054
3263
3464
3655
3838
4014
4183
4346
4502
4654
4800
4942
5079
5211
5340
5465
5587
5705
5821
5933
6042
6149
6253
6355
6454
6551
6646
6739
6830
6920
7007
7093
7177
7259
7340
2
3
0128
0531
0899
1239
1553
1847
2122
2380
2625
2856
3075
3284
3483
3674
3856
4031
4200
4362
4518
4669
4814
4955
5092
5224
5353
5478
5599
5717
5832
5944
6053
6160
6263
6365
6464
6561
6656
6749
6839
6928
7016
7101
7185
7267
7348
3
4
0170
0569
0934
1271
1584
1875
2148
2405
2648
2878
3096
3304
3502
3692
3874
4048
4216
4378
4533
4683
4829
4969
5105
5237
5366
5490
5611
5729
5843
5955
6064
6170
6274
6375
6474
6571
6665
6758
6848
6937
7024
7110
7193
7275
7356
4
5
0212
0607
0969
1303
1614
1903
2175
2430
2672
2900
3118
3324
3522
3711
3892
4065
4232
4393
4548
4698
4843
4983
5119
5250
5378
5502
5623
5740
5855
5966
6075
6180
6284
6385
6484
6580
6675
- 6767
6857
6946
7033
7118
7202
7284
7364
5
6
0253
0645
1004
1335
1644
1931
2201
2455
2695
2923
3139
3345
3541
3729
3909
4082
4249
4409
4564
4713
4857
4997
5132
5263
5391
5514
5635
5752
5866
5977
6085
6191
6294
6395
6493
6590
6684
6776
6866
6955
7042
7126
7210
7292
7372
6
7
0294
0682
1038
1367
1673
1959
2227
2480
2718
2945
3160
3365
3560
3747
3927
4099
4265
4425
4579
4728
4871
5011
5145
5276
5403
5527
5647
5763
5877
5988
6096
6201
6304
6405
6503
6599
6693
6785
6875
6964
7050
7135
7218
7300
7380
7
8
0334
0719
1072
1399
1703
1987
2253
2504
2742
2967
3181
3385
3579
3766
3945
4116
4281
4440
4594
4742
4886
5024
5159
5289
5416
5539
5658
5775
5888
5999
6107
6212
6314
6415
6513
6609
6702
6794
6884
6972
7059
7143
7226
7308
7388
8
9
0374
0755
1106
1430
1732
2014
2279
2529
2765
2989
3201
3404
3598
3784
3962
4133
4298
4456
4609
4757
4900
5038
5172
5302
5428
5551
5670
5786
5899
6010
6117
6222
6325
6425
6522
6618
6712
6803
6893
6981
7067
7152
7235
7316
7396
9
1
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
8
8
7
6
6
6
5
5
5
4
4
4
4
4
4
3
3
3
3,
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Proportional Parts
3 4 5 6 7
12
11
10
10
9
8
8
7
7
7
6
6
6
6
5
5
5
5
5
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
3
17
15
14
13
12
11
11
10
9
9
8
8
8
7
7
7
7
6
6
6
6
6
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
21
19
17
16
15
14
13
12
12
11
11
10
10
9
9
9
8
8
8
7
7
7
7
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
5
25
23
21
19
18
17
16
15
14
13
13
12
12
11
11
10
10
9
9
9
9
8
8
8
8
7
7
7
7
7
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
6
29
26
24
23
21
20
18
17
16
16
15
14
14
13
12
12
11
11
11
10
10
10
9
9
9
9
8
8
8
8
8
7
7
7
7
7
7
6
6
6
6
6
6
6
6
7
8
33
30
28
26
24
22
21
20
19
18
17
16
15
15
14
14
13
13
12
12
11
11
11
10
10
10
10
9
9
9
9
8
8
8
8
8
7
7
7
7
7
7
7
6
6
8
9
37
34
31
29
27
25
24
22
21
20
19
18
17
17
16
15
15
14
14
13
13
12
12
12
11
11
11
10
10
10
10
9
9
9
9
9
8
8
8
8
8
8
7
7
7
9
202

Table T
(continued)
FOUR PLACE COMMON LOGARITHMS
logiV
N
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
N
0
7404
7482
7559
7634
7709
7782
7853
7924
7993
8062
8129
8195
8261
8325
8388
8451
8513
8573
8633
8692
8751
8808
8865
8921
8976
9031
9085
9138
9191
9243
9294
9345
9395
9445
9494
9542
9590
9638
9685
9731
9777
9823
9868
9912
9956
0
1
7412
7490
7566
7642
7716
7789
7860
7931
8000
8069
8136
8202
8267
8331
8395
8457
8519
8579
8639
8698
8756
8814
8871
8927
8982
9036
9090
9143
9196
9248
9299
9350
9400
9450
9499
9547
9595
9643
9689
9736
9782
9827
9872
9917
9961
1
2
7419
7497
7574
7649
7723
7796
7868
7938
8007
8075
8142
8209
8274
8338
8401
8463
8525
8585
8645
8704
8762
8820
8876
8932
8987
9042
9096
9149
9201
9253
9304
9355
9405
9455
9504
9552
9600
9647
9694
9741
9786
9832
9877
9921
9965
2
3
7427
7505
7582
7657
7731
7803
7875
7945
8014
8082
8149
8215
8280
8344
8407
8470
8531
8591
8651
8710
8768
8825
8882
8938
8993
9047
9101
9154
9206
9258
9309
9360
9410
9460
9509
9557
9605
9652
9699
9745
9791
9836
9881
9926
9969
3
4
7435
7513
7589
7664
7738
7810
7882
7952
8021
8089
8156
8222
8287
8351
8414
8476
8537
8597
8657
8716
8774
8831
8887
8943
8998
9053
9106
9159
9212
9263
9315
9365
9415
9465
9513
9562
9609
9657
9703
9750
9795
9841
9886
9930
9974
4
5
7443
7520
7597
7672
7745
7818
7889
7959
8028
8096
8162
8228
8293
8357
8420
8482
8543
8603
8663
8722
8779
8837
8893
8949
9004
9058
9112
9165
9217
9269
9320
9370
9420
9469
9518
9566
9614
9661
9708
9754
9800
9845
9890
9934
9978
5
6
7451
7528
7604
7679
7752
7825
7896
7966
8035
8102
8169
8235
8299
8363
8426
8488
8549
8609
8669
8727
8785
8842
8899
8954
9009
9063
9117
9170
9222
9274
9325
9375
9425
9474
9523
9571
9619
9666
9713
9759
9805
9850
9894
9939
9983
6
7
7459
7536
7612
7686
7760
7832
7903
7973
8041
8109
8176
8241
8306
8370
8432
8494
8555
8615
8675
8733
8791
8848
8904
8960
9015
9069
9122
9175
9227
9279
9330
9380
9430
9479
9528
9576
9624
9671
9717
9763
9809
9854
9899
9943
9987
7
8
7466
7543
7619
7694
7767
7839
7910
7980
8048
8116
8182
8248
8312
8376
8439
8500
8561
8621
8681
8739
8797
8854
8910
8965
9020
9074
9128
9180
9232
9284
9335
9385
9435
9484
9533
9581
9628
9675
9722
9768
9814
9859
9903
9948
9991
8
9
7474
7551
7627
7701
7774
7846
7917
7987
8055
8122
8189
8254
8319
8382
8445
8506
8567
8627
8686
8745
8802
8859
8915
8971
9025
9079
9133
9186
9238
9289
9340
9390
9440
9489
9538
9586
9633
9680
9727
9773
9818
9863
9908
9952
9996
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
Proportional Parts
3 4 5 6 7
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
6
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
7
8
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
8
9
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
9
203

TABLE
2
FOUR PLACE COMMON ANTILOGARITHMS
10' or antilog p
V
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.33
.34
.35
.36
.37
.38
.39
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
V
0
1000
1023
1047
1072
1096
1122
1148
1175
1202
1230
1259
1288
1318
1349
1380
1413
1445
1479
1514
1549
1585
1622
1660
1698
1738
1778
1820
1862
1905
1950
1995
2042
2089
2138
2188
2239
2291
2344
2399
2455
2512
2570
2630
2692
2754
2818
2884
2951
3020
3090
0
1
1002
1026
1050
1074
1099
1125
1151
1178
1205
1233
1262
1291
1321
1352
1384
1416
1449
1483
1517
1552
1589
1626
1663
1702
1742
1782
1824
1866
1910
1954
2000
2046
2094
2143
2193
2244
2296
2350
2404
2460
2518
2576
2636
2698
2761
2825
2891
2958
3027
3097
1
2
1005
1028
1052
1076
1102
1127
1153
1180
1208
1236
1265
1294
1324
1355
1387
1419
1452
1486
1521
1556
1592
1629
1667
1706
1746
1786
1828
1871
1914
1959
2004
2051
2099
2148
2198
2249
2301
2355
2410
2466
2523
2582
2642
2704
2767
2831
2897
2965
3034
3105
2
3
1007
1030
1054
1079
1104
1130
1156
1183
1211
1239
1268
1297
1327
1358
1390
1422
1455
1489
1524
1560
1596
1633
1671
1710
1750
1791
1832
1875
1919
1963
2009
2056
2104
2153
2203
2254
2307
2360
2415
2472
2529
2588
2649
2710
2773
2838
2904
2972
3041
3112
3
4
1009
1033
1057
1081
1107
1132
1159
1186
1213
1242
1271
1300
1330
1361
1393
1426
1459
1493
1528
1563
1600
1637
1675
1714
1754
1795
1837
1879
1923
1968
2014
2061
2109
2158
2208
2259
2312
2366
2421
2477
2535
2594
2655
2716
2780
2844
2911
2979
3048
3119
4
5
1012
1035
1059
1084
1109
1135
1161
1189
1216
1245
1274
1303
1334
1365
1396
1429
1462
1496
1531
1567
1603
1641
1679
1718
1758
1799
1841
1884
1928
1972
2018
2065
2113
2163
2213
2265
2317
2371
2427
2483
2541
2600
2661
2723
2786
2851
2917
2985
3055
3126
5
6
1014
1038
1062
1086
1112
1138
1164
1191
1219
1247
1276
1306
1337
1368
1400
1432
1466
1500
1535
1570
1607
1644
1683
1722
1762
1803
1845
1888
1932
1977
2023
2070
2118
2168
2218
2270
2323
2377
2432
2489
2547
2606
2667
2729
2793
2858
2924
2992
3062
3133
6
7
1016
1040
1064
1089
1114
1140
1167
1194
1222
1250
1279
1309
1340
1371
1403
1435
1469
1503
1538
1574
1611
1648
1687
1726
1766
1807
1849
1892
1936
1982
2028
2075
2123
2173
2223
2275
2328
2382
2438
2495
2553
2612
2673
2735
2799
2864
2931
2999
3069
3141
7
8
1019
1042
1067
1091
1117
1143
1169
1197
1225
1253
1282
1312
1343
1374
1406
1439
1472
1507
1542
1578
1614
1652
1690
1730
1770
1811
1854
1897
1941
1986
2032
2080
2128
2178
2228
2280
2333
2388
2443
2500
2559
2618
2679
2742
2805
2871
2938
3006
3076
3148
8
9
1021
1045
1069
1094
1119
1146
1172
1199
1227
1256
1285
1315
1346
1377
1409
1442
1476
1510
1545
1581
1618
1656
1694
1734
1774
1816
1858
1901
1945
1991
2037
2084
2133
2183
2234
2286
2339
2393
2449
2506
2564
2624
2685
2748
2812
2877
2944
3013
3083
3155
9
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
Proportional Parts
3 4 5 6 7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
5
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
6
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
7
8
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
8
9
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
Э
204

Table 2
(continued)
FOUR PLACE COMMON ANTILOGARITHMS
10 or antilog p
V
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
V
0
3162
3236
3311
3388
3467
3548
3631
3715
3802
3890
3981
4074
4169
4266
4365
4467
4571
4677
4786
4898
5012
5129
5248
5370
5495
5623
5754
5888
6026
6166
6310
6457
6607
6761
6918
7079
7244
7413
7586
7762
7943
8128
8318
8511
8710
8913
9120
9333
9550
9772
0
1
3170
3243
3319
3396
3475
3556
3639
3724
3811
3899
3990
4083
4178
4276
4375
4477
4581
4688
4797
4909
5023
5140
5260
5383
5508
5636
5768
5902
6039
6180
6324
6471
6622
6776
6934
7096
7261
7430
7603
7780
7962
8147
8337
8531
8730
8933
9141
9354
9572
9795
1
2
3177
3251
3327
3404
3483
3565
3648
3733
3819
3908
3999
4093
4188
4285
4385
4487
4592
4699
4808
4920
5035
5152
5272
5395
5521
5649
5781
5916
6053
6194
6339
6486
6637
6792
6950
7112
7278
7447
7621
7798
7980
8166
8356
8551
8750
8954
9162
9376
9594
9817
2
3
3184
3258
3334
3412
3491
3573
3656
3741
3828
3917
4009
4102
4198
4295
4395
4498
4603
4710
4819
4932
5047
5164
5284
5408
5534
5662
5794
5929
6067
6209
6353
6501
6653
6808
6966
7129
7295
7464
7638
7816
7998
8185
8375
8570
8770
8974
9183
9397
9616
9840
3
4
3192
3266
3342
3420
3499
3581
3664
3750
3837
3926
4018
4111
4207
4305
4406
4508
4613
4721
4831
4943
5058
5176
5297
5420
5546
5675
5808
5943
6081
6223
6368
6516
6668
6823
6982
7145
7311
7482
7656
7834
8017
8204
8395
8590
8790
8995
9204
9419
9638
9863
4
5
3199
3273
3350
3428
3508
3589
3673
3758
3846
3936
4027
4121
4217
4315
4416
4519
4624
4732
4842
4955
5070
5188
5309
5433
5559
5689
5821
5957
6095
6237
6383
6531
6683
6839
6998
7161
7328
7499
7674
7852
8035
8222
8414
8610
8810
9016
9226
9441
9661
9886
5
6
3206
3281
3357
3436
3516
3597
3681
3767
3855
3945
4036
4130
4227
4325
4426
4529
4634
4742
4853
4966
5082
5200
5321
5445
5572
5702
5834
5970
6109
6252
6397
6546
6699
6855
7015
7178
7345
7516
7691
7870
8054
8241
8433
8630
8831
9036
9247
9462
9683
9908
6
7
3214
3289
3365
3443
3524
3606
3690
3776
3864
3954
4046
4140
4236
4335
4436
4539
4645
4753
4864
4977
5093
5212
5333
5458
5585
5715
5848
5984
6124
6266
6412
6561
6714
6871
7031
7194
7362
7534
7709
7889
8072
8260
8453
8650
8851
9057
9268
9484
9705
9931
7
8
3221
3296
3373
3451
3532
3614
3698
3784
3873
3963
4055
4150
4246
4345
4446
4550
4656
4764
4875
4989
5105
5224
5346
5470
5598
5728
5861
5998
6138
6281
6427
6577
6730
6887
7047
7211
7379
7551
7727
7907
8091
8279
8472
8670
8872
9078
9290
9506
9727
9954
8
9
3228
3304
3381
3459
3540
3622
3707
3793
3882
3972
4064
4159
4256
4355
4457
4560
4667
4775
4887
5000
5117
5236
5358
5483
5610
5741
5875
6012
6152
6295
6442
6592
6745
6902
7063
7228
7396
7568
7745
7925
8110
8299
8492
8690
8892
9099
9311
9528
9750
9977
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
2
Proportional Parts
3 4 5 6 7
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
10
10
10
10
11
11
11
11
5
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
10
10
10
10
11
11
11
11
12
12
12
12
13
13
13
14
6
5
5
5
6
6
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
10
10
10
10
11
11
11
11
12
12
12
12
13
13
13
14
14
14
15
15
15
16
16
7
8
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
10
10
10
10
10
11
11
11
11
12
12
12
13
13
13
13
14
14
14
15
15
15
16
16
17
17
17
18
18
8
9
7
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
10
10
10
10
11
11
11
11
12
12
12
12
13
13
13
14
14
14
15
15
15
16
16
16
17
17
17
18
18
19
19
20
20
20
9
205

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
.0000
.0175
.0349
.0523
.0698
.0872
.1045
.1219
.1392
.1564
.1736
.1908
.2079
.2250
.2419
.2588
.2756
.2924
.3090
.3256
.3420
.3584
.3746
.3907
.4067
.4226
.4384
.4540
.4695
.4848
.5000
.5150
.5299
.5446
.5592
.5736
.5878
.6018
.6157
.6293
.6428
.6561
.6691
.6820
.6947
.7071
10'
.0029
.0204
.0378
.0552
.0727
.0901
.1074
.1248
.1421
.1593
.1765
.1937
.2108
.2278
.2447
.2616
.2784
.2952
.3118
.3283
.3448
.3611
.3773
.3934
.4094
.4253
.4410
.4566
.4720
.4874
.5025
.5175
.5324
.5471
.5616
.5760
.5901
.6041
.6180
.6316
.6450
.6583
.6713
.6841
.6967
.7092
20'
.0058
.0233
.0407
.0581
.0756
.0929
.1103
.1276
.1449
.1622
.1794
.1965
.2136
.2306
.2476
.2644
.2812
.2979
.3145
.3311
.3475
.3638
.3800
.3961
.4120
.4279
.4436
.4592
.4746
.4899
.5050
.5200
.5348
.5495
.5640
.5783
.5925
.6065
.6202
.6338
.6472
.6604
.6734
.6862
.6988
.7112
30'
.0087
.0262
.0436
.0610
.0785
.0958
.1132
.1305
.1478
.1650
.1822
.1994
.2164
.2334
.2504
.2672
.2840
.3007
.3173
.3338
.3502
.3665
.3827
.3987
.4147
.4305
.4462
.4617
.4772
.4924
.5075
.5225
.5373
.5519
.5664
.5807
.5948
.6088
.6225
.6361
.6494
.6626
.6756
.6884
.7009
.7133
40'
.0116
.0291
.0465
.0640
.0814
.0987
.1161
.1334
.1507
.1679
.1851
.2022
.2193
.2363
.2532
.2700
.2868
.3035
.3201
.3365
.3529
.3692
.3854
.4014
.4173
.4331
.4488
.4643
.4797
.4950
.5100
.5250
.5398
.5544
.5688
.5831
.5972
.6111
.6248
.6383
.6517
.6648
.6777
.6905
.7030
.7153
50'
.0145
.0320
.0494
.0669
.0843
.1016
.1190
.1363
.1536
.1708
.1880
.2051
.2221
.2391
.2560
.2728
.2896
.3062
.3228
.3393
.3557
.3719
.3881
.4041
.4200
.4358
.4514
.4669
.4823
.4975
.5125
.5275
.5422
.5568
.5712
.5854
.5995
.6134
.6271
.6406
.6539
.6670
.6799
.6926
.7050
.7173
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
.7071
.7193
.7314
.7431
.7547
.7660
.7771
.7880
.7986
.8090
.8192
.8290
.8387
.8480
.8572
.8660
.8746
.8829
.8910
.8988
.9063
.9135
.9205
.9272
.9336
.9397
.9455
.9511
.9563
.9613
.9659
.9703
.9744
.9781
.9816
.9848
.9877
.9903
.9925
.9945
.9962
.9976
.9986
.9994
.9998
1.0000
10'
.7092
.7214
.733,4
.7451
.7566
.7679
.7790
.7898
.8004
.8107
.8208
.8307
.8403
.8496
.8587
.8675
.8760
.8843
.8923
.9001
.9075
.9147
.9216
.9283
.9346
.9407
.9465
.9520
.9572
.9621
.9667
.9710
.9750
.9787
.9822
.9853
.9881
.9907
.9929
.9948
.9964
.9978
.9988
.9995
.9999
20'
.7112
.7234
.7353
.7470
.7585
.7698
.7808
.7916
.8021
.8124
.8225
.8323
.8418
.8511
.8601
.8689
.8774
.8857
.8936
.9013
.9088
.9159
.9228
.9293
.9356
.9417
.9474
.9528
.9580
.9628
.9674
.9717
.9757
.9793
.9827
.9858
.9886
.9911
.9932
.9951
.9967
.9980
.9989
.9996
.9999
30'
.7133
.7254
.7373
.7490
.7604
.7716
.7826
.7934
.8039
.8141
.8241
.8339
.8434
.8526
.8616
.8704
.8788
.8870
.8949
.9026
.9100
.9171
.9239
.9304
.9367
.9426
.9483
.9537
.9588
.9636
.9681
.9724
.9763
.9799
.9833
.9863
.9890
.9914
.9936
.9954
.9969
.9981
.9990
.9997
1.0000
40'
.7153
.7274
.7392
.7509
.7623
.7735
.7844
.7951
.8056
.8158
.8258
.8355
.8450
.8542
.8631
.8718
.8802
.8884
.8962
.9038
.9112
.9182
.9250
.9315
.9377
.9436
.9492
.9546
.9596
.9644
.9689
.9730
.9769
.9805
.9838
.9868
.9894
.9918
.9939
.9957
.9971
.9983
.9992
.9997
1.0000
50'
.7173
.7294
.7412
.7528
.7642
.7753
.7862
.7969
.8073
.8175
.8274
.8371
.8465
.8557
.8646
.8732
.8816
.8897
.8975
.9051
.9124
.9194
.9261
.9325
.9387
.9446
.9502
.9555
.9605
.9652
.9696
.9737
.9775
.9811
.9843
.9872
.9899
.9922
.9942
.9959
.9974
.9985
.9993
.9998
1.0000
206

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
1.0000
.9998
.9994
.9986
.9976
.9962
.9945
.9925
.9903
.9877
.9848
.9816
.9781
.9744
.9703
.9659
.9613
.9563
.9511
.9455
.9397
.9336
.9272
.9205
.9135
.9063
.8988
.8910
.8829
.8746
.8660
.8572
.8480
.8387
.8290
.8192
.8090
.7986
.7880
.7771
.7660
.7547
.7431
.7314
.7193
.7071
10'
1.0000
.9998
.9993
.9985
.9974
.9959
.9942
.9922
.9899
.9872
.9843
.9811
.9775
.9737
.9696
.9652
.9605
.9555
.9502
.9446
.9387
.9325
.9261
.9194
.9124
.9051
.8975
.8897
.8816
.8732
.8646
.8557
.8465
.8371
.8274
.8175
.8073
.7969
.7862
.7753
.7642
.7528
.7412
.7294
.7173
.7050
20'
1.0000
.9997
.9992
.9983
.9971
.9957
.9939
.9918
.9894
.9868
.9838
.9805
.9769
.9730
.9689
.9644
.9596
.9546
.9492
.9436
.9377
.9315
.9250
.9182
.9112
.9038
.8962
.8884
.8802
.8718
.8631
.8542
.8450
.8355
.8258
.8158
.8056
.7951
.7844
.7735
.7623
.7509
.7392
.7274
.7153
.7030
30'
1.0000
.9997
.9990
.9981
.9969
.9954
.9936
.9914
.9890
.9863
.9833
.9799
.9763
.9724
.9681
.9636
.9588
.9537
.9483
.9426
.9367
.9304
.9239
.9171
.9100
.9026
.8949
.8870
.8788
.8704
.8616
.8526
.8434
.8339
.8241
.8141
.8039
.7934
.7826
.7716
.7604
.7490
.7373
.7254
.7133
.7009
40'
.9999
.9996
.9989
.9980
.9967
.9951
.9932
.9911
.9886
.9858
.9827
.9793
.9757
.9717
.9674
.9628
.9580
.9528
.9474
.9417
.9356
.9293
.9228
.9159
.9088
.9013
.8936
.8857
.8774
.8689
.8601
.8511
.8418
.8323
.8225
.8124
.8021
.7916
.7808
.7698
.7585
.7470
.7353
.7234
.7112
.6988
50'
.9999
.9995
.9988
.9978
.9964
.9948
.9929
.9907
.9881
.9853
.9822
.9787
.9750
.9710
.9667
.9621
.9572
.9520
.9465
.9407
.9346
.9283
.9216
.9147
.9075
.9001
.8923
.8843
.8760
.8675
.8587
.8496
.8403
.8307
.8208
.8107
.8004
.7898
.7790
.7679
.7566
.7451
.7333
.7214
.7092
.6967
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
.7071
.6947
.6820
.6691
.6561
.6428
.6293
.6157
.6018
.5878
.5736
.5592
.5446
.5299
.5150
.5000
.4848
.4695
.4540
.4384
.4226
.4067
.3907
.3746
.3584
.3420
.3256
.3090
.2924
.2756
.2588
.2419
.2250
.2079
.1908
.1736
.1564
.1392
.1219
.1045
.0872
.0698
.0523
.0349
.0175
.0000
10'
.7050
.6926
.6799
.6670
.6539
.6406
.6271
.6134
.5995
.5854
.5712
.5568
.5422
.5275
.5125
.4975
.4823
.4669
.4514
.4358
.4200
.4041
.3881
.3719
.3557
.3393
.3228
.3062
.2896
.2728
.2560
.2391
.2221
.2051
.1880
.1708
.1536
.1363
.1190
.1016
.0843
.0669
.0494
.0320
.0145
20'
.7030
.6905
.6777
.6648
.6517
.6383
.6248
.6111
.5972
.5831
.5688
.5544
.5398
.5250
.5100
.4950
.4797
.4643
.4488
.4331
.4173
.4014
.3854
.3692
.3529
.3365
.3201
.3035
.2868
.2700
.2532
.2363
.2193
.2022
.1851
.1679
.1507
.1334
.1161
.0987
.0814
.0640
.0465
.0291
.0116
30'
.7009
.6884
.6756
.6626
.6494
.6361
.6225
.6088
.5948
.5807
.5664
.5519
.5373
.5225
.5075
.4924
.4772
.4617
.4462
.4305
.4147
.3987
.3827
.3665
.3502
.3338
.3173
.3007
.2840
.2672
.2504
.2334
.2164
.1994
.1822
.1650
.1478
.1305
.1132
.0958
.0785
.0610
.0436
.0262
.0087
40'
.6988
.6862
.6734
.6604
.6472
.6338
.6202
.6065
.5925
.5783
.5640
.5495
.5348
.5200
.5050
.4899
.4746
.4592
.4436
.4279
.4120
.3961
.3800
.3638
.3475
.3311
.3145
.2979
.2812
.2644
.2476
.2306
.2136
.1965
.1794
.1622
.1449
.1276
.1103
.0929
.0756
.0581
.0407
.0233
.0058
50'
.6967
.6841
.6713
.6583
.6450
.6316
.6180
.6041
.5901
.5760
.5616
.5471
.5324
.5175
.5025
.4874
.4720
.4566
.4410
.4253
.4094
.3934
.3773
.3611
.3448
.3283
.3118
.2952
.2784
.2616
.2447
.2278
.2108
.1937
.1765
.1593
.1421
.1248
.1074
.0901
.0727
.0552
.0378
.0204
.0029
207

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
.0000
.0175
.0349
.0524
.0699
.0875
.1051
.1228
.1405
.1584
.1763
.1944
.2126
.2309
.2493
.2679
.2867
.3057
.3249
.3443
.3640
.3839
.4040
.4245
.4452
.4663
.4877
.5095
.5317
.5543
.5774
.6009
.6249
.6494
.6745
.7002
.7265
.7536
.7813
.8098
.8391
.8693
.9004
.9325
.9657
1.0000
10'
.0029
.0204
.0378
.0553
.0729
.0904
.1080
.1257
.1435
.1614
.1793
.1974
.2156
.2339
.2524
.2711
.2899
.3089
.3281
.3476
.3673
.3872
.4074
.4279
.4487
.4699
.4913
.5132
.5354
.5581
.5812
.6048
.6289
.6536
.6787
.7046
.7310
.7581
.7860
.8146
.8441
.8744
.9057
.9380
.9713
1.0058
20'
.0058
.0233
.0407
.0582
.0758
.0934
.1110
.1287
.1465
.1644
.1823
.2004
.2186
.2370
.2555
.2742
.2931
.3121
.3314
.3508
.3706
.3906
.4108
.4314
.4522
.4734
.4950
.5169
.5392
.5619
.5851
.6088
.6330
.6577
.6830
.7089
.7355
.7627
.7907
.8195
.8491
.8796
.9110
.9435
.9770
1.0117
30'
.0087
.0262
.0437
.0612
.0787
.0963
.1139
.1317
.1495
.1673
.1853
.2035
.2217
.2401
.2586
.2773
.2962
.3153
.3346
.3541
.3739
.3939
.4142
.4348
.4557
.4770
.4986
.5206
.5430
.5658
.5890
.6128
.6371
.6619
.6873
.7133
.7400
.7673
.7954
.8243
.8541
.8847
.9163
.9490
.9827
1.0176
40'
.0116
.0291
.0466
.0641
.0816
.0992
.1169
.1346
.1524
.1703
.1883
.2065
.2247
.2432
.2617
.2805
.2994
.3185
.3378
.3574
.3772
.3973
.4176
.4383
.4592
.4806
.5022
.5243
.5467
.5696
.5930
.6168
.6412
.6661
.6916
.7177
.7445
.7720
.8002
.8292
.8591
.8899
.9217
.9545
.9884
1.0235
50'
.0145
.0320
.0495
.0670
.0846
.1022
.1198
.1376
.1554
.1733
.1914
.2095
.2278
.2462
.2648
.2836
.3026
.3217
.3411
.3607
.3805
.4006
.4210
.4417
.4628
.4841
.5059
.5280
.5505
.5735
.5969
.6208
.6453
.6703
.6959
.7221
.7490
.7766
.8050
.8342
.8642
.8952
.9271
.9601
.9942
1.0295
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
1.0000
1.0355
1.0724
1.1106
1.1504
1.1918
1.2349
1.2799
1.3270
1.3764
1.4281
1.4826
1.5399
1.6003
1.6643
1.7321
1.8040
1.8807
1.9626
2.0503
2.1445
2.2460
2.3559
2.4751
2.6051
2.7475
2.9042
3.0777
3.2709
3.4874
3.7321
4.0108
4.3315
4.7046
5.1446
5.6713
6.3138
7.1154
8.1443
9.5144
11.430
14.301
19.081
28.636
57.290
со
10'
1.0058
1.0416
1.0786
1.1171
1.1571
1.1988
1.2423
1.2876
1.3351
1.3848
1.4370
1.4919
1.5497
1.6107
1.6753
1.7437
1.8165
1.8940
1.9768
2.0655
2.1609
2.2637
2.3750
2.4960
2.6279
2.7725
2.9319
3.1084
3.3052
3.5261
3.7760
4.0611
4.3897
4.7729
5.2257
5.7694
6.4348
7.2687
8.3450
9.7882
11.826
14.924
20.206
31.242
68.750
20'
1.0117
1.0477
1.0850
1.1237
1.1640
1.2059
1.2497
1.2954
1.3432
1.3934
1.4460
1.5013
1.5597
1.6212
1.6864
1.7556
1.8291
1.9074
1.9912
2.0809
2.1775
2.2817
2.3945
2.5172
2.6511
2.7980
2.9600
3.1397
3.3402
3.5656
3.8208
4.1126
4.4494
4.8430
5.3093
5.8708
6.5606
7.4287
8.5555
10.078
12.251
15.605
21.470
34.368
85.940
30'
1.0176
1.0538
1.0913
1.1303
1.1708
1.2131
1.2572
1.3032
1.3514
1.4019
1.4550
1.5108
1.5697
1.6319
1.6977
1.7675
1.8418
1.9210
2.0057
2.0965
2.1943
2.2998
2.4142
2.5386
2.6746
2.8239
2.9887
3.1716
3.3759
3.6059
3.8667
4.1653
4.5107
4.9152
5.3955
5.9758
6.6912
7.5958
8.7769
10.385
12.706
16.350
22.904
38.188
114.59
40'
1.0235
1.0599
1.0977
1.1369
1.1778
1.2203
1.2647
1.3111
1.3597
1.4106
1.4641
1.5204
1.5798
1.6426
1.7090
1.7796
1.8546
1.9347
2.0204
2.1123
2.2113
2.3183
2.4342
2.5605
2.6985
2.8502
3.0178
3.2041
3.4124
3.6470
3.9136
4.2193
4.5736
4.9894
5.4845
6.0844
6.8269
7.7704
9.0098
10.712
13.197
17.169
24.542
42.964
171.89
50'
1.0295
1.0661
1.1041
1.1436
1.1847
1.2276
1.2723
1.3190
1.3680
1.4193
1.4733
1.5301
1.5900
1.6534
1.7205
1.7917
1.8676
1.9486
2.0353
2.1283
2.2286
2.3369
2.4545
2.5826
2.7228
2.8770
3.0475
3.2371
3.4495
3.6891
3.9617
4.2747
4.6382
5.0658
5.5764
6.1970
6.9682
7.9530
9.2553
11.059
13.727
18.075
26.432
49.104
343.77
208

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
57.290
28.636
19.081
14.301
11.430
9.5144
8.1443
7.1154
6.3138
5.6713
5.1446
4.7046
4.3315
4.0108
3.7321
3.4874
3.2709
3.0777
2.9042
2.7475
2.6051
2.4751
2.3559
2.2460
2.1445
2.0503
1.9626
1.8807
1.8040
1.7321
1.6643
1.6003
1.5399
1.4826
1.4281
1.3764
1.3270
1.2799
1.2349
1.1918
1.1504
1.1106
1.0724
1.0355
1.0000
10'
343.77
49.104
26.432
18.075
13.727
11.059
9.2553
7.9530
6.9682
6.1970
5.5764
5.0658
4.6382
4.2747
3.9617
3.6891
3.4495
3.2371
3.0475
2.8770
2.7228
2.5826
2.4545
2.3369
2.2286
2.1283
2.0353
1.9486
1.8676
1.7917
1.7205
1.6534
1.5900
1.5301
1.4733
1.4193
1.3680
1.3190
1.2723
1.2276
1.1847
1.1436
1.1041
1.0661
1.0295
.9942
20'
171.89
42.964
24.542
17.169
13.197
10.712
9.0098
7.7704
6.8269
6.0844
5.4845
4.9894
4.5736
4.2193
3-.9136
3.6470
3.4124
3.2041
3.0178
2.8502
2.6985
2.5605
2.4342
3.3183
2.2113
2.1123
2.0204
1.9347
1.8546
1.7796
1.7090
1.6426
1.5798
1.5204
1.4641
1.4106
1.3597
1.3111
1.2647
1.2203
1.1778
1.1369
1.0977
1.0599
1.0235
.9884
30'
114.59
38.188
22.904
16.350
12.706
10.385
8.7769
7.5958
6.6912
5.9758
5.3955
4.9152
4.5107
4.1653
3.8667
3.6059
3.3759
3.1716
2.9887
2.8239
2.6746
2.5386
2.4142
2.2998
2.1943
2.0965
2.0057
1.9210
1.8418
1.7675
1.6977
1.6319
1.5697
1.5108
1.4550
1.4019
1.3514
1.3032
1.2572
1.2131
1.1708
1.1303
1.0913
1.0538
1.0176
.9827
40'
85.940
34.368
21.470
15.605
12.251
10.078
8.5555
7.4287
6.5606
5.8708
5.3093
4.8430
4.4494
4.1126
3.8208
3.5656
3.3402
3.1397
2.9600
2.7980
2.6511
2.5172
2.3945
2.2817
2.1775
2.0809
1.9912
1.9074
1.8291
1.7556
1.6864
1.6212
1.5597
1.5013
1.4460
1.3934
1.3432
1.2954
1.2497
1.2059
1.1640
1.1237
1.0850
1.0477
1.0117
.9770
50'
68.750
31.242
20.206
14.924
11.826
9.7882
8.3450
7.2687
6.4348
5.7694
5.2257
4.7729
4.3897
4.0611
3.7760
3.5261
3.3052
3.1084
2.9319
2.7725
2.6279
2.4960
2.3750
2.2637
2.1609
2.0655
1.9768
1.8940
1.8165
1.7437
1.6753
1.6107
1.5497
1.4919
1.4370
1.3848
1.3351
1.2876
1.2423
1.1988
1.1571
1.1171
1.0786
1.0416
1.0058
.9713
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
1.000
.9657
.9325
.9004
.8693
.8391
.8098
.7813
.7536
.7265
.7002
.6745
.6494
.6249
.6009
.5774
.5543
.5317
.5095
.4877
.4663
.4452
.4245
.4040
.3839
.3640
.3443
.3249
.3057
.2867
.2679
.2493
.2309
.2126
.1944
.1763
.1584
.1405
.1228
.1051
.0875
.0699
.0524
.0349
.0175
.0000
10'
.9942
.9601
.9271
.8952
.8642
.8342
.8050
.7766
.7490
.7221
.6959
.6703
.6453
.6208
.5969
.5735
.5505
.5280
.5059
.4841
.4628
.4417
.4210
.4006
.3805
.3607
.3411
.3217
.3026
.2836
.2648
.2462
.2278
.2095
.1914
.1733
.1554
.1376
.1198
.1022
.0846
.0670
.0495
.0320
.0145
20'
.9884
.9545
.9217
.8899
.8591
.8292
.8002
.7720
.7445
.7177
.6916
.6661
.6412
.6168
.5930
.5696
.5467
.5243
.5022
.4806
.4592
.4383
.4176
.3973
.3772
.3574
.3378
.3185
.2994
.2805
.2617
.2432
.2247
.2065
.1883
.1703
.1524
.1346
.1169
.0992
.0816
.0641
.0466
.0291
.0116
30'
.9827
.9490
.9163
.8847
.8541
.8243
.7954
.7673
.7400
.7133
.6873
.6619
.6371
.6128
.5890
.5658
.5430
.5206
.4986
.4770
.4557
.4348
.4142
.3939
.3739
.3541
.3346
.3153
.2962
.2773
.2586
.2401
.2217
.2035
.1853
.1673
.1495
.1317
.1139
.0963
.0787
.0612
.0437
.0262
.0087
40'
.9770
.9435
.9110
.8796
.8491
.8195
.7907
.7627
.7355
.7089
.6830
.6577
.6330
.6088
.5851
.5619
.5392
.5169
.4950
.4734
.4522
.4314
.4108
.3906
.3706
.3508
.3314
.3121
.2931
.2742
.2555
.2370
.2186
.2004
.1823
.1644
.1465
.1287
.1110
.0934
.0758
.0582
.0407
.0233
.0058
50'
.9713
.9380
.9057
.8744
.8441
.8146
.7860
.7581
.7310
.7046
.6787
.6536
.6289
.6048
.5812
.5581
.5354
.5132
.4913
.4699
.4487
.4279
.4074
.3872
.3673
.3476
.3281
.3089
.2899
.2711
.2524
.2339
.2156
.1974
.1793
.1614
.1435
.1257
.1080
.0904
.0729
.0553
.0378
.0204
.0029
209

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
1.000
1.000
1.001
1.001
1.002
1.004
1.006
1.008
1.010
1.012
1.015
1.019
1.022
1.026
1.031
1.035
1.040
1.046
1.051
1.058
1.064
1.071
1.079
1.086
1.095
1.103
1.113
1.122
1.133
1.143
1.155
1.167
1.179
1.192
1.206
1.221
1.236
1.252
1.269
1.287
1.305
1.325
1.346
1.367
1.390
1.414
10'
1.000
1.000
1.001
1.002
1.003
1.004
1.006
1.008
1.010
1.013
1.016
1.019
1.023
1.027
1.031
1.036
1.041
1.047
1.052
1.059
1.065
1.072
1.080
1.088
1.096
1.105
1.114
1.124
1.134
1.145
1.157
1.169
1.181
1.195
1.209
1.223
1.239
1.255
1.272
1.290
1.309
1.328
1.349
1.371
1.394
1.418
20'
1.000
1.000
1.001
1.002
1.003
1.004
1.006
1.008
1.011
1.013
1.016
1.020
1.024
1.028
1.032
1.037
1.042
1.048
1.053
1.060
1.066
1.074
1.081
1.089
1.097
1.106
1.116
1.126
1.136
1.147
1.159
1.171
1.184
1.197
1.211
1.226
1.241
1.258
1.275
1.293
1.312
1.332
1.353
1.375
1.398
1.423
30'
1.000
1.000
1.001
1.002
1.003
1.005
1.006
1.009
1.011
1.014
1.017
1.020
1.024
1.028
1.033
1.038
1.043
1.048
1.054
1.061
1.068
1.075
1.082
1.090
1.099
1.108
1.117
1.127
1.138
1.149
1.161
1.173
1.186
1.199
1.213
1.228
1.244
1.260
1.278
1.296
1.315
1.335
1.356
1.379
1.402
1.427
40'
1.000
1.000
1.001
1.002
1.003
1.005
1.007
1.009
1.012
1.014
1.018
1.021
1.025
1.029
1.034
1.039
1.044
1.049
1.056
1.062
1.069
1.076
1.084
1.092
1.100
1.109
1.119
1.129
1.140
1.151
1.163
1.175
1.188
1.202
1.216
1.231
1.247
1.263
1.281
1.299
1.318
1.339
1.360
1.382
1.406
1.431
50'
1.000
1.001
1.001
1.002
1.004
1.005
1.007
1.009
1.012
1.015
1.018
1.022
1.026
1.030
1.034
1.039
1.045
1.050
1.057
1.063
1.070
1.077
1.085
1.093
1.102
1.111
1.121
1.131
1.142
1.153
1.165
1.177
1.190
1.204
1.218
1.233
1.249
1.266
1.284
1.302
1.322
1.342
1.364
1.386
1.410
1.435
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
1.414
1.440
1.466
1.494
1.524
1.556
1.589
1.624
1.662
1.701
1.743
1.788
1.836
1.887
1.942
2.000
2.063
2.130
2.203
2.281
2.366
2.459
2.559
2.669
2.790
2.924
3.072
3.236
3.420
3.628
3.864
4.134
4.445
4.810
5.241
5.759
6.392
7.185
8.206
9.567
11.47
14.34
19.11
28.65
57.30
со
10'
1.418
1.444
1.471
1.499
1.529
1.561
1.595
1.630
1.668
1.708
1.751
1.796
1.844
1.896
1.951
2.010
2.074
2.142
2.215
2.295
2.381
2.475
2.577
2.689
2.812
2.947
3.098
3.265
3.453
3.665
3.906
4.182
4.502
4.876
5.320
5.855
6.512
7.337
8.405
9.839
11.87
14.96
20.23
31.26
68.76
20'
1.423
1.448
1.476
1.504
1.535
1.567
1.601
1.636
1.675
1.715
1.758
1.804
1.853
1.905
1.961
2.020
2.085
2.154
2.228
2.309
2.396
2.491
2.595
2.709
2.833
2.971
3.124
3.295
3.487
3.703
3.950
4.232
4.560
4.945
5.403
5.955
6.636
7.496
8.614
10.13
12.29
15.64
21.49
34.38
85.95
30'
1.427
1.453
1.480
1.509
1.540
1.572
1.606
1.643
1.681
1.722
1.766
1.812
1.861
1.914
1.970
2.031
2.096
2.166
2.241
2.323
2.411
2.508
2.613
2.729
2.855
2.996
3.152
3.326
3.521
3.742
3.994
4.284
4.620
5.016
5.487
6.059
6.765
7.661
8.834
10.43
12.75
16.38
22.93
38.20
114.6
40'
1.431
1.457
1.485
1.514
1.545
1.578
1.612
1.649
1.688
1.729
1.773
1.820
1.870
1.923
1.980
2.041
2.107
2.178
2.254
2.337
2.427
2.525
2.632
2.749
2.878
3.021
3.179
3.357
3.556
3.782
4.039
4.336
4.682
5.089
5.575
6.166
6.900
7.834
9.065
10.76
13.23
17.20
24.56
42.98
171.9
50'
1.435
1.462
1.490
1.519
1.550
1.583
1.618
1.655
1.695
1.736
1.781
1.828
1.878
1.932
1.990
2.052
2.118
2.190
2.268
2.352
2.443
2.542
2.650
2.769
2.901
3.046
3.207
3.388
3.592
3.822
4.086
4.390
4.745
5.164
5.665
6.277
7.040
8.016
9.309
11.10
13.76
18.10
26.45
49.11
343.8
210

X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
CO
57.30
28.65
19.11
14.34
11.47
9.567
8.206
7.185
6.392
5.759
5.241
4.810
4.445
4.134
3.864
3.628
3.420
3.236
3.072
2.924
2.790
2.669
2.559
2.459
2.366
2.281
2.203
2.130
2.063
2.000
1.942
1.887
1.836
1.788
1.743
1.701
1.662
1.624
1.589
1.556
1.524
1.494
1.466
1.440
1.414
10'
343.8
49.11
26.45
18.10
13.76
11.10
9.309
8.016
7.040
6.277
5.665
5.164
4.745
4.390
4.086
3.822
3.592
3.388
3.207
3.046
2.901
2.769
2.650
2.542
2.443
2.352
2.268
2.190
2.118
2.052
1.990
1.932
1.878
1.828
1.781
1.736
1.695
1.655
1.618
1.583
1.550
1.519
1.490
1.462
1.435
1.410
20'
171.9
42.98
24.56
17.20
13.23
10.76
9.065
7.834
6.900
6.166
5.575
5.089
4.682
4.336
4.039
3.782
3.556
3.357
3.179
3.021
2.878
2.749
2.632
2.525
2.427
2.337
2.254
2.178
2.107
2.041
1.980
1.923
1.870
1.820
1.773
1.729
1.688
1.649
1.612
1.578
1.545
1.514
1.485
1.457
1.431
1.406
30'
114.6
38.20
22.93
16.38
12.75
10.43
8.834
7.661
6.765
6.059
5.487
5.016
4.620
4.284
3.994
3.742
3.521
3.326
3.152
2.996
2.855
2.729
2.613
2.508
2.411
2.323
2.241
2.166
2.096
2.031
1.970
1.914
1.861
1.812
1.766
1.722
1.681
1.643
1.606
1.572
1.540
1.509
1.480
1.453
1.427
1.402
40'
85.95
34.38
21.49
15.64
12.29
10.13
8.614
7.496
6.636
5.955
5.403
4.945
4.560
4.232
3.950
3.703
3.487
3.295
3.124
2.971
2.833
2.709
2.595
2.491
2.396
2.309
2.228
2.154
2.085
2.020
1.961
1.905
1.853
1.804
1.758
1.715
1.675
1.636
1.601
1.567
1.535
1.504
1.476
1.448
1.423
1.398
50'
68.76
31.26
20.23
14.96
11.87
9.839
8.405
7.337
6.512
5.855
5.320
4.876
4.502
4.182
3.906
3.665
3.453
3.265
3.098
2.947
2.812
2.689
2.577
2.475
2.381
2.295
2.215
2.142
2.074
2.010
1.951
1.896
1.844
1.796
1.751
1.708
1.668
1.630
1.595
1.561
1.529
1.499
1.471
1.444
1.418
1.394
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
1.414
1.390
1.367
1.346
1.325
1.305
1.287
1.269
1.252
1.236
1.221
1.206
1.192
1.179
1.167
1.155
1.143
1.133
1.122
1.113
1.103
1.095
1.086
1.079
1.071
1.064
1.058
1.051
1.046
1.040
1.035
1.031
1.026
1.022
1.019
1.015
1.012
1.010
1.008
1.006
1.004
1.002
1.001
1.001
1.000
1.000
10'
1.410
1.386
1.364
1.342
1.322
1.302
1.284
1.266
1.249
1.233
1.218
1.204
1.190
1.177
1.165
1.153
1.142
1.131
1.121
1.111
1.102
1.093
1.085
1.077
1.070
1.063
1.057
1.050
1.045
1.039
1.034
1.030
1.026
1.022
1.018
1.015
1.012
1.009
1.007
1.005
1.004
1.002
1.001
1.001
1.000
20'
1.406
1.382
1.360
1.339
1.318
1.299
1.281
1.263
1.247
1.231
1.216
1.202
1.188
1.175
1.163
1.151
1.140
1.129
1.119
1.109
1.100
1.092
1.084
1.076
1.069
1.062
1.056
1.049
1.044
1.039
1.034
1.029
1.025
1.021
1.018
1.014
1.012
1.009
1.007
1.005
1.003
1.002
1.001
1.000
1.000
30'
1.402
1.379
1.356
1.335
1.315
1.296
1.278
1.260
1.244
1.228
1.213
1.199
1.186
1.173
1.161
1.149
1.138
1.127
1.117
1.108
1.099
1.090
1.082
1.075
1.068
1.061
1.054
1.048
1.043
1.038
1.033
1.028
1.024
1.020
1.017
1.014
1.011
1.009
1.006
1.005
1.003
1.002
1.001
1.000
1.000
40'
1.398
1.375
1.353
1.332
1.312
1.293
1.275
1.258
1.241
1.226
1.211
1.197
1.184
1.171
1.159
1.147
1.136
1.126
1.116
1.106
1.097
1.089
1.081
1.074
1.066
1.060
1.053
1.048
1.042
1.037
1.032
1.028
1.024
1.020
1.016
1.013
1.011
1.008
1.006
1.004
1.003
1.002
1.001
1.000
1.000
50'
1.394
1.371
1.349
1.328
1.309
1.290
1.272
1.255
1.239
1.223
1.209
1.195
1.181
1.169
1.157
1.145
1.134
1.124
1.114
1.105
1.096
1.088
1.080
1.072
1.065
1.059
1.052
1.047
1.041
1.036
1.031
1.027
1.023
1.019
1.016
1.013
1.010
1.008
1.006
1.004
1.003
1.002
1.001
1.000
1.000
211

X
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.33
.34
.35
.36
.37
.38
.39
.40
Sin a;
.00000
.01000
.02000
.03000
.03999
.04998
.05996
.06994
.07991
.08988
.09983
.10978
.11971
.12963
.13954
.14944
.15932
.16918
.17903
.18886
.19867
.20846
.21823
.22798
.23770
.24740
.25708
.26673
.27636
.28595
.29552
.30506
.31457
.32404
.33349
.34290
.35227
.36162
.37092
.38019
.38942
Cos x
1.00000
.99995
.99980
.99955
.99920
.99875
.99820
.99755
.99680
.99595
.99500
.99396
.99281
.99156
.99022
.98877
.98723
.98558
.98384
.98200
.98007
.97803
.97590
.97367
.97134
.96891
.96639
.96377
.96106
.95824
.95534
.95233
.94924
.94604
.94275
.93937
.93590
.93233
.92866
.92491
.92106
Tana;
.00000
.01000
.02000
.03001
.04002
.05004
.06007
.07011
.08017
.09024
.10033
.11045
.12058
.13074
.14092
.15114
.16138
.17166
.18197
.19232
.20271
.21314
.22362
.23414
.24472
.25534
.26602
.27676
.28755
.29841
.30934
.32033
.33139
.34252
.35374
.36503
.37640
.38786
.39941
.41105
.42279
Cot a;
00
99.9967
49.9933
33.3233
24.9867
19.9833
16.6467
14.2624
12.4733
11.0811
9.9666
9.0542
8.2933
7.6489
7.0961
6.6166
6.1966
5.8256
5.4954
5.1997
4.9332
4.6917
4.4719
4.2709
4.0864
3.9163
3.7591
3.6133
3.4776
3.3511
3.2327
3.1218
3.0176
2.9195
2.8270
2.7395
2.6567
2.5782
2.5037
2.4328
2.3652
Sec a;
1.00000
1.00005
1.00020
1.00045
1.00080
1.00125
1.00180
1.00246
1.00321
1.00406
1.00502
1.00608
1.00724
1.00851
1.00988
1.01136
1.01294
1.01463
1.01642
1.01833
1.02034
1.02246
1.02470
1.02705
1.02951
1.03209
1.03478
1.03759
1.04052
1.04358
1.04675
1.05005
1.05348
1.05704
1.06072
1.06454
1.06849
1.07258
1.07682
1.08119
1.08570
Csca;
00
100.0017
50.0033
33.3383
25.0067
20.0083
16.6767
14.2974
12.5133
11.1261
10.0167
9.1093
8.3534
7.7140
7.1662
6.6917
6.2767
5.9108
5.5857
5.2950
5.0335
4.7971
4.5823
4.3864
4.2069
4.0420
3.8898
3.7491
3.6185
3.4971
3.3839
3.2781
3.1790
3.0860
2.9986
2.9163
2.8387
2.7654
2.6960
2.6303
2.5679
212

Table 9
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
X
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
Sin a;
.38942
.39861
.40776
.41687
.42594
.43497
.44395
.45289
.46178
.47063
.47943
.48818
.49688
.50553
.51414
.52269
.53119
.53963
.54802
.55636
.56464
.57287
.58104
.58914
.59720
.60519
.61312
.62099
.62879
.63654
.64422
.65183
.65938
.66687
.67429
.68164
.68892
.69614
.70328
.71035
.71736
Cos x
.92106
.91712
.91309
.90897
.90475
.90045
.89605
.89157
.88699
.88233
.87758
.87274
.86782
.86281
.85771
.85252
.84726
.84190
.83646
.83094
.82534
.81965
.81388
.80803
.80210
.79608
.78999
.78382
.77757
.77125
.76484
.75836
.75181
.74517
.73847
.73169
.72484
.71791
.71091
.70385
.69671
Tana;
.42279
.43463
.44657
.45862
.47078
.48306
.49545
.50797
.52061
.53339
.54630
.55&36
.57256
.58592
.59943
.61311
.62695
.64097
.65517
.66956
.68414
.69892
.71391
.72911
.74454
.76020
.77610
.79225
.80866
.82534
.84229
.85953
.87707
.89492
.91309
.93160
.95045
.96967
.98926
1.0092
1.0296
Cot a;
2.3652
2.3008
2.2393
2.1804
2.1241
2.0702
2.0184
1.9686
1.9208
1.8748
1.8305
1.7878
1.7465
1.7067
1.6683
1.6310
1.5950
1.5601
1.5263
1.4935
1.4617
1.4308
1.4007
1.3715
1.3431
1.3154
1.2885
1.2622
1.2366
1.2116
1.1872
1.1634
1.1402
1.1174
1.0952
1.0734
1.0521
1.0313
1.0109
.99084
.97121
Sec a;
1.0857
1.0904
1.0952
1.1002
1.1053
1.1106
1.1160
1.1216
1.1274
1.1334
1.1395
1.1458
1.1523
1.1590
1.1659
1.1730
1.1803
1.1878
1.1955
1.2035
1.2116
1.2200
1.2287
1.2376
1.2467
1.2561
1.2658
1.2758
1.2861
1.2966
1.3075
1.3186
1.3301
1.3420
1.3542
1.3667
1.3796
1.3929
1.4066
1.4208
1.4353
Csc x
2.5679
2.5087
2.4524
2.3988
2.3478
2.2990
2.2525
2.2081
2.1655
2.1248
2.0858
2.0484
2.0126
1.9781
1.9450
1.9132
1.8826
1.8531
1.8247
1.7974
1.7710
1.7456
1.7211
1.6974
1.6745
1.6524
1.6310
1.6103
1.5903
1.5710
1.5523
1.5341
1.5166
1.4995
1.4830
1.4671
1.4515
1.4365
1.4219
1.4078
1.3940
213

Table 9
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
Ж
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
Sin x
.71736
.72429
.73115
.73793
.74464
.75128
.75784
.76433
.77074
.77707
.78333
.78950
.79560
.80162
.80756
.81342
.81919
.82489
.83050
.83603
.84147
.84683
.85211
.85730
.86240
.86742
.87236
.87720
.88196
.88663
.89121
.89570
.90010
.90441
.90863
.91276
.91680
.92075
.92461
.92837
.93204
Cos ж
.69671
.68950
.68222
.67488
.66746
.65998
.65244
.64483
.63715
.62941
.62161
.61375
.60582
.59783
.58979
.58168
.57352
.56530
.55702
.54869
.54030
.53186
.52337
.51482
.50622
.49757
.48887
.48012
.47133
.46249
.45360
.44466
.43568
.42666
.41759
.40849
.39934
.39015
.38092
.37166
.36236
Tan ж
1.0296
1.0505
1.0717
1.0934
1.1156
1.1383
1.1616
1.1853
1.2097
1.2346
1.2602
1.2864
1.3133
1.3409
1.3692
1.3984
1.4284
1.4592
1.4910
1.5237
1.5574
1.5922
1.6281
1.6652
1.7036
1.7433
1.7844
1.8270
1.8712
1.9171
1.9648
2.0143
2.0660
2.1198
2.1759
2.2345
2.2958
2.3600
2.4273
2.4979
2.5722
Cot ж
.97121
.95197
.93309
.91455
.89635
.87848
.86091
.84365
.82668
.80998
.79355
.77738
.76146
.74578
.73034
.71511
.70010
.68531
.67071
.65631
.64209
.62806
.61420
.60051
.58699
.57362
.56040
.54734
.53441
.52162
.50897
.49644
.48404
.47175
.45959
.44753
.43558
.42373
.41199
.40034
.38878
Sec ж
1.4353
1.4503
1.4658
1.4818
1.4982
1.5152
1.5327
1.5508
1.5695
1.5888
1.6087
1.6293
1.6507
1.6727
1.6955
1.7191
1.7436
1.7690
1.7953
1.8225
1.8508
1.8802
1.9107
1.9424
1.9754
2.0098
2.0455
2.0828
2.1217
2.1622
2.2046
2.2489
2.2952
2.3438
2.3947
2.4481
2.5041
2.5631
2.6252
2.6906
2.7597
Сэсж
1.3940
1.3807
1.3677
1.3551
1.3429
1.3311
1.3195
1.3083
1.2975
1.2869
1.2766
1.2666
1.2569
1.2475
1.2383
1.2294
1.2207
1.2123
1.2041
1.1961
1.1884
1.1809
1.1736
1.1665
1.1595
1.1528
1.1463
1.1400
1.1338
1.1279
1.1221
1.1164
1.1110
1.1057
1.1006
1.0956
1.0907
1.0861
1.0815
1.0772
1.0729
214

Table 9
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
X
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
Sin x
.93204
.93562
.93910
.94249
.94578
.94898
.95209
.95510
.95802
.96084
.96356
.96618
.96872
.97115
.97348
.97572
.97786
.97991
.98185
.98370
.98545
.98710
.98865
.99010
.99146
.99271
.99387
.99492
.99588
.99674
.99749
.99815
.99871
.99917
.99953
.99978
.99994
1.00000
.99996
.99982
.99957
Cos a;
.36236
.35302
.34365
.33424
.32480
.31532
.30582
.29628
.28672
.27712
.26750
.25785
.24818
.23848
.22875
.21901
.20924
.19945
.18964
.17981
.16997
.16010
.15023
.14033
.13042
.12050
.11057
.10063
.09067
.08071
.07074
.06076
.05077
.04079
.03079
.02079
.01080
.00080
-.00920
-.01920
-.02920
Tan ж
2.5722
2.6503
2.7328
2.8198
2.9119
3.0096
3.1133
3.2236
3.3414
3.4672
3.6021
3.7471
3.9033
4.0723
4.2556
4.4552
4.6734
4.9131
5.1774
5.4707
5.7979
6.1654
6.5811
7.0555
7.6018
8.2381
8.9886
9.8874
10.9834
12.3499
14.1014
16.4281
19.6695
24.4984
32.4611
48.0785
92.6205
1255.77
-108.649
-52.0670
-34.2325
Cot a;
.38878
.37731
.36593
.35463
.34341
.33227
.32121
.31021
.29928
.28842
.27762
.26687
.25619
.24556
.23498
.22446
.21398
.20354
.19315
.18279
.17248
.16220
.15195
.14173
.13155
.12139
.11125
.10114
.09105
.08097
.07091
.06087
.05084
.04082
.03081
.02080
.01080
.00080
-.00920
-.01921
-.02921
Sec a;
2.7597
2.8327
2.9100
2.9919
3.0789
3.1714
3.2699
3.3752
3.4878
3.6085
3.7383
3.8782
4.0294
4.1933
4.3715
4.5661
4.7792
5.0138
5.2731
5.5613
5.8835
6.2459
6.6567
7.1260
7.6673
8.2986
9.0441
9.9378
11.0288
12.3903
14.1368
16.4585
19.6949
24.5188
32.4765
48.0889
92.6259
1255.77
-108.654
-52.0766
-34.2471
Csca;
1.07292
1.06881
1.06485
1.06102
1.05732
1.05376
1.05032
1.04701
1.04382
1,04076
1.03782
1.03500
1.03230
1.02971
1.02724
1.02488
1.02264
1.02050
1.01848
1.01657
1.01477
1.01307
1.01148
1.00999
1.00862
1.00734
1.00617
1.00510
1.00414
1.00327
1.00251
1.00185
1.00129
1.00083
1.00047
1.00022
1.00006
1.00000
1.00004
1.00018
1.00043
215

TABLE
10
1O9 Slfl X (ж in degrees and minutes)
[subtract 10 from each entry]
X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
_
8.2419
8.5428
8.7188
8.8436
8.9403
9.0192
9.0859
9.1436
9.1943
9.2397
9.2806
9.3179
9.3521
9.3837
9.4130
9.4403
9.4659
9.4900
9.5126
9.5341
9.5543
9.5736
9.5919
9.6093
9.6259
9.6418
9.6570
9.6716
9.6856
9.6990
9.7118
9.7242
9.7361
9.7476
9.7586
9.7692
9.7795
9.7893
9.7989
9.8081
9.8169
9.8255
9.8338
9.8418
9.8495
10'
7.4637
8.3088
8.5776
8.7423
8.8613
8.9545
9.0311
9.0961
9.1525
9.2022
9.2468
9.2870
9.3238
9.3575
9.3887
9.4177
9.4447
9.4700
9.4939
9.5163
9.5375
9.5576
9.5767
9.5948
9.6121
9.6286
9.6444
9.6595
9.6740
9.6878
9.7012
9.7139
9.7262
9.7380
9.7494
9.7604
9.7710
9.7811
9.7910
9.8004
9.8096
9.8184
9.8269
9.8351
9.8431
9.8507
20'
7.7648
8.3668
8.6097
8.7645
8.8783
8.9682
9.0426
9.1060
9.1612
9.2100
9.2538
9.2934
9.3296
9.3629
9.3937
9.4223
9.4491
9.4741
9.4977
9.5199
9.5409
9.5609
9.5798
9.5978
9.6149
9.6313
9.6470
9.6620
9.6763
9.6901
9.7033
9.7160
9.7282
9.7400
9.7513
9.7622
9.7727
9.7828
9.7926
9.8020
9.8111
9.8198
9.8283
9.8365
9.8444
9.8520
30'
7.9408
8.4179
8.6397
8.7857
8.8946
8.9816
9.0539
9.1157
9.1697
9.2176
9.2606
9.2997
9.3353
9.3682
9.3986
9.4269
9.4533
9.4781
9.5015
9.5235
9.5443
9.5641
9.5828
9.6007
9.6177
9.6340
9.6495
9.6644
9.6787
9.6923
9.7055
9.7181
9.7302
9.7419
9.7531
9.7640
9.7744
9.7844
9.7941
9.8035
9.8125
9.8213
9.8297
9.8378
9.8457
9.8532
40'
8.0658
8.4637
8.6677
8.8059
8.9104
8.9945
9.0648
9.1252
9.1781
9.2251
9.2674
9.3058
9.3410
9.3734
9.4035
9.4314
9.4576
9.4821
9.5052
9.5270
9.5477
9.5673
9.5859
9.6036
9.6205
9.6366
9.6521
9.6668
9.6810
9.6946
9.7076
9.7201
9.7322
9.7438
9.7550
9.7657
9.7761
9.7861
9.7957
9.8050
9.8140
9.8227
9.8311
9.8391
9.8469
9.8545
50'
8.1627
8.5050
8.6940
8.8251
8.9256
9.0070
9.0755
9.1345
9.1863
9.2324
9.2740
9.3119
9.3466
9.3786
9.4083
9.4359
9.4618
9.4861
9.5090
9.5306
9.5510
9.5704
9.5889
9.6065
9.6232
9.6392
9.6546
9.6692
9.6833
9.6968
9.7097
9.7222
9.7342
9.7457
9.7568
9.7675
9.7778
9.7877
9.7973
9.8066
9.8155
9.8241
9.8324
9.8405
9.8482
9.8557
216

Table 10
(continued)
¦ •
IO9 Slfl д: (ж in degrees and minutes)
[subtract 10 from each entry]
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
9.8495
9.8569
9.8641
9.8711
9.8778
9.8843
9.8905
9.8965
9.9023
9.9080
9.9134
9.9186
9.9236
9.9284
9.9331
9.9375
9.9418
9.9459
9.9499
9.9537
9.9573
9.9607
9.9640
9.9672
9.9702
9.9730
9.9757
9.9782
9.9806
9.9828
9.9849
9.9869
9.9887
9.9904
9.9919
9.9934
9.9946
9.9958
9.9968
9.9976
9.9983
9.9989
9.9994
9.9997
9.9999
10.0000
10'
9.8507
9.8582
9.8653
9.8722
9.8789
9.8853
9.8915
9.8975
9.9033
9.9089
9.9142
9.9194
9.9244
9.9292
9.9338
9.9383
9.9425
9.9466
9.9505
9.9543
9.9579
9.9613
9.9646
9.9677
9.9706
9.9734
9.9761
9.9786
9.9810
9.9832
9.9853
9.9872
9.9890
9.9907
9.9922
9.9936
9.9948
9.9959
9.9969
9.9977
9.9985
9.9990
9.9995
9.9998
10.0000
20'
9.8520
9.8594
9.8665
9.8733
9.8800
9.8864
9.8925
9.8985
9.9042
9.9098
9.9151
9.9203
9.9252
9.9300
9.9346
9.9390
9.9432
9.9473
9.9512
9.9549
9.9584
9.9618
9.9651
9.9682
9.9711
9.9739
9.9765
9.9790
9.9814
9.9836
9.9856
9.9875
9.9893
9.9909
9.9924
9.9938
9.9950
9.9961
9.9971
9.9979
9.9986
9.9991
9.9995
9.9998
10.0000
30'
9.8532
9.8606
9.8676
9.8745
9.8810
9.8874
9.8935
9.8995
9.9052
9.9107
9.9160
9.9211
9.9260
9.9308
9.9353
9.9397
9.9439
9.9479
9.9518
9.9555
9.9590
9.9624
9.9656
9.9687
9.9716
9.9743
9.9770
9.9794
9.9817
9.9839
9.9859
9.9878
9.9896
9.9912
9.9927
9.9940
9.9952
9.9963
9.9972
9.9980
9.9987
9.9992
9.9996
9.9999
10.0000
40'
9.8545
9.8618
9.8688
9.8756
9.8821
9.8884
9.8945
9.9004
9.9061
9.9116
9.9169
9.9219
9.9268
9.9315
9.9361
9.9404
9.9446
9.9486
9.9524
9.9561
9.9596
9.9629
9.9661
9.9692
9.9721
9.9748
9.9774
9.9798
9.9821
9.9843
9.9863
9.9881
9.9899
9.9914
9.9929
9.9942
9.9954
9.9964
9.9973
9.9981
9.9988
9.9993
9.9996
9.9999
10.0000
50'
9.8557
9.8629
9.8699
9.8767
9.8832
9.8895
3.8955
9.9014
9.9070
9.9125
9.9177
9.9228
9.9276
9.9323
9.9368
9.9411
9.9453
9.9492
9.9530
9.9567
9.9602
9.9635
9.9667
9.9697
9.9725
9.9752
9.9778
9.9802
9.9825
9.9846
9.9866
9.9884
9.9901
9.9917
9.9931
9.9944
9.9956
9.9966
9.9975
9.9982
9.9989
9.9993
9.9997
9.9999
10.0000
217

TABLE
11
IO9 COS X (x in degrees and minutes)
[subtract 10 from each entry]
X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
10.0000
9.9999
9.9997
9.9994
9.9989
9.9983
9.9976
9.9968
9.9958
9.9946
9.9934
9.9919
9.9904
9.9887
9.9869
9.9849
9.9828
9.9806
9.9782
9.9757
9.9730
9.9702
9.9672
9.9640
9.9607
9.9573
9.9537
9.9499
9.9459
9.9418
9.9375
9.9331
9.9284
9.9236
9.9186
9.9134
9.9080
9.9023
9.8965
9.8905
9.8843
9.8778
9.8711
9.8641
9.8569
9.8495
10'
10.0000
9.9999
9.9997
9.9993
9.9989
9.9982
9.9975
9.9966
9.9956
9.9944
9.9931
9.9917
9.9901
9.9884
9.9866
9.9846
9.9825
9.9802
9.9778
9.9752
9.9725
9.9697
9.9667
9.9635
9.9602
9.9567
9.9530
9.9492
9.9453
9.9411
9.9368
9.9323
9.9276
9.9228
9.9177
9.9125
9.9070
9.9014
9.8955
9.8895
9.8832
9.8767
9.8699
9.8629
9.8557
9.8482
20'
10.0000
9.9999
9.9996
9.9993
9.9988
9.9981
9.9973
9.9964
9.9954
9.9942
9.9929
9.9914
9.9899
9.9881
9.9863
9.9843
9.9821
9.9798
9.9774
9.9748
9.9721
9.9692
9.9661
9.9629
9.9596
9.9561
9.9524
9.9486
9.9446
9.9404
9.9361
9.9315
9.9268
9.9219
9.9169
9.9116
9.9061
9.9004
9.8945
9.8884
9.8821
9.8756
9.8688
9.8618
9.8545
9.8469
30'
10.0000
9.9999
9.9996
9.9992
9.9987
9.9980
9.9972
9.9963
9.9952
9.9940
9.9927
9.9912
9.9896
9.9878
9.9859
9.9839
9.9817
9.9794
9.9770
9.9743
9.9716
9.9687
9.9656
9.9624
9.9590
9.9555
9.9518
9.9479
9.9439
9.9397
9.9353
9.9308
9.9260
9.9211
9.9160
9.9107
9.9052
9.8995
9.8935
9.8874
9.8810
9.8745
9.8676
9.8606
9.8532
9.8457
40'
10.0000
9.9998
9.9995
9.9991
9.9986
9.9979
9.9971
9.9961
9.9950
9.9938
9.9924
9.9909
9.9893
9.9875
9.9856
9.9836
9.9814
9.9790
9.9765
9.9739
9.9711
9.9682
9.9651
9.9618
9.9584
9.9549
9.9512
9.9473
9.9432
9.9390
9.9346
9.9300
9.9252
9.9203
9.9151
9.9098
9.9042
9.8985
9.8925
9.8864
9.8800
9.8733
9.8665
9.8594
9.8520
9.8444
50'
10.0000
9.9998
9.9995
9.9990
9.9985
9.9977
9.9969
9.9959
9.9948
9.9936
9.9922
9.9907
9.9890
9.9872
9.9853
9.9832
9.9810
9.9786
9.9761
9.9734
9.9706
9.9677
9.9646
9.9613
9.9579
9.9543
9.9505
9.9466
9.9425
9.9383
9.9338
9.9292
9.9244
9.9194
9.9142
9.9089
9.9033
9.8975
9.8915
9.8853
9.8789
9.8722
9.8653
9.8582
9.8507
9.8431
218

Table 11
(continued)
COS X (x in degrees and minutes)
[subtract 10 from each entrj'l
X
45°
46
47
48
49
50°
51
52
53
54
56°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
9.8495
9.8418
9.8338
9.8255
9.8169
9.8081
9.7989
9.7893
9.7795
9.7692
9.7586
9.7476
9.7361
9.7242
9.7118
9.6990
9.6856
9.6716
9.6570
9.6418
9.6259
9.6093
9.5919
9.5736
9.5543
9.5341
9.5126
9.4900
9.4659
9.4403
9.4130
9.3837
9.3521
9.3179
9.2806
9.2397
9.1943
9.1436
9.0859
9.0192
8.9403
8.8436
8.7188
8.5428
8.2419
10'
9.8482
9.8405
9.8324
9.8241
9.8155
9.8066
9.7973
9.7877
9.7778
9.7675
9.7568
9.7457
9.7342
9.7222
9.7097
9.6968
9.6833
9.6692
9.6546
9.6392
9.6232
9.6065
9.5889
9.5704
9.5510
9.5306
9.5090
9.4861
9.4618
9.4359
9.4083
9.3786
9.3466
9.3119
9.2740
9.2324
9.1863
9.1345
9.0755
9.0070
8.9256
8.8251
8.6940
8.5050
8.1627
20'
9.8469
9.8391
9.8311
9.8227
9.8140
9.8050
9.7957
9.7861
9.7761
9.7657
9.7550
9.7438
9.7322
9.7201
9.7076
9.6946
9.6810
9.6668
9.6521
9.6366
9.6205
9.6036
9.5859
9.5673
9.5477
9.5270
9.5052
9.4821
9.4576
9.4314
9.4035
9.3734
9.3410
9.3058
9.2674
9.2251
9.1781
9.1252
9.0648
8.9945
8.9104
8.8059
8.6677
8.4637
8.0658
30'
9.8457
9.8378
9.8297
9.8213
9.8125
9.8035
9.7941
9.7844
9.7744
9.7640
9.7531
9.7419
9.7302
9.7181
9.7055
9.6923
9.6787
9.6644
9.6495
9.6340
9.6177
9.6007
9.5828
9.5641
9.5443
9.5235
9.5015
9.4781
9.4533
9.4269
9.3986
9.3682
9.3353
9.2997
9.2606
9.2176
9.1697
9.1157
9.0539
8.9816
8.8946
8.7857
8.6397
8.4179
7.9408
40'
9.8444
9.8365
9.8283
9.8198
9.8111
9.8020
9.7926
9.7828
9.7727
9.7622
9.7513
9.7400
9.7282
9.7160
9.7033
9.6901
9.6763
9.6620
9.6470
9.6313
9.6149
9.5978
9.5798
9.5609
9.5409
9.5199
9.4977
9.4741
9.4491
9.4223
9.3937
9.3629
9.3296
9.2934
9.2538
9.2100
9.1612
9.1060
9.0426
8.9682
8.8783
8.7645
8.6097
8.3668
7.7648
50'
9.8431
9.8351
9.8269
9.8184
9.8096
9.8004
9.7910
9.7811
9.7710
9.7604
9.7494
9.7380
9.7262
9.7139
9.7012
9.6878
9.6740
9.6595
9.6444
9.6286
9.6121
9.5948
9.5767
9.5576
9.5375
9.5163
9.4939
9.4700
9.4447
9.4177
9.3887
9.3575
9.3238
9.2870
9.2468
9.2022
9.1525
9.0961
9.0311
8.9545
8.8613
8.7423
8.5776
8.3088
7.4637
219

TABLE
12
1O9 ТОП X (x in degrees and minutes)
[subtract 10 from each entry]
X
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45°
0'
—
8.2419
8.5431
8.7194
8.8446
8.9420
9.0216
9.0891
9.1478
9.1997
9.2463
9.2887
9.3275
9.3634
9.3968
9.4281
9.4575
9.4853
9.5118
9.5370
9.5611
9.5842
9.6064
9.6279
9.6486
9.6687
9.6882
9.7072
9.7257
9.7438
9.7614
9.7788
9.7958
9.8125
9.8290
9.8452
9.8613
9.8771
9.8928
9.9084
9.9238
9.9392
9.9544
9.9697
9.9848
10.0000
10'
7.4637
8.3089
8.5779
8.7429
8.8624
8.9563
9.0336
9.0995
9.1569
9.2078
9.2536
9.2953
9.3336
9.3691
9.4021
9.4331
9.4622
9.4898
9.5161
9.5411
9.5650
9.5879
9.6100
9.6314
9.6520
9.6720
9.6914
9.7103
9.7287
9.7467
9.7644
9.7816
9.7986
9.8153
9.8317
9.8479
9.8639
9.8797
9.8954
9.9110
9.9264
9.9417
9.9570
9.9722
9.9874
10.0025
20'
7.7648
8.3669
8.6101
8.7652
8.8795
8.9701
9.0453
9.1096
9.1658
9.2158
9.2609
9.3020
9.3397
9.3748
9.4074
9.4381
9.4669
9.4943
9.5203
9.5451
9.5689
9.5917
9.6136
9.6348
9.6553
9.6752
9.6946
9.7134
9.7317
9.7497
9.7673
9.7845
9.8014
9.8180
9.8344
9.8506
9.8666
9.8824
9.8980
9.9135
9.9289
9.9443
9.9595
9.9747
9.9899
10.0051
30'
7.9409
8.4181
8.6401
8.7865
8.8960
8.9836
9.0567
9.1194
9.1745
9.2236
9.2680
9.3085
9.3458
9.3804
9.4127
9.4430
9.4716
9.4987
9.5245
9.5491
9.5727
9.5954
9.6172
9.6383
9.6587
9.6785
9.6977
9.7165
9.7348
9.7526
9.7701
9.7873
9.8042
9.8208
9.8371
9.8533
9.8692
9.8850
9.9006
9.9161
9.9315
9.9468
9.9621
9.9772
9.9924
10.0076
40'
8.0658
8.4638
8.6682
8.8067
8.9118
8.9966
9.0678
9.1291
9.1831
9.2313
9.2750
9.3149
9.3517
9.3859
9.4178
9.4479
9.4762
9.5031
9.5287
9.5531
9.5766
9.5991
9.6208
9.6417
9.6620
9.6817
9.7009
9.7196
9.7378
9.7556
9.7730
9.7902
9.8070
9.8235
9.8398
9.8559
9.8718
9.8876
9.9032
9.9187
9.9341
9.9494
9.9646
9.9798
9.9949
10.0101
50'
8.1627
8.5053
8.6945
8.8261
8.9272
9.0093
9.0786
9.1385
9.1915
9.2389
9.2819
9.3212
9.3576
9.3914
9.4230
9.4527
9.4808
9.5075
9.5329
9.5571
9.5804
9.6028
9.6243
9.6452
9.6654
9.6850
9.7040
9.7226
9.7408
9.7585
9.7759
9.7930
9.8097
9.8263
9.8425
9.8586
9.8745
9.8902
9.9058
9.9212
9.9366
9.9519
9.9671
9.9823
9.9975
10.0126
220

Table 12
(continued)
tan X (x in degrees and minutes)
[subtract 10 from each entry!
X
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
10.0000
10.0152
10.0303
10.0456
10.0608
10.0762
10.0916
10.1072
10.1229
10.1387
10.1548
10.1710
10.1875
10.2042
10.2212
10.2386
10.2562
10.2743
10.2928
10.3118
10.3313
10.3514
10.3721
10.3936
10.4158
10.4389
10.4630
10.4882
10.5147
10.5425
10.5719
10.6032
10.6366
10.6725
10.7113
10.7537
10.8003
10.8522
10.9109
10.9784
11.0580
11.1554
11.2806
11.4569
11.7581
10'
10.0025
10.0177
10.0329
10.0481
10.0634
10.0788
10.0942
10.1098
10.1255
10.1414
10.1575
10.1737
10.1903
10.2070
10.2241
10.2415
10.2592
10.2774
10.2960
10.3150
10.3346
10.3548
10.3757
10.3972
10.4196
10.4429
10.4671
10.4925
10.5192
10.5473
10.5770
10.6086
10.6424
10.6788
10.7181
10.7611
10.8085
10.8615
10.9214
10.9907
11.0728
11.1739
11.3055
11.4947
11.8373
20'
10.0051
10.0202
10.0354
10.0506
10.0659
10.0813
10.0968
10.1124
10.1282
10.1441
10.1602
10.1765
10.1930
10.2098
10.2270
10.2444
10.2622
10.2804
10.2991
10.3183
10.3380
10.3583
10.3792
10.4009
10.4234
10.4469
10.4713
10.4969
10.5238
10.5521
10.5822
10.6141
10.6483
10.6851
10.7250
10.7687
10.8169
10.8709
10.9322
11.0034
11.0882
11.1933
11.3318
11.5362
11.9342
30'
10.0076
10.0228
10.0379
10.0532
10.0685
10.0839
10.0994
10.1150
10.1308
10.1467
10.1629
10.1792
10.1958
10.2127
10.2299
10.2474
10.2652
10.2835
10.3023
10.3215
10.3413
10.3617
10.3828
10.4046
10.4273
10.4509
10.4755
10.5013
10.5284
10.5570
10.5873
10.6196
10.6542
10.6915
10.7320
10.7764
10.8255
10.8806
10.9433
11.0164
11.1040
11.2135
11.3599
11.5819
12.0591
40'
10.0101
10.0253
10.0405
10.0557
10.0711
10.0865
10.1020
10.1176
10.1334
10.1494
10.1656
10.1820
10.1986
10.2155
10.2327
10.2503
10.2683
10.2866
10.3054
10.3248
10.3447
10.3652
10.3864
10.4083
10.4311
10.4549
10.4797
10.5057
10.5331
10.5619
10.5926
10.6252
10.6603
10.6980
10.7391
10.7842
10.8342
10.8904
10.9547
11.0299
11.1205
11.2348
11.3899
11.6331
12.2352
50'
10.0126
10.0278
10.0430
10.0583
10.0736
10.0890
10.1046
10.1203
10.1361
10.1521
10.1683
10.1847
10.2014
10.2184
10.2356
10.2533
10.2713
10.2897
10.3086
10.3280
10.3480
10.3686
10.3900
10.4121
10.4350
10.4589
10.4839
10.5102
10.5378
10.5669
10.5979
10.6309
10.6664
10.7047
10.7464
10.7922
10.8431
10.9005
10.9664
11.0437
11.1376
11.2571
11.4221
11.6911
12.5363
221

TABLE
13
CONVERSION OF RADIANS TO DEGREES,
MINUTES AND SECONDS OR FRACTIONS OF DEGREES
Radians
1
2
3
4
5
6
7
8
9
10
.1
.2
.3
.4
.5
.6
.7
.8
.9
.01
.02
.03
.04
.05
.06
.07
.08
.09
.001
.002
.003
.004
.005
.006
.007
.008
.009
.0001
.0002
.0003
.0004
.0005
.0006
.0007
.0008
.0009
Deg.
57°
114°
171°
229°
286°
343°
401°
458°
515°
572°
5°
11°
17°
22°
28°
34°
40°
45°
51°
0°
1°
1°
2°
2°
3°
4°
4°
5°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
0°
Min.
17'
35'
53'
10'
28'
46'
4'
21'
39'
57'
43'
27'
11'
55'
38'
22'
6'
50'
33'
34'
8'
43'
17'
51'
26'
0'
35'
9'
3'
6'
10'
13'
17'
20'
24'
27'
30'
0'
0'
1'
1'
1'
2'
2'
2'
3'
Sec.
44.8"
29.6"
14.4"
59.2"
44.0"
28.8"
13.6"
58.4"
43.3"
28.1"
46.5"
33.0"
19.4"
5.9"
52.4"
38.9"
25.4"
11.8"
58.3"
22.6"
45.3"
7.9"
30.6"
53.2"
15.9"
38.5"
1.2"
23.8"
26.3"
52.5"
18.8"
45.1"
11.3"
37.6"
3.9"
30.1"
56.4"
20.6"
41.3"
1.9"
22.5"
43.1"
3.8"
24.4"
45.0"
5.6"
Fractions of
Degrees
57.2958°
114.5916°
171.8873°
229.1831°
286.4789°
343.7747°
401.0705°
458.3662°
515.6620°
572.9578°
222

TABLE
14
CONVERSION OF DEGREES, MINUTES
AND SECONDS TO RADIANS
Degrees
1°
2°
3°
4°
5°
6°
7°
8°
9°
10°
Radians
.0174533
.0349066
.0523599
.0698132
.0872665
.1047198
.1221730
.1396263
.1570796
.1745329
Minutes
1'
2'
3'
4'
5'
6'
7'
8'
9'
10'
Radians
.00029089
.00058178
.00087266
.00116355
.00145444
.00174533
.00203622
.00232711
.00261800
.00290888
Seconds
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
Radians
.0000048481
.0000096963
.0000145444
.0000193925
.0000242407
.0000290888
.0000339370
.0000387851
.0000436332
.0000484814
223

X
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0
.00000
.09531
.18232
.26236
.33647
.40547
.47000
.53063
.58779
.64185
.69315
.74194
.78846
.83291
.87547
.91629
.95551
.99325
1.02962
1.06471
1.09861
1.13140
1.16315
1.19392
1.22378
1.25276
1.28093
1.30833
1.33500
1.36098
1.38629
1.41099
1.43508
1.45862
1.48160
1.50408
1.52606
1.54756
1.56862
1.58924
1
.00995
.10436
.19062
.27003
.34359
.41211
.47623
.53649
.59333
.64710
.69813
.74669
.79299
.83725
.87963
.92028
.95935
.99695
1.03318
1.06815
1.10194
1.13462
1.16627
1.19695
1.22671
1.25562
1.28371
1.31103
1.33763
1.36354
1.38879
1.41342
1.43746
1.46094
1.48387
1.50630
1.52823
1.54969
1.57070
1.59127
2
.01980
.11333
.19885
.27763
.35066
.41871
.48243
.54232
.59884
.65233
.70310
.75142
.79751
.84157
.88377
.92426
.96317
1.00063
1.03674
1.07158
1.10526
1.13783
1.16938
1.19996
1.22964
1.25846
1.28647
1.31372
1.34025
1.36609
1.39128
1.41585
1.43984
1.46326
1.48614
1.50851
1.53039
1.55181
1.57277
1.59331
3
.02956
.12222
.20701
.28518
.35767
.42527
.48858
.54812
.60432
.65752
.70804
.75612
.80200
.84587
.88789
.92822
.96698
1.00430
1.04028
1.07500
1.10856
1.14103
1.17248
1.20297
1.23256
1.26130
1.28923
1.31641
1.34286
1.36864
1.39377
1.41828
1.44220
1.46557
1.48840
1.51072
1.53256
1.55393
1.57485
1.59534
4
.03922
.13103
.21511
.29267
.36464
.43178
.49470
.55389
.60977
.66269
.71295
.76081
.80648
.85015
.89200
.93216
.97078
1.00796
1.04380
1.07841
1.11186
1.14422
1.17557
1.20597
1.23547
1.26413
1.29198
1.31909
1.34547
1.37118
1.39624
1.42070
1.44456
1.46787
1.49065
1.51293
1.53471
1.55604
1.57691
1.59737
5
.04879
.13976
.22314
.30010
.37156
.43825
.50078
.55962
.61519
.66783
.71784
.76547
.81093
.85442
.89609
.93609
.97456
1.01160
1.04732
1.08181
1.11514
1.14740
1.17865
1.20896
1.23837
1.26695
1.29473
1.32176
1.34807
1.37372
1.39872
1.42311
1.44692
1.47018
1.49290
1.51513
1.53687
1.55814
1.57898
1.59939
6
.05827
.14842
.23111
.30748
.37844
.44469
.50682
.56531
.62058
.67294
.72271
.77011
.81536
.85866
.90016
.94001
.97833
1.01523
1.05082
1.08519
1.11841
1.15057
1.18173
1.21194
1.24127
1.26976
1.29746
1.32442
1.35067
1.37624
1.40118
1.42552
1.44927
1.47247
1.49515
1.51732
1.53902
1.56025
1.58104
1.60141
7
.06766
.15700
.23902
.31481
.38526
.45108
.51282
.57098
.62594
.67803
.72755
.77473
.81978
.86289
.90422
.94391
.98208
1.01885
1.05431
1.08856
1.12168
1.15373
1.18479
1.21491
1.24415
1.27257
1.30019
1.32708
1.35325
1.37877
1.40364
1.42792
1.45161
1.47476
1.49739
1.51951
1.54116
1.56235
1.58309
1.60342
8
.07696
.16551
.24686
.32208
.39204
.45742
.51879
.57661
.63127
.68310
.73237
.77932
.82418
.86710
.90826
.94779
.98582
1.02245
1.05779
1.09192
1.12493
1.15688
1.18784
1.21788
1.24703
1.27536
1.30291
1.32972
1.35584
1.38128
1.40610
1.43031
1.45395
1.47705
1.49962
1.52170
1.54330
1.56444
1.58515
1.60543
9
.08618
.17395
.25464
.32930
.39878
.46373
.52473
.58222
.63658
.68813
.73716
.78390
.82855
.87129
.91228
.95166
.98954
1.02604
1.06126
1.09527
1.12817
1.16002
1.19089
1.22083
1.24990
1.27815
1.30563
1.33237
1.35841
1.38379
1.40854
1.43270
1.45629
1.47933
1.50185
1.52388
1.54543
1.56653
1.58719
1.60744
In 10 = 2.30259
2 In 10 = 4.60517
3 In 10 = 6.90776
4 In 10 =
5 In 10 =
6 In 10 =
9.21034
11.51293
13.81551
7 In 10
8 In 10
9 In 10
= 16.11810
= 18.42068
- 20.72327
224

Table 15
(continued)
NATURAL OR NAPIERIAN LOGARITHMS
log x or In л
X
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
0
1.60944
1.62924
1.64866
1.66771
1.68640
1.70475
1.72277
1.74047
1.75786
1.77495
1.79176
1.80829
1.82455
1.84055
1.85630
1.87180
1.88707
1.90211
1.91692
1.93152
1.94591
1.96009
1.97408
1.98787
2.00148
2.01490
2.02815
2.04122
2.05412
2.06686
2.07944
2.09186
2.10413
2.11626
2.12823
2.14007
2.15176
2.16332
2.17475
2.18605
2.19722
2.20827
2.21920
2.23001
2.24071
2.25129
2.26176
2.27213
2.28238
2.29253
1
1.61144
1.63120
1.65058
1.66959
1.68825
1.70656
1.72455
1.74222
1.75958
1.77665
1.79342
1.80993
1.82616
1.84214
1.85786
1.87334
1.88858
1.90360
1.91839
1.93297
1.94734
1.96150
1.97547
1.98924
2.00283
2.01624
2.02946
2.04252
2.05540
2.06813
2.08069
2.09310
2.10535
2.11746
2.12942
2.14124
2.15292
2.16447
2.17589
2.18717
2.19834
2.20937
2.22029
2.23109
2.24177
2.25234
2.26280
2.27316
2.28340
2.29354
2
1.61343
1.63315
1.65250
1.67147
1.69010
1.70838
1.72633
1.74397
1.76130
1.77834
1.79509
1.81156
1.82777
1.84372
1.85942
1.87487
1.89010
1.90509
1.91986
1.93442
1.94876
1.96291
1.97685
1.99061
2.00418
2.01757
2.03078
2.04381
2.05668
2.06939
2.08194
2.09433
2.10657
2.11866
2.13061
2.14242
2.15409
2.16562
2.17702
2.18830
2.19944
2.21047
2.22138
2.23216
2.24284
2.25339
2.26384
2.27419
2.28442
2.29455
3
1.61542
1.63511
1.65441
1.67335
1.69194
1.71019
1.72811
1.74572
1.76302
1.78002
1.79675
1.81319
1.82938
1.84530
1.86097
1.87641
1.89160
1.90658
1.92132
1.93586
1.95019
1.96431
1.97824
1.99198
2.00553
2.01890
2.03209
2.04511
2.05796
2.07065
2.08318
2.09556
2.10779
2.11986
2.13180
2.14359
2.15524
2.16677
2.17816
2.18942
2.20055
2.21157
2.22246
2.23324
2.24390
2.25444
2.26488
2.27521
2.28544
2.29556
4
1.61741
1.63705
1.65632
1.67523
1.69378
1.71199
1.72988
1.74746
1.76473
1.78171
1.79840
1.81482
1.83098
1.84688
1.86253
1.87794
1.89311
1.90806
1.92279
1.93730
1.95161
1.96571
1.97962
1.99334
2.00687
2.02022
2.03340
2.04640
2.05924
2.07191
2.08443
2.09679
2.10900
2.12106
2.13298
2.14476
2.15640
2.16791
2.17929
2.19054
2.20166
2.21266
2.22354
2.23431
2.24496
2.25549
2.26592
2.27624
2.28646
2.29657
5
1.61939
1.63900
1.65823
1.67710
1.69562
1.71380
1.73166
1.74920
1.76644
1.78339
1.80006
1.81645
1.83258
1.84845
1.86408
1.87947
1.89462
1.90954
1.92425
1.93874
1.95303
1.96711
1.98100
1.99470
2.00821
2.02155
2.03471
2.04769
2.06051
2.07317
2.08567
2.09802
2.11021
2.12226
2.13417
2.14593
2.15756
2.16905
2.18042
2.19165
2.20276
2.21375
2.22462
2.23538
2.24601
2.25654
2.26696
2.27727
2.28747
2.29757
6
1.62137
1.64094
1.66013
1.67896
1.69745
1.71560
1.73342
1.75094
1.76815
1.78507
1.80171
1.81808
1.83418
1.85003
1.86563
1.88099
1.89612
1.91102
1.92571
1.94018
1.95445
1.96851
1.98238
1.99606
2.00956
2.02287
2.03601
2.04898
2.06179
2.07443
2.08691
2.09924
2.11142
2.12346
2.13535
2.14710
2.15871
2.17020
2.18155
2.19277
2.20387
2.21485
2.22570
2.23645
2.24707
2.25759
2.26799
2.27829
2.28849
2.29858
7
1.62334
1.64287
1.66203
1.68083
1.69928
1.71740
1.73519
1.75267
1.76985
1.78675
1.80336
1.81970
1.83578
1.85160
1.86718
1.88251
1.89762
1.91250
1.92716
1.94162
1.95586
1.96991
1.98376
1.99742
2.01089
2.02419
2.03732
2.05027
2.06306
2.07568
2.08815
2.10047
2.11263
2.12465
2.13653
2.14827
2.15987
2.17134
2.18267
2.19389
2.20497
2.21594
2.22678
2.23751
2.24813
2.25863
2.26903
2.27932
2.28950
2.29958
8
1.62531
1.64481
1.66393
1.68269
1.70111
1.71919
1.73695
1.75440
1.77156
1.78842
1.80500
1.82132
1.83737
1.85317
1.86872
1.88403
1.89912
1.91398
1.32862
1.94305
1.95727
1.97130
1.98513
1.99877
2.01223
2.02551
2.03862
2.05156
2.06433
2.07694
2.08939
2.10169
2.11384
2.12585
2.13771
2.14943
2.16102
2.17248
2.18380
2.19500
2.20607
2.21703
2.22786
2.23858
2.24918
2.25968
2.27006
2.28034
2.29051
2.30058
9
1.62728
1.64673
1.66582
1.68455
1.70293
1.72098
1.73871
1.75613
1.77326
1.79009
1.80665
1.82294
1.83896
1.85473
1.87026
1.88555
1.90061
1.91545
1.93007
1.94448
1.95869
1.97269
1.98650
2.00013
2.01357
2.02683
2.03992
2.05284
2.06560
2.07819
2.09063
2.10291
2.11505
2.12704
2.13889
2.15060
2.16217
2.17361
2.18493
2.19611
2.20717
2.21812
2.22894
2.23965
2.25024
2.26072
2.27109
2.28136
2.29152
2.30158
225

X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.
5.
6.
7.
8.
9.
10.
0
1.0000
1.1052
1.2214
1.3499
1.4918
1.6487
1.8221
2.0138
2.2255
2.4596
2.7183
3.0042
3.3201
3.6693
4.0552
4.4817
4.9530
5.4739
6.0496
6.6859
7.3891
8.1662
9.0250
9.9742
11.023
12.182
13.464
14.880
16.445
18.174
20.086
22.198
24.533
27.113
29.964
33.115
36.598
40.447
44.701
49.402
54.598
148.41
403.43
1096.6
2981.0
8103.1
22026
1
1.0101
1.1163
1.2337
1.3634
1.5068
1.6653
1.8404
2.0340
2.2479
2.4843
2.7456
3.0344
3.3535
3.7062
4.0960
4.5267
5.0028
5.5290
6.1104
6.7531
7.4633
8.2482
9.1157
10.074
11.134
12.305
13.599
15.029
16.610
18.357
20.287
22.421
24.779
27.385
30.265
33.448
36.966
40.854
45.150
49.899
60.340
164.02
445.86
1212.0
3294.5
8955.3
2
1.0202
1.1275
1.2461
1.3771
1.5220
1.6820
1.8589
2.0544
2.2705
2.5093
2.7732
3.0649
3.3872
3.7434
4.1371
4.5722
5.0531
5.5845
6.1719
6.8210
7.5383
8.3311
9.2073
10.176
11.246
12.429
13.736
15.180
16.777
18.541
20.491
22.646
25.028
27.660
30.569
33.784
37.338
41.264
45.604
50.400
66.686
181.27
492.75
1339.4
3641.0
9897.1
3
1.0305
1.1388
1.2586
1.3910
1.5373
1.6989
1.8776
2.0751
2.2933
2.5345
2.8011
3.0957
3.4212
3.7810
4.1787
4.6182
5.1039
5.6407
6.2339
6.8895
7.6141
8.4149
9.2999
10.278
11.359
12.554
13.874
15.333
16.945
18.728
20.697
22.874
25.280
27.938
30.877
34.124
37.713
41.679
46.063
50.907
73.700
200.34
544.57
1480.3
4023.9
10938
4
1.0408
1.1503
1.2712
1.4049
1.5527
1.7160
1.8965
2.0959
2.3164
2.5600
2.8292
3.1268
3.4556
3.8190
4.2207
4.6646
5.1552
5.6973
6.2965
6.9588
7.6906
8.4994
9.3933
10.381
11.473
12.680
14.013
15.487
17.116
18.916
20.905
23.104
25.534
28.219
31.187
34.467
38.092
42.098
46.525
51.419
81.451
221.41
601.85
1636.0
4447.1
12088
б
1.0513
1.1618
1.2840
1.4191
1.5683
1.7333
1.9155
2.1170
2.3396
2.5857
2.8577
3.1582
3.4903
3.8574
4.2631
4.7115
5.2070
5.7546
6.3598
7.0287
7.7679
8.5849
9.4877
10.486
11.588
12.807
14.154
15.643
17.288
19.106
21.115
23.336
25.790
28.503
31.500
34.813
38.475
42.521
46.993
51.935
90.017
244.69
665.14
1808.0
4914.8
13360
6
1.0618
1.1735
1.2969
1.4333
1.5841
1.7507
1.9348
2.1383
2.3632
2.6117
2.8864
3.1899
3.5254
3.8962
4.3060
4.7588
5.2593
5.8124
6.4237
7.0993
7.8460
8.6711
9.5831
10.591
11.705
12.936
14.296
15.800
17.462
19.298
21.328
23.571
26.050
28.789
31.817
35.163
38.861
42.948
47.465
52.457
99.484
270.43
735.10
1998.2
5431.7
14765
7
1.0725
1.1853
1.3100
1.4477
1.6000
1.7683
1.9542
2.1598
2.3869
2.6379
2.9154
3.2220
3.5609
3.9354
4.3492
4.8066
5.3122
5.8709
6.4883
7.1707
7.9248
8.7583
9.6794
10.697
11.822
13.066
14.440
15.959
17.637
19.492
21.542
23.807
26.311
29.079
32.137
35.517
39.252
43.380
47.942
52.985
109.95
298.87
812.41
2208.3
6002.9
16318
8
1.0833
1.1972
1.3231
1.4623
1.6161
1.7860
1.9739
2.1815
2.4109
2.6645
2.9447
3.2544
3.5966
3.9749
4.3929
4.8550
5.3656
5.9299
6.5535
7.2427
8.0045
8.8463
9.7767
10.805
11.941
13.197
14.585
16.119
17.814
19.688
21.758
24.047
26.576
29.371
32.460
35.874
39.646
43.816
48.424
53.517
121.51
330.30
897.85
2440.6
6634.2
18034
9
1.0942
1.2092
1.3364
1.4770
1.6323
1.8040
1.9937
2.2034
2.4351
2.6912
2.9743
3.2871
3.6328
4.0149
4.4371
4.9037
5.4195
5.9895
6.6194
7.3155
8.0849
8.9352
9.8749
10.913
12.061
13.330
14.732
16.281
17.993
19.886
21.977
24.288
26.843
29.666
32.786
36.234
40.045
44.256
48.911
54.055
134.29
365.04
992.27
2697.3
7332.0
19930
226

X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.
5.
6.
7.
8.
9.
10.
0
1.00000
.90484
.81873
.74082
.67032
.60653
.54881
.49659
.44933
.40657
.36788
.33287
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.27253
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.16530
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.12246
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.08208
.07427
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.05502
.04979
.04505
.04076
.03688
.03337
.03020
.02732
.02472
.02237
.02024
.018316
.0267379
.0224788
.0391188
.0333546
.0312341
.0445400
1
.99005
.89583
.81058
.73345
.66365
.60050
.54335
.49164
.44486
.40252
.36422
.32956
.29820
.26982
.24414
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.19989
.18087
.16365
.14808
.13399
.12124
.10970
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.08127
.07353
.06654
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.04929
.04460
.04036
.03652
.03304
.02990
.02705
.02448
.02215
.02004
.016573
.0260967
.0222429
.0382510
.0330354
.0311167
2
.98020
.88692
.80252
.72615
.65705
.59452
.53794
.48675
.44043
.39852
.36060
.32628
.29523
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.03615
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.02960
.02678
.02423
.02193
.01984
.014996
.0255166
.0220294
.0374659
.0327465
.0310104
3
.97045
.87810
.79453
.71892
.65051
.58860
.53259
.48191
.43605
.39455
.35701
.32303
.29229
.26448
.23931
.21654
.19593
.17728
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.14515
.13134
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.10753
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.07208
.06522
.05901
.05340
.04832
.04372
.03956
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.02930
.02652
.02399
.02171
.01964
.013569
.0249916
.0218363
.0367554
.0324852
.0491424
4
.96079
.86936
.78663
.71177
.64404
.58275
.52729
.47711
.43171
.39063
.35345
.31982
.28938
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.23693
.21438
.19398
.17552
.15882
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.13003
.11765
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.05287
.04783
.04328
.03916
.03544
.03206
.02901
.02625
.02375
.02149
.01945
.012277
.0245166
.0216616
.0361125
.0322487
.0482724
5
.95123
.86071
.77880
.70469
.63763
.57695
.52205
.47237
.42741
.38674
.34994
.31664
.28650
.25924
.23457
.21225
.19205
.17377
.15724
.14227
.12873
.11648
.10640
.09637
.08629
.07808
.07065
.06393
.05784
.05234
.04736
.04285
.03877
.03508
.03175
.02872
.02599
.02352
.02128
.01925
.011109
.0240868
.0215034
.0355308
.0320347
.0*74852
6
.94176
.85214
.77105
.69768
.63128
.57121
.51685
.46767
.42316
.38289
.34646
.31349
.28365
.25666
.23224
.21014
.19014
.17204
¦.15567
.14086
.12745
.11533
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.06329
.05727
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.03839
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.02844
.02573
.02328
.02107
.01906
.010052
.0236979
.0213604
.0350045
.0318411
.0467729
7
.93239
.84366
.76338
.69073
.62500
.56553
.51171
.46301
.41895
.37908
.34301
.31037
.28083
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.02816
.02548
.02305
.02086
.01887
.0290953
.0233460
.0П2309
.0345283
.0316659
.0461283
8
.92312
.83527
.75578
.68386
.61878
.55990
.50662
.45841
.41478
.37531
.33960
.30728
.27804
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.02282
.02065
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.0211138
.0340973
.0315073
.0455452
9
.91393
.82696
.74826
.67706
.61263
.55433
.50158
.45384
.41066
.37158
.33622
.30422
.27527
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.18452
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.02760
.02497
.02260
.02045
.01850
.0274466
.0227394
.02W078
.0337074
.0313639
.0450175
227

X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0
.0000
.1002
.2013
.3045
.4108
.5211
.6367
.7586
.8881
1.0265
1.1752
1.3356
1.5095
1.6984
1.9043
2.1293
2.3766
2.6456
2.9422
3.2682
3.6269
4.0219
4.4571
4.9370
5.4662
6.0502
6.6947
7.4063
8.1919
9.0596
1
.0100
.1102
.2115
.3150
.4216
.5324
.6485
.7712
.9015
1.0409
1.1907
1.3524
1.5276
1.7182
1.9259
2.1529
2.4015
2.6740
2.9734
3.3025
3.6647
4.0635
4.5030
4.9876
5.5221
6.1118
6.7628
7.4814
8.2749
9.1512
2
.0200
.1203
.2218
.3255
.4325
.5438
.6605
.7838
.9150
1.0554
1.2063
1.3693
1.5460
1.7381
1.9477
2.1768
2.4276
2.7027
3.0049
3.3372
3.7028
4.1056
4.5494
5.0387
5.5785
6.1741
6.8315
7.5572
8.3586
9.2437
3
.0300
.1304
.2320
.3360
.4434
.5552
.6725
.7966
.9286
1.0700
1.2220
1.3863
1.5645
1.7583
1.9697
2.2008
2.4540
2.7317
3.0367
3.3722
3.7414
4.1480
4.5962
5.0903
5.6354
6.2369
6.9008
7.6338
8.4432
9.3371
4
.0400
.1405
.2423
.3466
.4543
.5666
.6846
.8094
.9423
1.0847
1.2379
1.4035
1.5831
1.7786
1.9919
2.2251
2.4806
2.7609
3.0689
3.4075
3.7803
4.1909
4.6434
5.1425
5.6929
6.3004
6.9709
7.7112
8.5287
9.4315
5
.0500
.1506
.2526
.3572
.4653
.5782
.6967
.8223
.9561
1.0995
1.2539
1.4208
1.6019
1.7991
2.0143
2.2496
2.5075
2.7904
3.1013
3.4432
3.8196
4.2342
4.6912
5.1951
5.7510
6.3645
7.0417
7.7894
8.6150
9.5268
6
.0600
.1607
.2629
.3678
.4764
.5897
.7090
.8353
.9700
1.1144
1.2700
1.4382
1.6209
1.8198
2.0369
2.2743
2.5346
2.8202
3.1340
3.4792
3.8593
4.2779
4.7394
5.2483
5.8097
6.4293
7.1132
7.8683
8.7021
9.6231
7
.0701
.1708
.2733
.3785
.4875
.6014
.7213
.8484
.9840
1.1294
1.2862
1.4558
1.6400
1.8406
2.0597
2.2993
2.5620
2.8503
3.1671
3.5156
3.8993
4.3221
4.7880
5.3020
5.8689
6.4946
7.1854
7.9480
8.7902
9.7203
8
.0801
.1810
.2837
.3892
.4986
.6131
.7336
.8615
.9981
1.1446
1.3025
1.4735
1.6593
1.8617
2.0827
2.3245
2.5896
2.8806
3.2005
3.5523
3.9398
4.3666
4.8372
5.3562
5.9288
6.5607
7.2583
8.0285
8.8791
9.8185
9
.0901
.1911
.2941
.4000
.5098
.6248
.7461
.8748
1.0122
1.1598
1.3190
1.4914
1.6788
1.8829
2.1059
2.3499
2.6175
2.9112
3.2341
3.5894
3.9806
4.4116
4.8868
5.4109
5.9892
6.6274
7.3319
8.1098
8.9689
9.9177
228

Table 18a
(continued)
HYPERBOLIC FUNCTIONS
sinh x
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
0
10.018
11.076
12.246
13.538
14.965
16.543
18.285
20.211
22.339
24.691
27.290
30.162
33.336
36.843
40.719
45.003
49.737
54.969
60.751
67.141
74.203
82.008
90.633
100.17
110.70
122.34
135.21
149.43
165.15
182.52
1
10.119
11.188
12.369
13.674
15.116
16.709
18.470
20.415
22.564
24.939
27.564
30.465
33.671
37.214
41.129
45.455
50.237
55.522
61.362
67.816
74.949
82.832
91.544
101.17
111.81
123.57
136.57
150.93
166.81
184.35
2
10.221
11.301
12.494
13.812
15.268
16.877
18.655
20.620
22.791
25.190
27.842
30.772
34.009
37.588
41.542
45.912
50.742
56.080
61.979
68.498
75.702
83.665
92.464
102.19
112.94
124.82
137.94
152.45
168.48
186.20
3
10.324
11.415
12.620
13.951
15.422
17.047
18.843
20.828
23.020
25.444
28.122
31.081
34.351
37.965
41.960
46.374
51.252
56.643
62.601
69.186
76.463
84.506
93.394
103.22
114.07
126.07
139.33
153.98
170.18
188.08
4
10.429
11.530
12.747
14.092
15.577
17.219
19.033
21.037
23.252
25.700
28.404
31.393
34.697
38.347
42.382
46.840
51.767
57.213
63.231
69.882
77.232
85.355
94.332
104.25
115.22
127.34
140.73
155.53
171.89
189.97
5
10.534
11.647
12.876
14.234
15.734
17.392
19.224
21.249
23.486
25.958
28.690
31.709
35.046
38.733
42.808
47.311
52.288
57.788
63.866
70.584
78.008
86.213
95.281
105.30
116.38
128.62
142.14
157.09
173.62
191.88
6
10.640
11.764
13.006
14.377
15.893
17.567
19.418
21.463
23.722
26.219
28.979
32.028
35.398
39.122
43.238
47.787
52.813
58.369
64.508
71.293
78.792
87.079
96.238
106.36
117.55
129.91
143.57
158.67
175.36
193.80
7
10.748
11.883
13.137
14.522
16.053
пли
19.613
21.679
23.961
26.483
29.270
32.350
35.754
39.515
43.673
48.267
53.344
58.995
65.157
72.010
79.584
87.955
97.205
107.43
118.73
131.22
145.02
160.27
177.12
195.75
8
10.856
12.003
13.269
14.668
16.215
17.923
19.811
21.897
24.202
26.749
29.564
32.675
36.113
39.913
44.112
48.752
53.880
59.548
65.812
72.734
80.384
88.839
98.182
108.51
119.92
132.53
146.47
161.88
178.90
197.72
9
10.966
12.124
13.403
14.816
16.378
18.103
20.010
22.117
24.445
27.018
29.862
33.004
36.476
40.314
44.555
49.242
54.422
60.147
66.473
73.465
81.192
89.732
99.169
109.60
121.13
133.87
147.95
163.51
180.70
199.71
229

X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0
1.0000
1.0050
1.0201
1.0453
1.0811
1.1276
1.1855
1.2552
1.3374
1.4331
1.5431
1.6685
1.8107
1.9709
2.1509
2.3524
2.5775
2.8283
3.1075
3.4177
3.7622
4.1443
4.5679
5.0372
5.5569
6.1323
6.7690
7.4735
8.2527
9.1146
1
1.0001
1.0061
1.0221
1.0484
1.0852
1.1329
1.1919
1.2628
1.3464
1.4434
1.5549
1.6820
1.8258
1.9880
2.1700
2.3738
2.6013
2.8549
3.1371
3.4506
3.7987
4.1847
4.6127
5.0868
5.6119
6.1931
6.8363
7.5479
8.3351
9.2056
2
1.0002
1.0072
1.0243
1.0516
1.0895
1.1383
1.1984
1.2706
1.3555
1.4539
1.5669
1.6956
1.8412
2.0053
2.1894
2.3955
2.6255
2.8818
3.1669
3.4838
3.8355
4.2256
4.6580
5.1370
5.6674
6.2545
6.9043
7.6231
8.4182
9.2976
3
1.0005
1.0085
1.0266
1.0549
1.0939
1.1438
1.2051
1.2785
1.3647
1.4645
1.5790
1.7093
1.8568
2.0228
2.2090
2.4174
2.6499
2.9090
3.1972
3.5173
3.8727
4.2669
4.7037
5.1876
5.7235
6.3166
6.9729
7.6991
8.5022
9.3905
4
1.0008
1.0098
1.0289
1.0584
1.0984
1.1494
1.2119
1.2865
1.3740
1.4753
1.5913
1.7233
1.8725
2.0404
2.2288
2.4395
2.6746
2.9364
3.2277
3.5512
3.9103
4.3085
4.7499
5.2388
5.7801
6.3793
7.0423
7.7758
8.5871
9.4844
5
1.0013
1.0113
1.0314
1.0619
1.1030
1.1551
1.2188
1.2947
1.3835
1.4862
1.6038
1.7374
1.8884
2.0583
2.2488
2.4619
2.6995
2.9642
3.2585
3.5855
3.9483
4.3507
4.7966
5.2905
5.8373
6.4426
7.1123
7.8533
8.6728
9.5791
6
1.0018
1.0128
1.0340
1.0655
1.1077
1.1609
1.2258
1.3030
1.3932
1.4973
1.6164
1.7517
1.9045
2.0764
2.2691
2.4845
2.7247
2.9922
3.2897
3.6201
3.9867
4.3932
4.8437
5.3427
5.8951
6.5066
7.1831
7.9316
8.7594
9.6749
7
1.0025
1.0145
1.0367
1.0692
1.1125
1.1669
1.2330
1.3114
1.4029
1.5085
1.6292
1.7662
1.9208
2.0947
2.2896
2.5073
2.7502
3.0206
3.3212
3.6551
4.0255
4.4362
4.8914
5.3954
5.9535
6.5712
7.2546
8.0106
8.8469
9.7716
8
1.0032
1.0162
1.0395
1.0731
1.1174
1.1730
1.2402
1.3199
1.4128
1.5199
1.6421
1.7808
1.9373
2.1132
2.3103
2.5305
2.7760
3.0492
3.3530
3.6904
4.0647
4.4797
4.9395
5.4487
6.0125
6.6365
7.3268
8.0905
8.9352
9.8693
9
1.0041
1.0181
1.0423
1.0770
1.1225
1.1792
1.2476
1.3286
1.4229
1.5314
1.6552
1.7957
1.9540
2.1320
2.3312
2.5538
2.8020
3.0782
3.3852
3.7261
4.1043
4.5236
4.9881
5.5026
6.0721
6.7024
7.3998
8.1712
9.0244
9.9680
230

Table 18b
(continued)
HYPERBOLIC FUNCTIONS
cosh .v
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
0
10.068
11.121
12.287
13.575
14.999
16.573
18.313
20.236
22.362
24.711
27.308
30.178
33.351
36.857
40.732
45.014
49.747
54.978
60.759
67.149
74.210
82.014
90.639
100.17
110.71
122.35
135.22
149.44
165.15
182.52
1
10.168
11.233
12.410
13.711
15.149
16.739
18.497
20.439
22.586
24.959
27.583
30.482
33.686
37.227
41.141
45.466
50.247
55.531
61.370
67.823
74.956
82.838
91.550
101.18
111.82
123.58
136.57
150.94
166.81
184.35
2
10.270
11.345
12.534
13.848
15.301
16.907
18.682
20.644
22.813
25.210
27.860
30.788
34.024
37.601
41.554
45.923
50.752
56.089
61.987
68.505
75.709
83.671
92.470
102.19
112.94
124.82
137.95
152.45
168.49
186.21
3
10.373
11.459
12.660
13.987
15.455
17.077
18.870
20.852
23.042
25.463
28.139
31.097
34.366
37.979
41.972
46.385
51.262
56.652
62.609
69.193
76.470
84.512
93.399
103.22
114.08
126.07
139.33
153.99
170.18
188.08
4
10.476
11.574
12.786
14.127
15.610
17.248
19.059
21.061
23.273
25.719
28.422
31.409
34.711
38.360
42.393
46.851
51.777
57.221
63.239
69.889
77.238
85.361
94.338
104.26
115.22
127.34
140.73
155.53
171.89
189.97
5
10.581
11.689
12.915
14.269
15.766
17.421
19.250
21.272
23.507
25.977
28.707
31.725
35.060
38.746
42.819
47.321
52.297
57.796
63.874
70.591
78.014
86.219
95.286
105.31
116.38
128.62
142.15
157.10
173.62
191.88
6
10.687
11.806
13.044
14.412
15.924
17.596
19.444
21.486
23.743
26.238
28.996
32.044
35.412
39.135
43.250
47.797
52.823
58.377
64.516
71.300
78.798
87.085
96.243
106.67
117.55
129.91
143.58
158.68
175.36
193.81
7
10.794
11.925
13.175
14.556
16.084
17.772
19.639
21.702
23.982
26.502
29.287
32.365
35.768
39.528
43.684
48.277
53.354
58.964
65.164
72.017
79.590
87.960
97.211
107.43
118.73
131.22
145.02
160.27
177.13
195.75
8
10.902
12.044
13.307
14.702
16.245
17.951
19.836
21.919
24.222
26.768
29.581
32.691
36.127
39.925
44.123
48.762
53.890
59.556
65.819
72.741
80.390
88.844
98.188
108.51
119.93
132.54
146.48
161.88
178.91
197.72
9
11.011
12.165
13.440
14.850
16.408
18.131
20.035
22.139
24.466
27.037
29.878
33.019
36.490
40.326
44.566
49.252
54.431
60.155
66.481
73.472
81.198
89.737
99.174
109.60
121.13
133.87
147.95
163.51
180.70
199.71
231

X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0
.00000
.09967
.19738
.29131
.37995
.46212
.53705
.60437
.66404
.71630
.76159
.80050
.83365
.86172
.88535
.90515
.92167
.93541
.94681
.95624
.96403
.97045
.97574
.98010
.98367
.98661
.98903
.99101
.99263
.99396
1
.01000
.10956
.20697
.30044
.38847
.46995
.54413
.61068
.66959
.72113
.76576
.80406
.83668
.86428
.88749
.90694
.92316
.93665
.94783
.95709
.96473
.97103
.97622
.98049
.98400
.98688
.98924
.99118
.99278
.99408
2
.02000
.11943
.21652
.30951
.39693
.47770
.55113
.61691
.67507
.72590
.76987
.80757
.83965
.86678
.88960
.90870
.92462
.93786
.94884
.95792
.96541
.97159
.97668
.98087
.98431
.98714
.98946
.99136
.99292
.99420
3
.02999
.12927
.22603
.31852
.40532
.48538
.55805
.62307
.68048
.73059
.77391
.81102
.84258
.86925
.89167
.91042
.92606
.93906
.94983
.95873
.96609
.97215
.97714
.98124
.98462
.98739
.98966
.99153
.99306
.99431
4
.03998
.13909
.23550
.32748
.41364
.49299
.56490
.62915
.68581
.73522
.77789
.81441
.84546
.87167
.89370
.91212
.92747
.94023
.95080
.95953
.96675
.97269
.97759
.98161
.98492
.98764
.98987
.99170
.99320
.99443
5
.04996
.14889
.24492
.33638
.42190
.50052
.57167
.63515
.69107
.73978
.78181
.81775
.84828
.87405
.89569
.91379
.92886
.94138
.95175
.96032
.96740
.97323
.97803
.98197
.98522
.98788
.99007
.99186
.99333
.99454
6
.05993
.15865
.25430
.34521
.43008
.50798
.57836
.64108
.69626
.74428
.78566
.82104
.85106
.87639
.89765
.91542
.93022
.94250
.95268
.96109
.96803
.97375
.97846
.98233
.98551
.98812
.99026
.99202
.99346
.99464
7
.06989
.16838
.26362
.35399
.43820
.51536
.58498
.64693
.70137
.74870
.78946
.82427
.85380
.87869
.89958
.91703
.93155
.94361
.95359
.96185
.96865
.97426
.97888
.98267
.98579
.98835
.99045
.99218
.99359
.99475
8
.07983
.17808
.27291
.36271
.44624
.52267
.59152
.65271
.70642
.75307
.79320
.82745
.85648
.88095
.90147
.91860
.93286
.94470
.95449
.96259
.96926
.97477
.97929
.98301
.98607
.98858
.99064
.99233
.99372
.99485
9
.08976
.18775
.28213
.37136
.45422
.52990
.59798
.65841
.71139
.75736
.79688
.83058
.85913
.88317
.90332
.92015
.93415
.94576
.95537
.96331
.96986
.97526
.97970
.98335
.98635
.98881
.99083
.99248
.99384
.99496
232

Table 18c
(continued)
HYPERBOLIC FUNCTIONS
tanh x
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
0
.99505
.99595
.99668
.99728
.99777
.99818
.99851
.99878
.99900
.99918
.99933
.99945
.99955
.99963
.99970
.99975
.99980
.99983
.99986
.99989
.99991
.99993
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99998
1
.99515
.99603
.99675
.99734
.99782
.99821
.99853
.99880
.99902
.99920
.99934
.99946
.99956
.99964
.99970
.99976
.99980
.99984
.99987
.99989
.99991
.99993
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99999
2
.99525
.99611
.99681
.99739
.99786
.99825
.99857
.99883
.99904
.99921
.99936
.99947
.99957
.99965
.99971
.99976
.99981
.99984
.99987
.99990
.99991
.99993
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99999
3
.99534
.99618
.99688
.99744
.99790
.99828
.99859
.99885
.99906
.99923
.99937
.99948
.99958
.99966
.99972
.99977
.99981
.99984
.99987
.99990
.99991
.99993
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99999
4
.99543
.99626
.99694
.99749
.99795
.99832
.99862
.99887
.99908
.99924
.99938
.99949
.99958
.99966
.99972
.99977
.99981
.99985
.99987
.99990
.99992
.99993
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99999
5
.99552
.99633
.99700
.99754
.99799
.99835
.99865
.99889
.99909
.99926
.99939
.99950
.99959
.99967
.99973
.99978
.99982
.99985
.99988
.99990
.99992
.99993
.99994
.99995
.99996
.99997
.99998
.99998
.99998
.99999
6
.99561
.99641
.99706
.99759
.99803
.99838
.99868
.99892
.99911
.99927
.99941
.99951
.99960
.99967
.99973
.99978
.99982
.99985
.99988
.99990
.99992
.99993
.99995
.99996
.99996
.99997
.99998
.99998
.99998
.99999
7
.99570
.99648
.99712
.99764
.99807
.99842
.99870
.99894
.99913
.99929
.99942
.99952
.99961
.99968
.99974
.99979
.99982
.99986
.99988
.99990
.99992
.99994
.99995
.99996
.99996
.99997
.99998
.99998
.99998
.99999
8
.99578
.99655
.99717
.99768
.99810
.99845
.99873
.99896
.99915
.99930
.99943
.99953
.99962
.99969
.99974
.99979
.99983
.99986
.99988
.99991
.99992
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99998
.99999
9
.99587
.99662
.99723
.99773
.99814
.99848
.99875
.99898
.99916
.99932
.99944
.99954
.99962
.99969
.99975
.99979
.99983
.99986
.99989
.99991
.99992
.99994
.99995
.99996
.99997
.99997
.99998
.99998
.99998
.99999
233

TABLE
19
FACTORIAL n
n! = l-2«3 n
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
n\
1 (by definition)
1
2
6
24
120
720
5040
40,320
362,880
3,628,800
39,916,800
479,001,600
6,227,020,800
87,178,291,200
1,307,674,368,000
20,922,789,888,000
355,687,428,096,000
6,402,373,705,728,000
121,645,100,408,832,000
2,432,902,008,176,640,000
51,090,942,171,709,440,000
1,124,000,727,777,607,680,000
25,852,016,738,884,976,640,000
620,448,401,733,239,439,360,000
15,511,210,043,330,985,984,000,000
403,291,461,126,605,635,584,000,000
10,888,869,450,418,352,160,768,000,000
304,888,344,611,713,860,501,504,000,000
8,841,761,993,739,701,954,543,616,000,000
265,252,859,812,191,058,636,308,480,000,000
8.22284 X 1033
2.63131 X 1035
8.68332 X 1036
2.95233 X 1038
1.03331 У 1040
3.71993 X iO41
1.37638 X 1043
Б.23023 X 1044
2.03979 X 1046
n
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
n\
8.15915 X
3.34525 X
1.40501 X
6.04153 X
2.65827 X
1.19622 X
5.50262 X
2.58623 X
1.24139 X
6.08282 X
3.04141 X
1.55112 X
8.06582 X
4.27488 X
2.30844 X
1.26964 X
7.10999 X
4.05269 X
2.35056 X
1.38683 X
8.32099 X
5.07580 X
3.14700 X
1.98261 X
1.26887 X
8.24765 X
5.44345 X
3.64711 X
2.48004 X
1.71122 X
1.19786 X
8.50479 X
6.12345 X
4.47012 X
3.30789 X
2.48091 X
1.88549 X
1.45183 X
1.13243 X
8.94618 X
1047
1Q49
1051
1Q52
1054
1O56
1057
1Q59
1081
1O62
1O64
Ю66
1O67
1O69
1071
Ю73
1074
1O76
1Q78
lO8O
10"
Ю83
Ю85
10»7
1O89
1090
Ю92
1O94
1O96
Ю98
10100
1Q101
1Q103
Ю105
Ю107
Ю109
10Ш
10ПЗ
Ю115
10U6
n
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
n\
7.15695
5.79713
4.75364
3.94552
3.31424
2.81710
2.42271
2.10776
1.85483
1.65080
1.48572
1.35200
1.24384
1.15677
1.08737
1.03300
9.91678
9.61928
9.42689
9.33262
9.33262
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Ю118
Ю120
10122
10124
1Q126
1Q128
10130
10132
lOis*
10136
1Q138
1Q140
1Q142
10'**
1Q146
Ю148
Ю149
10151
Ю153
Ю155
10157
234

TABLE
20
GAMMA FUNCTION
T(x) = | Ьх-Ч~гш for 1 s ж g 2
*s о
[For other values use the formula T(x + 1) = x Y(x)]
X
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
T(x)
1.00000
.99433
.98884
.98355
.97844
.97350
.96874
.96415
.95973
.95546
.95135
.94740
.94359
.93993
.93642
.93304
.92980
.92670
.92373
.92089
.91817
.91558
.91311
.91075
.90852
.90640
.90440
.90250
.90072
.89904
.89747
.89600
.89464
.89338
.89222
.89115
.89018
.88931
.88854
.88785
.88726
.88676
.88636
.88604
.88581
.88566
.88560
.88563
.88575
.88595
.88623
X
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.70
1.71
1.72
1.73
1.74
1.75
1.76
1.77
1.78
1.79
1.80
1.81
1.82
1.83
1.84
1.85
1.86
1.87
1.88
1.89
1.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2.00
Г(х)
.88623
.88659
.88704
.88757
.88818
.88887
.88964
.89049
.89142
.89243
.89352
.89468
.89592
.89724
.89864
.90012
.90167
.90330
.90500
.90678
.90864
.91057
.91258
.91467
.91683
.91906
.92137
.92376
.92623
.92877
.93138
.93408
.93685
.93969
.94261
.94561
.94869
.95184
.95507
.95838
.96177
.96523
.96877
.97240
.97610
.97988
.98374
.98768
.99171
.99581
1.00000
235

TABLE
21
BINOMIAL COEFFICIENTS
_i _
kl(n-k)l
/r!
n~k
0! =
Note that each number is the sum of two numbers in the row above; one of these numbers is in the same
column and the other is in the preceding column [e.g. 56 = 35 + 21]. The arrangement is often called
Pascal's triangle [see 3.6, page 4].
\ к
n \^
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
2
1
3
6
10
15
21
28
36
45
55
66
78
91
105
120
136
153
171
190
210
231
253
276
300
325
351
378
406
435
3
1
4
10
20
35
56
84
120
165
220
286
364
455
560
680
816
969
1140
1330
1540
1771
2024
2300
2600
2925
3276
3654
4060
4
1
5
15
35
70
126
210
330
495
715
1001
1365
1820
2380
3060
3876
4845
5985
7315
8865
10626
12650
14950
17550
20475
23751
27405
5
1
6
21
56
126
252
462
792
1287
2002
3003
4368
6188
8568
11628
15504
20349
26334
33649
42504
53130
65780
80730
98280
118755
142506
6
1
7
28
84
210
462
924
1716
3003
5005
8008
12376
18564
27132
38760
54264
74613
100947
134596
177100
230230
296010
376740
475020
593775
7
1
8
36
120
330
792
1716
3432
6435
11440
19448
31824
50388
77520
116280
170544
245157
346104
480700
657800
888030
1184040
1560780
2035800
8
1
9
45
165
495
1287
3003
6435
12870
24310
43758
75582
125970
203490
319770
490314
735471
1081575
1562275
2220075
3108105
4292145
5852925
9
1
10
55
220
715
2002
5005
11440
24310
48620
92378
167960
293930
497420
817190
1307504
2042975
3124550
4686825
6906900
10015005
14307150
236

Table 21
(continued)
BINOMIAL COEFFICIENTS
n! _ n(n — 1) • • • (n — к + 1)
kl(n-k)\ ~ fe!
n
n — к
0! =
\ к
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10
1
11
66
286
1001
3003
8008
19448
43758
92378
184756
352716
646646
1144066
1961256
3268760
5311735
8436285
13123110
20030010
30045015
11
1
12
78
364
1365
4368
12376
31824
75582
167960
352716
705432
1352078
2496144
4457400
7726160
13037895
21474180
34597290
54627300
12
1
13
91
455
1820
6188
18564
50388
125970
293930
646646
1352078
2704156
5200300
9657700
17383860
30421755
51895935
86493225
13
1
14
105
560
2380
8568
27132
77520
203490
497420
1144066
2496144
5200300
10400600
20058300
37442160
67863915
119759850
14
1
15
120
680
3060
11628
38760
116280
319770
817190
1961256
4457400
9657700
20058300
40116600
77558760
145422675
15
1
16
136
816
3876
15504
54264
170544
490314
1307504
3268760
7726160
17383860
37442160
77558760
155117520
For к > 15 use the fact that
n
n — к
237

и
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
729
784
841
900
961
1 024
1 089
1 156
1 225
1 296
1 369
1 444
1 521
1 600
1 681
1 764
1 849
1 936
2 025
2 116
2 209
2 304
2 401
2 500
п?
1
8
27
64
125
216
343
512
729
1 000
1 331
1 728
2 197
2 744
3 375
4 096
4 913
5 832
6 859
8 000
9 261
10 648
12 167
13 824
15 625
17 576
19 683
21 952
24 389
27 000
29 791
32 768
35 937
39 304
42 875
46 656
50 653
54 872
59 319
64 000
68 921
74 088
79 507
85 184
91 125
97 336
103 823
110 592
117 649
125 000
\рп
1.000 000
1.414 214
1.732 051
2.000 000
2.236 068
2.449 490
2.645 751
2.828 427
3.000 000
3.162 278
3.316 625
3.464 102
3.605 551
3.741 657
3.872 983
4.000 000
4.123 106
4.242 641
4.358 899
4.472 136
4.582 576
4.690 416
4.795 832
4.898 979
5.000 000
5.099 020
5.196 152
5.291 503
5.385 165
5.477 226
5.567 764
5.656 854
5.744 563
5.830 952
5.916 080
6.000 000
6.082 763
6.164 414
6.244 998
6.324 555
6.403 124
6.480 741
6.557 439
6.633 250
6.708 204
6.782 330
6.855 655
6.928 203
7.000 000
7.071 068
VlOw
3.162 278
4.472 136
5.477 226
6.324 555
7.071 068
7.745 967
8.366 600
8.944 272
9.486 833
10.000 00
10.488 09
10.954 45
11.401 75
11.832 16
12.247 45
12.649 11
13.038 40
13.416 41
13.784 05
14.142 14
14.491 38
14.832 40
15.165 75
15.491 93
15.811 39
16.124 52
16.431 68
16.733 20
17.029 39
17.320 51
17.606 82
17.888 54
18.165 90
18.439 09
18.708 29
18.973 67
19.235 38
19.493 59
19.748 42
20.000 00
20.248 46
20.493 90
20.736 44
20.976 18
21.213 20
21.447 61
21.679 48
21.908 90
22.135 94
22.360 68
3Г
уп
1.000 000
1.259 921
1.442 250
1.587 401
1.709 976
1.817 121
1.912 931
2.000 000
2.080 084
2.154 435
2.223 980
2.289 428
2.351 335
2.410 142
2.466 212
2.519 842
2.571 282
2.620 741
2.668 402
2.714 418
2.758 924
2.802 039
2.843 867
2.884 499
2.924 018
2.962 496
3.000 000
3.036 589
3.072 317
3.107 233
3.141 381
3.174 802
3.207 534
'3.239 612
3.271 066
3.301 927
3.332 222
3.361 975
3.391 211
3.419 952
3.448 217
3.476 027
3.503 398
3.530 348
3.556 893
3.583 048
3.608 826
3.634 241
3.659 306
3.684 031
з,
yiOn
2.154 435
2.714 418
3.107 233
3.419 952
3.684 031
3.914 868
4.121 285
4.308 869
4.481 405
4.641 589
4.791 420
4.932 424
5.065 797
5.192 494
5.313 293
5.428 835
5.539 658
5.646 216
5.748 897
5.848 035
5.943 922
6.036 811
6.126 926
6.214 465
6.299 605
6.382 504
6.463 304
6.542 133
6.619 106
6.694 330
6.767 899
6.839 904
6.910 423
6.979 532
7.047 299
7.113 787
7.179 054
7.243 156
7.306 144
7.368 063
7.428 959
7.488 872
7.547 842
7.605 905
7.663 094
7.719 443
7.774 980
7.829 735
7.883 735
7.937 005
3/
VIOOn
4.641 589
5.848 035
6.694 330
7.368 063
7.937 005
8.434 327
8.879 040
9.283 178
9.654 894
10.000 00
10.322 80
10.626 59
10.913 93
11.186 89
11.447 14
11.696 07
11.934 83
12.164 40
12.385 62
12.599 21
12.805 79
13.005 91
13.200 06
13.388 66
13.572 09
13.750 69
13.924 77
14.094 60
14.260 43
14.422 50
14.581 00
14.736 13
14.888 06
15.036 95
15.182 94
15.326 19
15.466 80
15.604 91
15.740 61
15.874 01
16.005 21
16.134 29
16.261 33
16.386 43
16.509 64
16.631 03
16.750 69
16.868 65
16.984 99
17.099 76
1/п
1.000 000
.500 000
.333 333
.250 000
.200 000
.166 667
.142 857
.125 000
.111 111
.100 000
.090 909
.083 333
.076 923
.071 429
.066 667
.062 500
.058 824
.055 556
.052 632
.050 000
.047 619
.045 455
.043 478
.041 667
.040 000
.038 462
.037 037
.035 714
.034 483
.033 333
.032 258
.031 250
.030 303
.029 412
.028 571
.027 778
.027 027
.026 316
.025 641
.025 000
.024 390
.023 810
.023 256
.022 727
.022 222
.021 739
.021 277
.020 833
.020 408
.020 000
238

Table 22
(continued)
SQUARES, CUBES, ROOTS AND RECIPROCALS
n
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
n2
2 500
2 601
2 704
2 809
2 916
3 025
3 136
3 249
3 364
3 481
3 600
3 721
3 844
3 969
4 096
4 225
4 356
4 489
4 624
4 761
4 900
5 041
5 184
5 329
5 476
5 625
5 776
5 929
6 084
6 241
6 400
6 561
6 724
6 889
7 056
7 225
7 396
7 569
7 744
7 921
8 100
8 281
8 464
8 649
8 836
9 025
9 216
9 409
9 604
9 801
10 000
125 000
132 651
140 608
148 877
157 464
166 375
175 616
185 193
195 112
205 379
216 000
226 981
238 328
250 047
262 144
274 625
287 496
300 763
314 432
328 509
343 000
357 911
373 248
389 017
405 224
421 875
438 976
456 533
474 552
493 039
512 000
531 441
551 368
571 787
592 704
614 125
636 056
658 503
681 472
704 969
729 000
753 571
778 688
804 357
830 584
857 375
884 736
912 673
941 192
970 299
1 000 000
7.071 068
7.141 428
7.211 103
7.280 110
7.348 469
7.416 198
7.483 315
7.549 834
7.615 773
7.681 146
7.745 967
7.810 250
7.874 008
7.937 254
8.000 000
8.062 258
8.124 038
8.185 353
8.246 211
8.306 624
8.366 600
8.426 150
8.485 281
8.544 004
8.602 325
8.660 254
8.717 798
8.774 964
8.831 761
8.888 194
8.944 272
9.000 000
9.055 385
9.110 434
9.165 151
9.219 544
9.273 618
9.327 379
9.380 832
9.433 981
9.486 833
9.539 392
9.591 663
9.643 651
9.695 360
9.746 794
9.797 959
9.848 858
9.899 495
9.949 874
10.00 000
VUn
22.360 68
22.583 18
22.803 51
23.021 73
23.237 90
23.452 08
23.664 32
23.874 67
24.083 19
24.289 92
24.494 90
24.698 18
24.899 80
25.099 80
25.298 22
25.495 10
25.690 47
25.884 36
26.076 81
26.267 85
26.457 51
26.645 83
26.832 82
27.018 51
27.202 94
27.386 13
27.568 10
27.748 87
27.928 48
28.106 94
28.284 27
28.460 50
28.635 64
28.809 72
28.982 75
29.154 76
29.325 76
29.495 76
29.664 79
29.832 87
30.000 00
30.166 21
30.331 50
30.495 90
30.659 42
30.822 07
30.983 87
31.144 82
31.304 95
31.464 27
31.622 78
zr
v»
3.684 031
3.708 430
3.732 511
3.756 286
3.779 763
3.802 952
3.825 862
3.848 501
3.870 877
3.892 996
3.914 868
3.936 497
3.957 892
3.979 057
4.000 000
4.020 726
4.041 240
4.061 548
4.081 655
4.101 566
4.121 285
4.140 818
4.160 168
4.179 339
4.198 336
4.217 163
4.235 824
4.254 321
4.272 659
4.290 840
4.308 869
4.326 749
4.344 481
4.362 071
4.379 519
4.396 830
4.414 005
4.431 048
4.447 960
4.464 745
4.481 405
4.497 941
4.514 357
4.530 655
4.546 836
4.562 903
4.578 857
4.594 701
4.610 436
4.626 065
4.641 589
3r
\/10n
7.937 005
7.989 570
8.041 452
8.092 672
8.143 253
8.193 213
8.242 571
8.291 344
8.339 551
8.387 207
8.434 327
8.480 926
8.527 019
8.572 619
8.617 739
8.662 391
8.706 588
8.750 340
8.793 659
8.836 556
8.879 040
8.921 121
8.962 809
9.004 113
9.045 042
9.085 603
9.125 805
9.165 656
9.205 164
9.244 335
9.283 178
9.321 698
9.359 902
9.397 796
9.435 388
9.472 682
9.509 685
9.546 403
9.582 840
9.619 002
9.654 894
9.690 521
9.725 888
9.761 000
9.795 861
9.830 476
9.864 848
9.898 983
9.932 884
9.966 555
10.00 000
V'lOOn
17.099 76
17.213 01
17.324 78
17.435 13
17.544 11
17.651 74
17.758 08
17.863 16
17.967 02
18.069 69
18.171 21
18.271 60
18.370 91
18.469 15
18.566 36
18.662 56
18.757 77
18.852 04
18.945 36
19.037 78
19.129 31
19.219 97
19.309 79
19.398 77
19.486 95
19.574 34
19.660 95
19.746 81
19.831 92
19.916 32
20.000 00
20.082 99
20.165 30
20.246 94
20.327 93
20.408 28
20.488 00
20.567 10
20.645 60
20.723 51
20.800 84
20.877 59
20.953 79
21.029 44
21.104 54
21.179 12
21.253 17
21.326 71
21.399 75
21.472 29
21.544 35
1/n
.020 000
.019 608
.019 231
.018 868
.018 519
.018 182
.017 857
.017 544
.017 241
.016 949
.016 667
.016 393
.016 129
.015 873
.015 625
.015 385
.015 152
.014 925
.014 706
.014 493
.014 286
.014 085
.013 889
.013 699
.013 514
.013 333
.013 158
.012 987
.012 821
.012 658
.012 500
.012 346
.012 195
.012 048
.011 905
.011 765
.011 628
.011 494
.011 364
.011 236
.011 111
.010 989
.010 870
.010 753
.010 638
.010 526
.010 417
.010 309
.010 204
.010 101
.010 000
239

COMPOUND AMOUNT: A + rf
If a principal P is deposited at interest rate r (in decimals) compounded
annually, then at the end of n years the accumulated amount A = P(l + r)n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1%
1.0100
1.0201
1.0303
1.0406
1.0510
1.0615
1.0721
1.0829
1.0937
1.1046
1.1157
1.1268
1.1381
1.1495
1.1610
1.1726
1.1843
1.1961
1.2081
1.2202
1.2324
1.2447
1.2572
1.2697
1.2824
1.2953
1.3082
1.3213
1.3345
1.3478
1.3613
1.3749
1.3887
1.4026
1.4166
1.4308
1.4451
1.4595
1.4741
1.4889
1.5038
1.5188
1.5340
1.5493
1.5648
1.580Б
1.5963
1.6122
1.6283
1.6446
4%
1.0125
1.0252
1.0380
1.0509
1.0641
1.0774
1.0909
1.1045
1.1183
1.1323
1.1464
1.1608
1.1753
1.1900
1.2048
1.2199
1.2351
1.2506
1.2662
1.2820
1.2981
1.3143
1.3307
1.3474
1.3642
1.3812
1.3985
1.4160
1.4337
1.4516
1.4698
1.4881
1.5067
1.5256
1.5446
1.5639
1.5835
1.6033
1.6233
1.6436
1.6642
1.6850
1.7060
1.7274
1.7489
1.7708
1.7929
1.8154
1.8380
1.8610
11%
1.0150
1.0302
1.0457
1.0614
1.0773
1.0934
1.1098
1.1265
1.1434
1.1605
1.1779
1.1956
1.2136
1.2318
1.2502
1.2690
1.2880
1.3073
1.3270
1.3469
1.3671
1.3876
1.4084
1.4295
1.4509
1.4727
1.4948
1.5172
1.5400
1.5631
1.5865
1.6103
1.6345
1.6590
1.6839
1.7091
1.7348
1.7608
1.7872
1.8140
1.8412
1.8688
1.8969
1.9253
1.9542
1.9835
2.0133
2.0435
2.0741
2.1052
2%
1.0200
1.0404
1.0612
1.0824
1.1041
1.1262
1.1487
1.1717
1.1951
1.2190
1.2434
1.2682
1.2936
1.3195
1.3459
1.3728
1.4002
1.4282
1.4568
1.4859
1.5157
1.5460
1.5769
1.6084
1.6406
1.6734
1.7069
1.7410
1.7758
1.8114
1.8476
1.8845
1.9222
1.9607
1.9999
2.0399
2.0807
2.1223
2.1647
2.2080
2.2522
2.2972
2.3432
2.3901
2.4379
2.4866
2.5363
2.5871
2.6388
2.6916
2-1%
1.0250
1.0506
1.0769
1.1038
1.1314
1.1597
1.1887
1.2184
1.2489
1.2801
1.3121
1.3449
1.3785
1.4130
1.4483
1.4845
1.5216
1.5597
1.5987
1.6386
1.6796
1.7216
1.7646
1.8087
1.8539
1.9003
1.9478
1.9965
2.0464
2.0976
2.1500
2.2038
2.2589
2.3153
2.3732
2.4325
2.4933
2.5557
2.6196
2.6851
2.7522
2.8210
2.8915
2.9638
3.0379
3.1139
3.1917
3.2715
3.3533
3.4371
3%.
1.0300
1.0609
1.0927
1.1255
1.1593
1.1941
1.2299
1.2668
1.3048
1.3439
1.3842
1.4258
1.4685
1.5126
1.5580
1.6047
1.6528
1.7024
1.7535
1.8061
1.8603
1.9161
1.9736
2.0328
2.0938
2.1566
2.2213
2.2879
2.3566
2.4273
2.5001
2.5751
2.6523
2.7319
2.8139
2.8983
2.9852
3.0748
3.1670
3.2620
3.3599
3.4607
3.5645
3.6715
3.7816
3.8950
4.0119
4.1323
4.2562
4.3839
4%
1.0400
1.0816
1.1249
1.1699
1.2167
1.2653
1.3159
1.3688
1.4233
1.4802
1.5395
1.6010
1.6651
1.7317
1.8009
1.8730
1.9479
2.0258
2.1068
2.1911
2.2788
2.3699
2.4647
2.5633
2.6658
2.7725
2.8834
2.9987
3.1187
3.2434
3.3731
3.5081
3.6484
3.7943
3.9461
4.1039
4.2681
4.4388
4.6164
4.8010
4.9931
5.1928
5.4005
5.6165
5.8412
6.0748
6.3178
6.5705
6.8333
7.1067
5%
1.0500
1.1025
1.1576
1.2155
1.2763
1.3401
1.4071
1.4775
1.5513
1.6289
1.7103
1.7959
1.8856
1.9799
2.0789
2.1829
2.2920
2.4066
2.5270
2.6533
2.7860
2.9253
3.0715
3.2251
3.3864
3.5557
3.7335
3.9201
4.1161
4.3219
4.5380
4.7649
5.0032
5.2533
5.5160
5.7918
6.0814
6.3855
6.7048
7.0400
7.3920
7.7616
8.1497
8.5572
8.9850
9.4343
9.9060
10.4013
10.9213
11.4674
6%
1.0600
1.1236
1.1910
1.2635
1.3382
1.4185
1.5036
1.5938
1.6895
1.7908
1.8983
2.0122
2.1329
2.2609
2.3966
2.5404
2.6928
2.8543
3.0256
3.2071
3.3996
3.6035
3.8197
4.0489
4.2919
4.5494
4.8223
5.1117
5.4184
5.7435
6.0881
6.4534
6.8406
7.2510
7.6861
8.1473
8.6361
9.1543
9.7035
10.2857
10.9029
11.5570
12.2505
12.9855
13.7646
14.5905
15.4659
16.3939
17.3775
18.4202
240

PRESENT VALUE OF AN AMOUNT: A+r)"
The present value P which will amount to A in и years at an in-
interest rate of r (in decimals) compounded annually is P = A(l + r)n.
N. Г
П N.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1%
.99010
.98030
.97059
.96098
.95147
.94205
.93272
.92348
.91434
.90529
.89632
.88745
.87866
.86996
.86135
.85282
.84438
.83602
.82774
.81954
.81143
.80340
.79544
.78757
.77977
.77205
.76440
.75684
.74934
.74192
.73458
.72730
.72010
.71297
.70591
.69892
.69200
.68515
.67837
.67165
.66500
.65842
.65190
.64545
.63905
.63273
.62646
.62026
.61412
.60804
4%
.98765
.97546
.96342
.95152
.93978
.92817
.91672
.90540
.89422
.88318
.87228
.86151
.85087
.84037
.82999
.81975
.80963
.79963
.78976
.78001
.77038
.76087
.75147
.74220
.73303
.72398
.71505
.70622
.69750
.68889
.68038
.67198
.66369
.65549
.64740
.63941
.63152
.62372
.61602
.60841
.60090
.59348
.58616
.57892
.57177
.56471
.55774
.55086
.54406
.53734
11%
.98522
.97066
.95632
.94218
.92826
.91454
.90103
.88771
.87459
.86167
.84893
.83639
.82403
.81185
.79985
.78803
.77639
.76491
.75361
.74247
.73150
.72069
.71004
.69954
.68921
.67902
.66899
.65910
.64936
.63976
.63031
.62099
.61182
.60277
.59387
.58509
.57644
.56792
.55953
.55126
.54312
.53509
.52718
.61939
.51171
.50415
.49670
.48936
.48213
.47500
2%
.98039
.96117
.94232
.92385
.90573
.88797
.87056
.85349
.83676
.82035
.80426
.78849
.77303
.75788
.74301
.72845
.71416
.70016
.68643
.67297
.65978
.64684
.63416
.62172
.60953
.59758
.58586
.57437
.56311
.55207
.54125
.53063
.52023
.51003
.50003
.49022
.48061
.47119
.46195
.45289
.44401
.43530
.42677
.41840
.41020
.40215
.39427
.38654
.37896
.37153
2i%
.97561
.95181
.92860
.90595
.88385
.86230
.84127
.82075
.80073
.78120
.76214
.74356
.72542
.70773
.69047
.67362
.65720
.64117
.62553
.61027
.59539
.58086
.56670
.55288
.53939
.52623
.51340
.50088
.48866
.47674
.46511
.45377
.44270
.43191
.42137
.41109
.40107
.39128
.38174
.37243
.36335
.35448
.34584
.33740
.32917
.32115
.31331
.30567
.29822
.29094
3%
.97087
.94260
.91514
.88849
.86261
.83748
.81309
.78941
.76642
.74409
.72242
.70138
.68095
.66112
.64186
.62317
.60502
.58739
.57029
.55368
.53755
.52189
.50669
.49193
.47761
.46369
.45019
.43708
.42435
.41199
.39999
.38834
.37703
.36604
.35538
.34503
.33498
.32523
.31575
.30656
.29763
.28896
.28054
.27237
.26444
.25674
.24926
.24200
.23495
.22811
4%
.96154
.92456
.88900
.85480
.82193
.79031
.75992
.73069
.70259
.67556
.64958
.62460
.60057
.57748
.55526
.53391
.51337
.49363
.47464
.45639
.43883
.42196
.40573
.39012
.37512
.36069
.34682
.33348
.32065
.30832
.29646
.28506
.27409
.26355
.25342
.24367
.23430
.22529
.21662
.20829
.20028
.19257
.18517
.17805
.17120
.16461
.15828
.15219
.14634
.14071
5%
.95238
.90703
.86384
.82270
.78353
.74622
.71068
.67684
.64461
.61391
.58468
.55684
.53032
.50507
.48102
.45811
.43630
.41552
.39573
.37689
.35894
.34185
.32557
.31007
.29530
.28124
.26785
.25509
.24295
.23138
.22036
.20987
.19987
.19035
.18129
.17266
.16444
.15661
.14915
.14205
.13528
.12884
.12270
.11686
.11130
.10600
.10095
.09614
.09156
.08720
6%
.94340
.89000
.83962
.79209
.74726
.70496
.66506
.62741
.59190
.55839
.52679
.49697
.46884
.44230
.41727
.39365
.37136
.35034
.33051
.31180
.29416
.27751
.26180
.24698
.23300
.21981
.20737
.19563
.18456
.17411
.16425
.15496
.14619
.13791
.13011
.12274
.11579
.10924
.10306
.09722
.09172
.08653
.08163
.07701
.07265
.06854
.06466
.06100
.05755
.05429
241

TABLE
25
AMOUNT OF AN ANNUITY:
A 4- r)" - 1
If a principal P is deposited at tho end of each year at interest rate r (in
decimals) compounded annually, then at the end of n years the accumulated
amount is P
The process is often called an annuity.
n\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1%
1.0000
2.0100
3.0301
4.0604
5.1010
6.1520
7.2135
8.2857
9.3685
10.4622
11.5668
12.6825
13.8093
14.9474
16.СЭ69
17.2579
18.4304
19.6147
20.8109
22.0190
23.2392
24.4716
25.7163
26.9735
28.2432
29.5256
30.8209
32.1291
33.4504
34.7849
36.1327
37.4941
38.8690
40.2577
41.6603
43.0769
44.5076
45.9527
47.4123
48.8864
50.3752
51.8790
53.3978
54.9318
56.4811
58.0459
59.6263
61.2226
62.8348
64.4632
4%
1.0000
2.0125
3.0377
4.0756
5.1266
6.1907
7.2680
8.3589
9.4634
10.5817
11.7139
12.8604
14.0211
15.1964
16.3863
17.5912
18.8111
20.0462
21.2968
22.5630
23.8450
25.1431
26.4574
27.7881
29.1354
30.4996
31.8809
33.2794
34.6954
36.1291
37.5807
39.0504
40.5386
42.0453
43.5709
45.1155
46.6794
48.2629
49.8662
51.4896
53.1332
54.7973
56.4823
58.1883
59.9157
61.6646
63.4354
65.2284
67.0437
68.8818
Ц%
1.0000
2.0150
3.0452
4.0909
5.1523
6.2296
7.3230
8.4328
9.5593
10.7027
11.8633
13.0412
14.2368
15.4504
16.6821
17.9324
19.2014
20.4894
21.7967
23.1237
24.4705
25.8376
27.2251
28.6335
30.0630
31.5140
32.9867
34.4815
35.9987
37.5387
39.1018
40.6883
42.2986
43.9331
45.5921
47.2760
48.9851
50.7199
52.4807
54.2679
56.0819
57.9231
59.7920
61.6889
63.6142
65.5684
67.5519
69.5652
71.6087
73.6828
2%
1.0000
2.0200
3.0604
4.1216
5.2040
6.3081
7.4343
8.5830
9.7546
10.9497
12.1687
13.4121
14.6803
15.9739
17.2934
18.6393
20.0121
21.4123
22.8406
24.2974
25.7833
27.2990
28.8450
30.4219
32.0303
33.6709
35.3443
37.0512
38.7922
40.5681
42.3794
44.2270
46.1116
48.0338
49.9945
51.9944
54.0343
56.1149
58.2372
60.4020
62.6100
64.8622
67.1595
69.5027
71.8927
74.3306
76.8172
79.3535
81.9406
84.5794
21%
1.0000
2.0250
3.0756
4.1525
5.2563
6.3877
7.5474
8.7361
9.9545
11.2034
12.4835
13.7956
15.1404
16.5190
17.9319
19.3802
20.8647
22.3863
23.9460
25.5447
27.1833
28.8629
30.5844
32.3490
34.1578
36.0117
37.9120
39.8598
41.8563
43.9027
46.0003
48.1503
50.3540
52.6129
54.9282
57.3014
59.7339
62.2273
64.7830
67.4026
70.0876
72.8398
75.6608
78.5523
81.5161
84.5540
87.6679
90.8596
94.1311
97.4843
3%
1.0000
2.0300
3.0909
4.1836
5.3091
6.4684
7.6625
8.8923
10.1591
11.4639
12.8078
14.1920
15.6178
17.0863
18.5989
20.1569
21.7616
23.4144
25.1169
26.8704
28.6765
30.5368
32.4529
34.4265
36.4593
38.5530
40.7096
42.9309
45.2189
47.5754
50.0027
52.5028
55.0778
57.7302
60.4621
63.2759
66.1742
69.1594
72.2342
75.4013
78.6633
82.0232
85.4839
89.0484
92.7199
96.5015
100.3965
104.4084
108.5406
112.7969
4%
1.0000
2.0400
3.1216
4.2465
5.4163
6.6330
7.8983
9.2142
10.5828
12.0061
13.4864
15.0258
16.6268
18.2919
20.0236
21.8245
23.6975
25.6454
27.6712
29.7781
31.9692
34.2480
36.6179
39.0826
41.6459
44.3117
47.0842
49.9676
52.9663
56.0849
59.3283
62.7015
66.2095
69.8579
73.6522
77.5983
81.7022
85.9703
90:4091
95.0255
99.8265
104.8196
110.0124
115.4129
121.0294
126.8706
132.9454
139.2632
145.8337
152.6671
5%
1.0000
2.0500
3.1525
4.3101
5.5256
6.8019
8.1420
9.5491
11.0266
12.5779
14.2068
15.9171
17.7130
19.5986
21.5786
23.6575
25.8404
28.1324
30.5390
33.0660
35.7193
38.5052
41.4305
44.5020
47.7271
51.1135
54.6691
58.4026
62.3227
66.4388
70.7608
75.2988
80.0638
85.0670
90.3203
95.8363
101.6281
107.7095
114.0950
120.7998
127.8398
135.2318
142.9933
151.1430
159.7002
168.6852
178.1194
188.0254
198.4267
209.3480
6%
1.0000
2.0600
3.1836
4.3746
5.6371
6.9753
8.3938
9.8975
11.4913
13.1808
14.9716
16.8699
18.8821
21.0151
23.2760
25.6725
28.2129
30.9057
33.7600
36.7856
39.9927
43.3923
46.9958
50.8156
54.8645
59.1564
63.7058
68.5281
73.6398
79.0582
84.8017
90.8898
97.3432
104.1838
111.4348
119.1209
127.2681
135.9042
145.0585
154.7620
165.0477
175.9505
187.5076
199.7580
212.7435
226.5081
241.0986
256.5645
272.9584
290.3359
242

TABLE
26
PRESENT VALUE OF AN ANNUITY:
An annuity in which the yearly payment at the end of each of n years is A at an
interest rate r (in decimals) compounded annually has present value A
^\ r
%^\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1%
0.9901
1.9704
2.9410
3.9020
4.8534
5.7955
6.7282
7.6517
8.5660
9.4713
10.3676
11.2551
12.1337
13.0037
13.8651
14.7179
15.5623
16.3983
17.2260
18.0456
18.8570
19.6604
20.4558
21.2434
22.0232
22.7952
23.5596
24.3164
25.0658
25.8077
26.5423
27.2696
27.9897
28.7027
29.4086
30.1075
30.7995
31.4847
32.1630
32.8347
33.4997
34.1581
34.8100
35.4555
36.0945
36.7272
37.3537
37.9740
38.5881
39.1961
Ц%
0.9877
1.9631
2.9265
3.8781
4.8178
5.7460
6.6627
7.5681
8.4623
9.3455
10.2178
11.0793
11.9302
12.7706
13.6005
14.4203
15.2299
16.0295
16.8193
17.5993
18.3697
19.1306
19.8820
20.6242
21.3573
22.0813
22.7963
23.5025
24.2000
24.8889
25.5693
26.2413
26.9050
27.5605
28.2079
28.8473
29.4788
30.1025
30.7185
31.3269
31.9278
32.5213
33.1075
33.6864
34.2582
34.8229
35.3806
35.9315
36.4755
37.0129
Ц%
0.9852
1.9559
2.9122
3.8544
4.7826
5.6972
6.5982
7.4859
8.3605
9.2222
10.0711
10.9075
11.7315
12.5434
13.3432
14.1313
14.9076
15.6726
16.4262
17.1686
17.9001
18.6208
19.3309
20.0304
20.7196
21.3986
22.0676
22.7267
23.3761
24.0158
24.6461
25.2671
25.8790
26.4817
27.0756
27.6607
28.2371
28.8051
29.3646
29.9158
30.4590
30.9941
31.5212
32.0406
32.5523
33.0565
33.5532
34.0426
34.5247
34.9997
2%
0.9804
1.9416
2.8839
3.8077
4.7135
5.6014
6.4720
7.3255
8.1622
8.9826
9.7868
10.5753
11.3484
12.1062
12.8493
13.5777
14.2919
14.9920
15.6785
16.3514
17.0112
17.6580
18.2922
18.9139
19.5235
20.1210
20.7069
21.2813
21.8444
22.3965
22.9377
23.4683
23.9886
24.4986
24.9986
25.4888
25.9695
26.4406
26.9026
27.3555
27.7995
28.2348
28.6616
29.0800
29.4902
29.8923
30.2866
30.6731
31.0521
31.4236
2i%
0.9756
1.9274
2.8560
3.7620
4.6458
5.5081
6.3494
7.1701
7.9709
8.7521
9.5142
10.2578
10.9832
11.6909
12.3814
13.0550
13.7122
14.3534
14.9789
15.5892
16.1845
16.7654
17.3321
17.8850
18.4244
18.9506
19.4640
19.9649
20.4535
20.9303
21.3954
21.8492
22.2919
22.7238
23.1452
23.5563
23.9573
24.3486
24.7303
25.1028
25.4661
25.8206
26.1664
26.5038
26.8330
27.1542
27.4675
27.7732
28.0714
28.3623
3%
0.9709
1.9135
2.8286
3.7171
4.5797
5.4172
6.2303
7.0197
7.7861
8.5302
9.2526
9.9540
10.6350
11.2961
11.9379
12.5611
13.1661
13.7535
14.3238
14.8775
15.4150
15.9369
16.4436
16.9355
17.4131
17.8768
18.3270
18.7641
19.1885
19.6004
20.0004
20.3888
20.7658
21.1318
21.4872
21.8323
22.1672
22.4925
22.8082
23.1148
23.4124
23.7014
23.9819
24.2543
24.5187
24.7754
25.0247
25.2667
25.5017
25.7298
4%
0.9615
1.8861
2.7751
3.6299
4.4518
5.2421
6.0021
6.7327
7.4353
8.1109
8.7605
9.3851
9.9856
10.5631
11.1184
11.6523
12.1657
12.6593
13.1339
13.5903
14.0292
14.4511
14.8568
15.2470
15.6221
15.9828
16.3296
16.6631
16.9837
17.2920
17.5885
17.8736
18.1476
18.4112
18.6646
18.9083
19.1426
19.3679
19.5845
19.7928
19.9931
20.1856
20.3708
20.5488
20.7200
20.8847
21.0429
21.1951
21.3415
21.4822
5%
0.9524
1.8594
2.7232
3.5460
4.3295
5.0757
5.7864
6.4632
7.1078
7.7217
8.3064
8.8633
9.3936
9.8986
10.3797
10.8378
11.2741
11.6896
12.0853
12.4622
12.8212
13.1630
13.4886
13.7986
14.0939
14.3752
14.6430
14.8981
15.1411
15.3725
15.5928
15.8027
16.0025
16.1929
16.3742
16.5469
16.7113
16.8679
17.0170
17.1591
17.2944
17.4232
17.5459
17.6628
17.7741
17.8801
17.9810
18.0772
18.1687
18.2559
6%
0.9434
1.8334
2.6730
3.4651
4.2124
4.9173
5.5824
6.2098
6.8017
7.3601
7.8869
8.3838
8.8527
9.2950
9.7122
10.1059
10.4773
10.8276
11.1581
11.4699
11.7641
12.0416
12.3034
12.5504
12.7834
13.0032
13.2105
13.4062
13.5907
13.7648
13.9291
14.0840
14.2302
14.3681
14.4982
14.6210
14.7368
14.8460
14.9491
15.0463
15.1380
15.2245
15.3062
15.3832
15.4558
15.5244
15.5890
15.6500
15.7076
15.7619
243

X
0.
1.
2.
3.
4.
Б.
6.
7.
8.
9.
0
1.0000
.7652
.2239
-.2601
-.3971
-.1776
.1506
.3001
.1717
-.0903
1
.9975
.7196
.1666
-.2921
-.3887
-.1443
.1773
.2991
.1475
-.1142
2
.9900
.6711
.1104
-.3202
-.3766
-.1103
.2017
.2951
.1222
-.1367
3
.9776
.6201
.0555
-.3443
-.3610
-.0758
.2238
.2882
.0960
-.1577
4
.9604
.5669
.0025
-.3643
-.3423
-.0412
.2433
.2786
.0692
-.1768
5
.9385
.5118
-.0484
-.3801
-.3205
-.0068
.2601
.2663
.0419
-.1939
6
.9120
.4554
-.0968
-.3918
-.2961
.0270
.2740
.2516
.0146
-.2090
7
.8812
.3980
-.1424
-.3992
-.2693
.0599
.2851
.2346
-.0125
-.2218
8
.8463
.3400
-.1850
-.4026
-.2404
.0917
.2931
.2154
-.0392
-.2323
9
.8075
.2818
-.2243
-.4018
-.2097
.1220
.2981
.1944
-.0653
-.2403
BESSEL FUNCTIONS
X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
.0000
.4401
.5767
.3391
-.0660
-.3276
-.2767
-.0047
.2346
.2453
1
.0499
.4709
.5683
.3009
-.1033
-.3371
-.2559
.0252
.2476
.2324
2
.0995
.4983
.5560
.2613
-.1386
-.3432
-.2329
.0543
.2580
.2174
3
.1483
.5220
.5399
.2207
-.1719
-.3460
-.2081
.0826
.2657
.2004
4
.1960
.5419
.5202
.1792
-.2028
-.3453
-.1816
.1096
.2708
.1816
5
.2423
.5579
.4971
.1374
-.2311
-.3414
-.1538
.1352
.2731
.1613
6
.2867
.5699
.4708
.0955
-.2566
-.3343
-.1250
.1592
.2728
.1396
7
.3290
.5778
.4416
.0538
-.2791
-.3241
-.0953
.1813
.2697
.1166
8
.3688
.5815
.4097
.0128
-.2985
-.3110
-.0652
.2014
.2641
.0928
9
.4059
.5812
.3754
-.0272
-.3147
-.2951
-.0349
.2192
.2559
.0684
244

X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
00
.0883
.5104
.3769
-.0169
-.3085
-.2882
-.0259
.2235
.2499
1
-1.5342
.1622
.5183
.3431
-.0561
-.3216
-.2694
.0042
.2381
.2383
2
-1.0811
.2281
.5208
.3071
-.0938
-.3313
-.2483
.0339
.2501
.2245
3
-.8073
.2865
.5181
.2691
-.1296
-.3374
-.2251
.0628
.2595
.2086
4
-.6060
.3379
.5104
.2296
-.1633
-.3402
-.1999
.0907
.2662
.1907
5
-.4445
.3824
.4981
.1890
-.1947
-.3395
-.1732
.1173
.2702
.1712
6
-.3085
.4204
.4813
.1477
-.2235
-.3354
-.1452
.1424
.2715
.1502
7
-.1907
.4520
.4605
.1061
-.2494
-.3282
-.1162
.1658
.2700
.1279
8
-.0868
.4774
.4359
.0645
-.2723
-.3177
-.0864
.1872
.2659
.1045
9
.0056
.4968
.4079
.0234
-.2921
-.3044
-.0563
.2065
.2592
.0804
BESSEL FUNCTIONS
X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
— 00
-.7812
-.1070
.3247
.3979
.1479
-.1750
-.3027
-.1581
.1043
1
-6.4590
-.6981
-.0517
.3496
.3846
.1137
-.1998
-.2995
-.1331
.1275
2
-3.3238
-.6211
.0015
.3707
.3680
.0792
-.2223
-.2934
-.1072
.1491
3
-2.2931
-.5485
.0523
.3879
.3484
.0445
-.2422
-.2846
-.0806
.1691
4
-1.7809
-.4791
.1005
.4010
.3260
.0101
-.2596
-.2731
-.0535
.1871
5
-1.4715
-.4123
.1459
.4102
.3010
-.0238
-.2741
-.2591
-.0262
.2032
6
-1.2604
-.3476
.1884
.4154
.2737
-.0568
-.2857
-.2428
.0011
.2171
7
-1.1032
-.2847
.2276
.4167
.2445
-.0887
-.2945
-.2243
.0280
.2287
8
-.9781
-.2237
.2635
.4141
.2136
-.1192
-.3002
-.2039
.0544
.2379
9
-.8731
-.1644
.2959
.4078
.1812
-.1481
-.3029
-.1817
.0799
.2447
245

X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
1.000
1.266
2.280
4.881
11.30
27.24
67.23
168.6
427.6
1094
1
1.003
1.326
2.446
5.294
12.32
29.79
73.66
185.0
469.Б
1202
2
1.010
1.394
2.629
5.747
13.44
32.58
80.72
202.9
515.6
1321
3
1.023
1.469
2.830
6.243
14.67
ЗБ.65
88.46
222.7
566.3
14Б1
4
1.040
1.553
3.049
6.78Б
16.01
39.01
96.96
244.3
621.9
1595
5
1.063
1.647
3.290
7.378
17.48
42.69
106.3
268.2
683.2
1753
6
1.092
1.750
3.553
8.028
19.09
46.74
116.5
294.3
750.5
1927
7
1.126
1.864
3.842
8.739
20.86
51.17
127.8
323.1
824.4
2119
8
1.167
1.990
4.157
9.517
22.79
56.04
140.1
354.7
905.8
2329
9
1.213
2.128
4.503
10.37
24.91
61.38
153.7
389.4
995.2
2561
BESSEL FUNCTIONS
X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
.0000
.5652
1.591
3.953
9.759
24.34
61.34
156.0
399.9
1031
1
.0501
.6375
1.745
4.326
10.69
26.68
67.32
171.4
439.5
1134
2
.1005
.7147
1.914
4.734
11.71
29.25
73.89
188.3
483.0
1247
3
.1517
.7973
2.098
5.181
12.82
32.08
81.10
206.8
531.0
1371
4
.2040
.8861
2.298
5.670
14.05
35.18
89.03
227.2
583.7
1508
5
.2579
.9817
2.517
6.206
15.39
38.59
97.74
249.6
641.6
1658
6
.3137
1.085
2.755
6.793
16.86
42.33
107.3
274.2
705.4
1824
7
.3719
1.196
3.016
7.436
18.48
46.44
117.8
301.3
775.5
2006
8
.4329
1.317
3.301
8.140
20.25
50.95
129.4
331.1
852.7
2207
9
.4971
1.448
3.613
8.913
22.20
55.90
142.1
363.9
937.5
2428
246

X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
00
.4210
.1139
.03474
.01116
.023691
.021244
.034248
.031465
.0*5088
1
2.4271
.3656
.1008
.03095
.029980
.023308
.021117
.033817
.031317
.044579
2
1.7527
.3185
.08927
.02759
.028927
.022966
.021003
.033431
.031185
.044121
3
1.3725
.2782
.07914
.02461
.027988
.022659
.039001
.033084
.031066
.043710
4
1.1145
.2437
.07022
.02196
.027149
.022385
.038083
.032772
.049588
.043339
5
.9244
.2138
.06235
.01960
.026400
.022139
.037259
.032492
.048626
.043006
6
.7775
.1880
.05540
.01750
.025730
.021918
.036520
.032240
.047761
.042706
7
.6605
.1655
.04926
.01563
.025132
.021721
.035857
.032014
.046983
.042436
8
.5653
.1459
.04382
.01397
.024597
.021544
.035262
.031811
.046283
.042193
9
.4867
.1288
.03901
.01248
.024119
.021386
.034728
.031629
.045654
.041975
BESSEL FUNCTIONS
X
0.
1.
2.
со'
4.
5.
6.
7.
8.
9.
0
00
.6019
.1399
.04016
.01248
.024045
.021344
.034542
.0П554
.045364
1
9.8538
.5098
.1227
.03563
.01114
.023619
.021205
.034078
.031396
.044825
2
4.7760
.4346
.1079
.03164
.029938
.023239
.02Ю81
.033662
.031255
.044340
3
3.0560
.3725
.09498
.02812
.028872
.022900
.039691
.033288
.0П128
.043904
4
2.1844
.3208
.08372
.02500
.027923
.022597
.038693
.032953
.03Ю14
.043512
5
1.6564
.2774
.07389
.02224
.027078
.022326
.037799
.032653
.049120
.043160
6
1.3028
.2406
.06528
.01979
.026325
.022083
.036998
.032383
.048200
.042843
7
1.0503
.2094
.05774
.01763
.025654
.021866
.036280
.032141
.047374
.042559
8
.8618
.1826
.05111
.01571
.025055
.021673
.035636
.031924
.046631
.042302
9
.7165
.1597
.04529
.01400
.024521
.021499
.035059
.031729
.045964
.042072
247

X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
1.0000
.9844
.7517
-.2214
-2.5634
-6.2301
-8.8583
-3.6329
20.974
73.936
1
1.0000
.9771
.6987
-.3855
-2.8843
-6.6107
-8.8491
-2.2571
24.957
80.576
2
1.0000
.9676
.6377
-.5644
-3.2195
-6.9803
-8.7561
-.6737
29.245
87.350
3
.9999
.9554
.5680
-.7584
-3.5679
-7.3344
-8.5688
1.1308
33.840
94.208
4
.9996
.9401
.4890
-.9680
-3.9283
-7.6674
-8.2762
3.1695
38.738
101.10
5
.9990
.9211
.4000
-1.1936
-4.2991
-7.9736
-7.8669
5.4550
43.936
107.95
6
.9980
.8979
.3001
-1.4353
-4.6784
-8.2466
-7.3287
7.9994
49.423
114.70
7
.9962
.8700
.1887
-1.6933
-5.0639
-8.4794
-6.6492
10.814
55.187
121.26
8
.9936
.8367
.06511
-1.9674
-5.4531
-8.6644
-5.8155
13.909
61.210
127.54
9
.9898
.7975
-.07137
-2.2576
-5.8429
-8.7937
-4.8146
17.293
67.469
133.43
BESSEL FUNCTIONS
X
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
0
.0000
.2496
.9723
1.9376
2.2927
.1160
-7.3347
-21.239
-35.017
-24.713
1
.022500
.3017
1.0654
2.0228
2.2309
-.3467
-8.4545
-22.848
-35.667
-20.724
2
.01000
.3587
1.1610
2.1016
2.1422
-.8658
-9.6437
-24.456
-36.061
-15.976
3
.02250
.4204
1.2585
2.1723
2.0236
-1.4443
-10.901
-26.049
-36.159
-10.412
4
.04000
.4867
1.3575
2.2334
1.8726
-2.0845
-12.223
-27.609
-35.920
-3.9693
5
.06249
.5576
1.4572
2.2832
1.6860
-2.7890
-13.607
-29.116
-35.298
3.4106
6
.08998
.6327
1.5569
2.3199
1.4610
-3.5597
-15.047
-30.548
-34.246
11.787
7
.1224
.7120
1.6557
2.3413
1.1946
-4.3986
-16.538
-31.882
-32.714
21.218
8
.1599
.7953
1.7529
2.3454
.8837
-5.3068
-18.074
-33.092
-30.651
31.758
9
.2023
.8821
1.8472
2.3300
.5251
-6.2854
-19.644
-34.147
-28.003
43.459
248

X
0.
1.
2.
3.
4.
Б.
6.
7.
8.
9.
0
.2867
-.04166
-.06703
-.03618
-.01151
-.036530
.021922
.021486
.036372
1
2.4205
.2228
-.05111
-.06468
-.03308
-.029865
-.031295
.021951
.021397
.035681
2
1.7331
.1689
-.05834
-.06198
-.03011
-.028359
.033191
.021956
.021306
.035030
3
1.3372
.1235
-.06367
-.05903
-.02726
-.026989
.036991
.021940
.021216
.034422
4
1.0626
.08513
-.06737
-.05590
-.02456
-.025749
.021017
.021907
.021126
.033855
5
.8559
.05293
-.06969
-.05264
-.02200
-.024632
.021278
.021860
.02Ю37
.033330
6
.6931
.02603
-.07083
-.04932
-.01960
-.023632
.021488
.021800
.039511
.032846
7
.5614
.023691
-.07097
-.04597
-.01734
-.022740
.021653
.021731
.038675
.032402
8
.4529
-.01470
-.07030
-.04265
-.01525
-.021952
.021777
.021655
.037871
.031996
9
.3625
-.02966
-.06894
-.03937
-.01330
-.0212Б8
.021866
.021Б72
.037102
.031628
BESSEL FUNCTIONS
X
0.
1.
2.
3.
4.
Б.
6.
7.
8.
9.
0
-.7854
-.4950
-.2024
-.05112
.022198
.01119
.027216
.022700
.033696
-.033192
1
-.7769
-.4601
-.1812
-.04240
.024386
.01105
.026696
.022366
.032440
-.033368
2
-.7581
-.4262
-.1614
-.03458
.026194
.01082
.026183
.022057
.031339
-.033486
3
-.7331
-.3933
-.1431
-.02762
.027661
.01051
.02Б681
.021770
.043809
-.033552
4
-.7038
-.3617
-.1262
-.02145
.028826
.01014
.025194
.021507
-.044449
-.033574
5
-.6716
-.3314
-.1107
-.01600
.029721
.029716
.024724
.021267
-.031149
-.033557
6
-.6374
-.3026
-.09644
-.01123
.01038
.029255
.024274
.021048
-.031742
-.033508
7
-.6022
-.2752
-.08342
-.027077
.01083
.028766
.023846
.038498
-.032233
-.033430
8
-.5664
-.2494
-.07157
-.023487
.01110
.028258
.023440
.036714
-.032632
-.033329
9
-.5305
-.2251
-.06083
-.034108
.01121
.027739
.023058
.035117
-.032949
-.033210
249

TABLE
39
VALUES FOR APPROXIMATE
ZEROS OF BESSEL FUNCTIONS
The following table lists the first few positive roots of various equations. Note that for all cases
listed the successive large roots differ approximately by v — 3.14159... .
7 /-Л Л
Yn(x) = 0
Т' (rf\ — П
Y'n(x) = 0
% = 0
2.4048
5.5201
8.6537
11.7915
14.9309
18.0711
0.8936
3.9577
7.0861
10.2223
13.3611
16.5009
0.0000
3.8317
7.0156
10.1735
13.3237
16.4706
2.1971
5.4297
8.5960
11.7492
14.8974
18.0434
n-1
3.8317
7.0156
10.1735
13.3237
16.4706
19.6159
2.1971
5.4297
8.5960
11.7492
14.8974
18.0434
1.8412
5.3314
8.5363
11.7060
14.8636
18.0155
3.6830
6.9415
10.1234
13.2858
16.4401
19.5902
и = 2
5.1356
8.4172
11.6198
14.7960
17.9598
21.1170
3.3842
6.7938
10.0235
13.2100
16.3790
19.5390
3.0542
6.7061
9.9695
13.1704
16.3475
19.5129
5.0026
8.3507
11.5742
14.7609
17.9313
21.0929
n - 3
6.3802
9.7610
13.0152
16.2235
19.4094
22.5827
4.5270
8.0976
11.3965
14.6231
17.8185
20.9973
4.2012
8.0152
11.3459
14.5859
17.7888
20.9725
6.2536
9.6988
12.9724
16.1905
19.3824
22.5598
n = 4
7.5883
11.0647
14.3725
17.6160
20.8269
24.0190
5.6452
9.3616
12.7301
15.9996
19.2244
22.4248
5.3176
9.2824
12.6819
15.9641
19.1960
22.4010
7.4649
11.0052
14.3317
17.5844
20.8011
23.9970
% = 5
8.7715
12.3386
15.7002
18.9801
22.2178
25.4303
6.7472
10.5972
14.0338
17.3471
20.6029
23.8265
6.4156
10.5199
13.9872
17.3128
20.5755
23.8036
8.6496
12.2809
15.6608
18.9497
22.1928
25.4091
n- 6
9.9361
13.5893
17.0038
20.3208
23.5861
26.8202
7.8377
11.8110
15.3136
18.6707
21.9583
25.2062
7.5013
11.7349
15.2682
18.6374
21.9317
25.1839
9.8148
13.5328
16.9655
20.2913
23.5619
26.7995
250

TABLE
40
EXPONENTIAL, SINE AND COSINE INTEGRALS
х) = Г ~du, Si(x) = С ^du, Ci{x) = С
COSM
U
du
X
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Ei(x)
.5598
.2194
.1000
.04890
.02491
.01305
.026970
.023779
.022073
.021148
.036409
.033601
.032034
.031155
.0*6583
.043767
.042162
.041245
.057185
.054157
Si(x)
.0000
.4931
.9461
1.3247
1.6054
1.7785
1.8487
1.8331
1.7582
1.6541
1.5499
1.4687
1.4247
1.4218
1.4546
1.5107
1.5742
1.6296
1.6650
1.6745
1.6583
Ci(x)
.1778
-.3374
-.4704
-.4230
-.2859
-.1196
.0321
.1410
.1935
.1900
.1421
.0681
-.0111
-.0767
-.1156
-.1224
-.09943
-.05535
-.022678
.04546
251

TABLE
41
LEGENDRE POLYNOMIALS Pn(x)
[Po(x) = 1, Pi (ж) - x]
X
.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
P2(*)
-.5000
-.4963
-.4850
-.4663
-.4400
-.4063
-.3650
-.3163
-.2600
-.1963
-.1250
-.0463
.0400
.1338
.2350
.3438
.4600
.5838
.7150
.8538
1.0000
P3(x)
.0000
-.0747
-.1475
-.2166
-.2800
-.3359
-.3825
-.4178
-.4400
-.4472
-.4375
-.4091
-.3600
-.2884
-.1925
-.0703
.0800
.2603
.4725
.7184
1.0000
Pi(x)
.3750
.3657
.3379
.2928
.2320
.1577
.0729
-.0187
-.1130
-.2050
-.2891
-.3590
-.4080
-.4284
-.4121
-.3501
-.2330
-.0506
.2079
.5541
1.0000
PS(*)
.0000
.0927
.1788
.2523
.3075
.3397
.3454
.3225
.2706
.1917
.0898
-.0282
-.1526
-.2705
-.3652
-.4164
-.3995
-.2857
-.0411
.3727
1.0000
252

LEGENDRE POLYNOMIALS Pn(cos0)
в
0°
5°
10°
15°
20°
25°
30°
35°
40°
45°
50°
55°
60°
65°
70°
75°
80°
85°
90°
P^cosff)
1.0000
.9962
.9848
.9659
.9397
.9063
.8660
.8192
.7660
.7071
.6428
.5736
.5000
.4226
.3420
.2588
.1737
.0872
.0000
P2 (cos в)
1.0000
.9886
.9548
.8995
.8245
.7321
.6250
.5065
.3802
.2500
.1198
-.0065
-.1250
-.2321
-.3245
-.3995
-.4548
-.4886
-.5000
Pg(cose)
1.0000
.9773
.9106
.8042
.6649
.5016
.3248
.1454
-.0252
-.1768
-.3002
-.3886
-.4375
-.4452
-.4130
-.3449
-.2474
-.1291
.0000
P4 (cos в)
1.0000
.9623
.8532
.6847
.4750
.2465
.0234
-.1714
-.3190
-.4063
-.4275
-.3852
-.2891
-.1552
-.0038
.1434
.2659
.3468
.3750
Ps(cose)
1.0000
.9437
.7840
.5471
.2715
.0009
-.2233
-.3691
-.4197
-.3757
-.2545
-.0868
.0898
.2381
.3281
.3427
.2810
.1577
.0000
253

TABLE
43
COMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS
К
X-/2
\/l - A;2 sin2 в '
E
/»~/2
= I y/1 - k2 sin2 0 de, к - sh
¦A
0°
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
К
1.5708
1.5709
1.5713
1.5719
1.5727
1.5738
1.5751
1.5767
1.5785
1.5805
1.5828
1.5854
1.5882
1.5913
1.5946
1.5981
1.6020
1.6061
1.6105
1.6151
1.6200
1.6252
1.6307
1.6365
1.6426
1.6490
1.6557
1.6627
1.6701
1.6777
1.6858
E
1.5708
1.5707
1.5703
1.5697
1.5689
1.5678
1.5665
1.5649
1.5632
1.5611
1.5589
1.5E64
1.5537
1.5507
1.5476
1.5442
1.5405
1.5367
1.5326
1.5283
1.5238
1.5191
1.5141
1.5090
1.5037
1.4981
1.4924
1.4864
1.4803
1.4740
1.4675
¦A
30°
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
К
1.6858
1.6941
1.7028
1.7119
1.7214
1.7312
1.7415
1.7522
1.7633
1.7748
1.7868
1.7992
1.8122
1.8256
1.8396
1.8541
1.8691
1.8848
1.9011
1.9180
1.9356
1.9539
1.9729
1.9927
2.0133
2.0347
2.0571
2.0804
2.1047
2.1300
2.1565
E
1.4675
1.4608
1.4539
1.4469
1.4397
1.4323
1.4248
1.4171
1.4092
1.4013
1.3931
1.3849
1.3765
1.3680
1.3594
1.3506
1.3418
1.3329
1.3238
1.3147
1.3055
1.2963
1.2870
1.2776
1.2681
1.2587
1.2492
1.2397
1.2301
1.2206
1.2111
¦A
60°
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
К
2.1565
2.1842
2.2132
2.2435
2.2754
2.3088
2.3439
2.3809
2.4198
2.4610
2.5046
2.5507
2.5998
2.6521
2.7081
2.7681
2.8327
2.9026
2.9786
3.0617
3.1534
3.2553
3.3699
3.5004
3.6519
3.8317
4.0528
4.3387
4.7427
5.4349
00
E
1.2111
1.2015
1.1920
1.1826
1.1732
1.1638
1.1545
1.1453
1.1362
1.1272
1.1184
1.1096
1.1011
1.0927
1.0844
1.0764
1.0686
1.0611
1.0538
1.0468
1.0401
1.0338
1.0278
1.0223
1.0172
1.0127
1.0086
1.0053
1.0026
1.0008
1.0000
254

TABLE
44
INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
'* de
= Г
Jo
о л/1 - кг sin2 в '
к = sin if>
ф\
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0°
0.0000
0.1745
0.3491
0.5236
0.6981
0.8727
1.0472
1.2217
1.3963
1.5708
10°
0.0000
0.1746
0.3493
0.5243
0.6997
0.8756
1.0519
1.2286
1.4056
1.5828
20°
0.0000
0.1746
0.3499
0.5263
0.7043
0.8842
1.0660
1.2495
1.4344
1.6200
30°
0.0000
0.1748
0.3508
0.5294
0.7116
0.8982
1.0896
1.2853
1.4846
1.6858
40°
0.0000
0.1749
0.3520
0.5334
0.7213
0.9173
1.1226
1.3372
1.5597
1.7868
50°
0.0000
0.1751
0.3533
0.5379
0.7323
0.9401
1.1643
1.4068
1.6660
1.9356
60°
0.0000
0.1752
0.3545
0.5422
0.7436
0.9647
1.2126
1.4944
1.8125
2.1565
70°
0.0000
0.1753
0.3555
0.5459
0.7535
0.9876
1.2619
1.5959
2.0119
2.5046
80°
0.0000
0.1754
0.3561
0.5484
0.7604
1.0044
1.3014
1.6918
2.2653
3.1534
90°
0.0000
0.1754
0.3564
0.5493
0.7629
1.0107
1.3170
1.7354
2.4362
CO
TABLE
45
INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND
Е(к,ф) = 1
Jo
sin2 в d9, к - sin
ф \
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0°
0.0000
0.1745
0.3491
0.5236
0.6981
0.8727
1.0472
1.2217
1.3963
1.5708
10°
0.0000
0.1745
0.3489
0.5229
0.6966
0.8698
1.0426
1.2149
1.3870
1.5589
20°
0.0000
0.1744
0.3483
0.5209
0.6921
0.8614
1.0290
1.1949
1.3597
1.5238
30°
0.0000
0.1743
0.3473
0.5179
0.6851
0.8483
1.0076
1.1632
1.3161
1.4675
40°
0.0000
0.1742
0.3462
0.5141
0.6763
0.8317
0.9801
1.1221
1.2590
1.3931
50°
0.0000
0.1740
0.3450
0.5100
0.6667
0.8134
0.9493
1.0750
1.1926
1.3055
60°
0.0000
0.1739
0.3438
0.5061
0.6575
0.7954
0.9184
1.0266
1.1225
1.2111
70°
0.0000
0.1738
0.3429
0.5029
0.6497
0.7801
0.8914
0.9830
1.0565
1.1184
80°
0.0000
0.1737
0.3422
0.5007
0.6446
0.7697
0.8728
0.9514
1.0054
1.0401
90°
0.0000
0.1736
0.3420
0.5000
0.6428
0.7660
0.8660
0.9397
0.9848
1.0000
255

TABLE
46
ORDINATES OF THE
STANDARD NORMAL CURVE
n, — 1 p-x2/2
y\
y\ \.
0 x
X
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0
.3989
.3970
.3910
.3814
.3683
.3521
.3332
.3123
.2897
.2661
.2420
.2179
.1942
.1714
.1497
.1295
.1109
.0940
.0790
.0656
.0540
.0440
.0355
.0283
.0224
.0175
.0136
.0104
.0079
.0060
.0044
.0033
.0024
.0017
.0012
.0009
.0006
.0004
.0003
.0002
1
.3989
.3965
.3902
.3802
.3668
.3503
.3312
.3101
.2874
.2637
.2396
.2155
.1919
.1691
.1476
.1276
.1092
.0925
.0775
.0644
.0529
.0431
.0347
.0277
.0219
.0171
.0132
.0101
.0077
.0058
.0043
.0032
.0023
.0017
.0012
.0008
.0006
.0004
.0003
.0002
2
.3989
.3961
.3894
.3790
.3653
.3485
.3292
.3079
.2850
.2613
.2371
.2131
.1895
.1669
.1456
.1257
.1074
.0909
.0761
.0632
.0519
.0422
.0339
.0270
.0213
.0167
.0129
.0099
.0075
.0056
.0042
.0031
.0022
.0016
.0012
.0008
.0006
.0004
.0003
.0002
3
.3988
.3956
.3885
.3778
.3637
.3467
.3271
.3056
.2827
.2589
.2347
.2107
.1872
.1647
.1435
.1238
.1057
.0893
.0748
.0620
.0508
.0413
.0332
.0264
.0208
.0163
.0126
.0096
.0073
.0055
.0040
.0030
.0022
.0016
.0011
.0008
.0005
.0004
.0003
.0002
4
.3986
.3951
.3876
.3765
.3621
.3448
.3251
.3034
.2803
.2565
.2323
.2083
.1849
.1626
.1415
.1219
.1040
.0878
.0734
.0608
.0498
.0404
.0325
.0258
.0203
.0158
.0122
.0093
.0071
.0053
.0039
.0029
.0021
.0015
.0011
.0008
.0005
.0004
.0003
.0002
5
.3984
.3945
.3867
.3752
.3605
.3429
.3230
.3011
.2780
.2541
.2299
.2059
.1826
.1604
.1394
.1200
.1023
.0863
.0721
.0596
.0488
.0396
.0317
.0252
.0198
.0154
.0119
.0091
.0069
.0051
.0038
.0028
.0020
.0015
.0010
.0007
.0005
.0004
.0002
.0002
6
.3982
.3939
.3857
.3739
.3589
.3410
.3209
.2989
.2756
.2516
.2275
.2036
.1804
.1582
.1374
.1182
.1006
.0848
.0707
.0584
.0478
.0387
.0310
.0246
.0194
.0151
.0116
.0088
.0067
.0050
.0037
.0027
.0020
.0014
.0010
.0007
.0005
.0003
.0002
.0002
7
.3980
.3932
.3847
.3725
.3572
.3391
.3187
.2966
.2732
.2492
.2251
.2012
.1781
.1561
.1354
.1163
.0989
.0833
.0694
.0573
.0468
.0379
.0303
.0241
.0189
.0147
.0113
.0086
.0065
.0048
.0036
.0026
.0019
.0014
.0010
.0007
.0005
.0003
.0002
.0002
8
.3977
.3925
.3836
.3712
.3555
.3372
.3166
.2943
.2709
.2468
.2227
.1989
.1758
.1539
.1334
.1145
.0973
.0818
.0681
.0562
.0459
.0371
.0297
.0235
.0184
.0143
.0110
.0084
.0063
.0047
.0035
.0025
.0018
.0013
.0009
.0007
.0005
.0003
.0002
.0001
9
.3973
.3918
.3825
.3697
.3538
.3352
.3144
.2920
.2685
.2444
.2203
.1965
.1736
.1518
.1315
.1127
.0957
.0804
.0669
.0551
.0449
.0363
.0290
.0229
.0180
.0139
.0107
.0081
.0061
.0046
.0034
.0025
.0018
.0013
.0009
.0006
.0004
.0003
.0002
.0001
256

TABLE
47
AREAS UNDER THE
STANDARD NORMAL CURVE
from —0° to x
erf (x) - —L f* e-'2'2 dt
Л
V
0 x
X
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0
.5000
.5398
.5793
.6179
.6554
.6915
.7258
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.9998
.9998
.9999
.9999
1.0000
1
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7612
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.9998
.9998
.9999
.9999
1.0000
2
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.9998
.9999
.9999
.9999
1.0000
3
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
4
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7996
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
5
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
6
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
7
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
8
.5319
.5714
.6103
.6480
.6844
.7190
.7518
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.9998
.9999
.9999
.9999
1.0000
9
.5359
.5754
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
.9998
.9999
.9999
.9999
1.0000
257

TABLE
48
PERCENTILE VALUES (V) FOR
STUDENT'S t DISTRIBUTION
with n degrees of freedom
(shaded area = p)
A
Y
tp
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
CO
«.995
63.66
9.92
5.84
4.60
4.03
3.71
3.50
3.36
3.25
3.17
3.11
3.06
3.01
2.98
2.95
2.92
2.90
2.88
2.86
2.84
2.83
2.82
2.81
2.80
2.79
2.78
2.77
2.76
2.76
2.75
2.70
2.66
2.62
2.58
«.99
31.82
6.96
4.54
3.75
3.36
3.14
3.00
2.90
2.82
2.76
2.72
2.68
2.65
2.62
2.60
2.58
2.57
2.55
2.54
2.53
2.52
2.51
2.50
2.49
2.48
2.48
2.47
2.47
2.46
2.46
2.42
2.39
2.36
2.33
«.975
12.71
4.30
3.18
2.78
2.57
2.45
2.36
2.31
2.26
2.23
2.20
2.18
2.16
2.14
2.13
2.12
2.11
2.10
2.09
2.09
2.08
2.07
2.07
2.06
2.06
2.06
2.05
2.05
2.04
2.04
2.02
2.00
1.98
1.96
«.95
6.31
2.92
2.35
2.13
2.02
1.94
1.90
1.86
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.75
1.74
1.73
1.73
1.72
1.72
1.72
1.71
1.71
1.71
1.71
1.70
1.70
1.70
1.70
1.68
1.67
1.66
1.645
«90
3.08
1.89
1.64
1.53
1.48
1.44
1.42
1.40
1.38
1.37
1.36
1.36
1.35
1.34
1.34
1.34
1.33
1.33
1.33
1.32
1.32
1.32
1.32
1.32
1.32
1.32
1.31
1.31
1.31
1.31
1.30
1.30
1.29
1.28
«.80
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.848
.845
.842
«.75
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.677
.674
«.70
.727
.617
.584
.569
.559
.553
.549
.546
.543
.542
.540
.539
.538
.537
.536
.535
.534
.534
.533
.533
.532
.532
.532
.531
.531
.531
.531
.530
.530
.530
.529
.527
.526
.524
«.60
.325
.289
.277
.271
.267
.265
.263
.262
.261
.260
.260
.259
.259
.258
.258
.258
.257
.257
.257
.257
.257
.256
.256
.256
.256
.256
.256
.256
.256
.256
.255
.254
.254
.253
«.55
.158
.142
.137
.134
.132
.131
.130
.130
.129
.129
.129
.128
.128
.128
.128
.128
.128
.127
.127
.127
.127
.127
.127
.127
.127
.127
.127
.127
.127
.127
.126
.126
.126
.126
Source: R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and
Medical Research Fth edition, 1963), Table III, Oliver and Boyd Ltd., Edin-
Edinburgh, by permission of the authors and publishers.
258

TABLE
49
PERCENTILE VALUES (x?) FOR
THE CHI-SQUARE DISTRIBUTION
with n degrees of freedom
(shaded area = p)
n
XP
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
*.995
7.88
10.6
12.8
14.9
16.7
18.5
20.3
22.0
23.6
25.2
26.8
28.3
29.8
31.3
32.8
34.3
35.7
37.2
38.6
40.0
41.4
42.8
44.2
45.6
46.9
48.3
49.6
51.0
52.3
53.7
66.8
79.5
92.0
104.2
116.3
128.3
140.2
V2
X.99
6.63
9.21
11.3
13.3
15.1
16.8
18.5
20.1
21.7
23.2
24.7
26.2
27.7
29.1
30.6
32.0
33.4
34.8
36.2
37.6
38.9
40.3
41.6
43.0
44.3
45.6
47.0
48.3
49.6
50.9
63.7
76.2
88.4
100.4
112.3
124.1
135.8
X.975
5.02
7.38
9.35
11.1
12.8
14.4
16.0
17.5
19.0
20.5
21.9
23.3
24.7
26.1
27.5
28.8
30.2
31.5
32.9
34.2
35.5
36.8
38.1
39.4
40.6
41.9
43.2
44.5
45.7
47.0
59.3
71.4
83.3
95.0
106.6
118.1
129.6
*.95
3.84
5.99
7.81
9.49
11.1
12.6
14.1
15.5
16.9
18.3
19.7
21.0
22.4
23.7
25.0
26.3
27.6
28.9
30.1
31.4
32.7
33.9
35.2
36.4
37.7
38.9
40.1
41.3
42.6
43.8
55.8
67.5
79.1
90.5
101.9
113.1
124.3
Y2
2.71
4.61
6.25
7.78
9.24
10.6
12.0
13.4
14.7
16.0
17.3
18.5
19.8
21.1
22.3
23.5
24.8
26.0
27.2
28.4
29.6
30.8
32.0
33.2
34.4
35.6
36.7
37.9
39.1
40.3
51.8
63.2
74.4
85.5
96.6
107.6
118.5
Y2
X.75
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.2
11.4
12.5
13.7
14.8
16.0
17.1
18.2
19.4
20.5
21.6
22.7
23.8
24.9
26.0
27.1
28.2
29.3
30.4
31.5
32.6
33.7
34.8
45.6
56.3
67.0
77.6
88.1
98.6
109.1
X.50
.455
1.39
2.37
3.36
4.35
5.35
6.35
7.34
8.34
9.34
10.3
11.3
12.3
13.3
14.3
15.3
16.3
17.3
18.3
19.3
20.3
21.3
22.3
23.3
24.3
25.3
26.3
27.3
28.3
29.3
39.3
49.3
59.3
69.3
79.3
89.3
99.3
X.25
.102
.575
1.21
1.92
2.67
3.45
4.25
5.07
5.90
6.74
7.58
8.44
9.30
10.2
11.0
11.9
12.8
13.7
14.6
15.5
16.3
17.2
18.1
19.0
19.9
20.8
21.7
22.7
23.6
24.5
33.7
42.9
52.3
61.7
71.1
80.6
90.1
Y2
x.10
.0158
.211
.584
1.06
1.61
2.20
2.83
3.49
4.17
4.87
5.58
6.30
7.04
7.79
8.55
9.31
10.1
10.9
11.7
12.4
13.2
14.0
14.8
15.7
16.5
17.3
18.1
18.9
19.8
20.6
29.1
37.7
46.5
55.3
64.3
73.3
82.4
*.O5
.0039
.103
.352
.711
1.15
1.64
2.17
2.73
3.33
3.94
4.57
5.23
5.89
6.57
7.26
7.96
8.67
9.39
10.1
10.9
11.6
12.3
13.1
13.8
14.6
15.4
16.2
16.9
17.7
18.5
26.5
34.8
43.2
51.7
60.4
69.1
77.9
X.025
.0010
.0506
.216
.484
.831
1.24
1.69
2.18
2.70
3.25
3.82
4.40
5.01
5.63
6.26
6.91
7.56
8.23
8.91
9.59
10.3
11.0
11.7
12.4
13.1
13.8
14.6
15.3
16.0
16.8
24.4
32.4
40.5
48.8
57.2
65.6
74.2
x.oi
.0002
.0201
.115
.297
.554
.872
1.24
1.65
2.09
2.56
3.05
3.57
4.11
4.66
5.23
5.81
6.41
7.01
7.63
8.26
8.90
9.54
10.2
10.9
11.5
12.2
12.9
13.6
14.3
15.0
22.2
29.7
37.5
45.4
53.5
61.8
70.1
X.005
.0000
.0100
.072
.207
.412
.676
.989
1.34
1.73
2.16
2.60
3.07
3.57
4.07
4.60
5.14
5.70
6.26
6.84
7.43
8.03
8.64
9.26
9.89
10.5
11.2
11.8
12.5
13.1
13.8
20.7
28.0
35.5
43.3
51.2
59.2
67.3
Source: Catherine M. Thompson, Table of percentage points of the x2 distribution,
Biometrika, Vol. 32 A941), by permission of the author and publisher.
259

95th PERCENTILE VALUES FOR
THE F DISTRIBUTION
«i = degrees of freedom for numerator
n-2. = degrees of freedom for denominator
(shaded area = .95)
«\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
40
50
60
70
80
100
150
200
400
CO
1
161.4
18.51
10.13
7.71
6.61
5.99
5.59
5.32
5.12
4.96
4.84
4.75
4.67
4.60
4.54
4.49
4.45
4.41
4.38
4.35
4.30
4.26
4.23
4.20
4.17
4.08
4.03
4.00
3.98
3.96
3.94
3.91
3.89
3.86
3.84
2
199.5
19.00
9.55
6.94
5.79
5.14
4.74
4.46
4.26
4.10
3.98
3.89
3.81
3.74
3.68
3.63
3.59
3.55
3.52
3.49
3.44
3.40
3.37
3.34
3.32
3.23
3.18
3.15
3.13
3.11
3.09
3.06
3.04
3.02
2.99
3
215.7
19.16
9.28
6.59
5.41
4.76
4.35
4.07
3.86
3.71
3.59
3.49
3.41
3.34
3.29
3.24
3.20
3.16
3.13
3.10
3.05
3.01
2.98
2.95
2.92
2.84
2.79
2.76
2.74
2.72
2.70
2.67
2.65
2.62
2.60
4
224.6
19.25
9.12
6.39
5.19
4.53
4.12
3.84
3.63
3.48
3.36
3.26
3.18
3.11
3.06
3.01
2.96
2.93
2.90
2.87
2.82
2.78
2.74
2.71
2.69
2.61
2.56
2.53
2.50
2.48
2.46
2.43
2.41
2.39
2.37
5
230.2
19.30
9.01
6.26
5.05
4.39
3.97
3.69
3.48
3.33
3.20
3.11
3.03
2.96
2.90
2.85
2.81
2.77
2.74
2.71
2.66
2.62
2.59
2.56
2.53
2.45
2.40
2.37
2.35
2.33
2.30
2.27
2.26
2.23
2.21
6
234.0
19.33
8.94
6.16
4.95
4.28
3.87
3.58
3.37
3.22
3.09
3.00
2.92
2.85
2.79
2.74
2.70
2.66
2.63
2.60
2.55
2.51
2.47
2.45
2.42
2.34
2.29
2.25
2.23
2.21
2.19
2.16
2.14
2.12
2.09
8
238.9
19.37
8.85
6.04
4.82
4.15
3.73
3.44
3.23
3.07
2.95
2.85
2.77
2.70
2.64
2.59
2.55
2.51
2.48
2.45
2.40
2.36
2.32
2.29
2.27
2.18
2.13
2.10
2.07
2.05
2.03
2.00
1.98
1.96
1.94
12
243.9
19.41
8.74
5.91
4.68
4.00
3.57
3.28
3.07
2.91
2.79
2.69
2.60
2.53
2.48
2.42
2.38
2.34
2.31
2.28
2.23
2.18
2.15
2.12
2.09
2.00
1.95
1.92
1.89
1.88
1.85
1.82
1.80
1.78
1.75
16
246.3
19.43
8.69
5.84
4.60
3.92
3.49
3.20
2.98
2.82
2.70
2.60
2.51
2.44
2.39
2.33
2.29
2.25
2.21
2.18
2.13
2.09
2.05
2.02
1.99
1.90
1.85
1.81
1.79
1.77
1.75
1.71
1.69
1.67
1.64
20
248.0
19.45
8.66
5.80
4.56
3.87
3.44
3.15
2.93
2.77
2.65
2.54
2.46
2.39
2.33
2.28
2.23
2.19
2.15
2.12
2.07
2.03
1.99
1.96
1.93
1.84
1.78
1.75
1.72
1.70
1.68
1.64
1.62
1.60
1.57
30
250.1
19.46
8.62
5.75
4.50
3.81
3.38
3.08
2.86
2.70
2.57
2.46
2.38
2.31
2.25
2.20
2.15
2.11
2.07
2.04
1.98
1.94
1.90
1.87
1.84
1.74
1.69
1.65
1.62
1.60
1.57
1.54
1.52
1.49
1.46
40
251.1
19.46
8.60
5.71
4.46
3.77
3.34
3.05
2.82
2.67
2.53
2.42
2.34
2.27
2.21
2.16
2.11
2.07
2.02
1.99
1.93
1.89
1.85
1.81
1.79
1.69
1.63
1.59
1.56
1.54
1.51
1.47
1.45
1.42
1.40
50
252.2
19.47
8.58
5.70
4.44
3.75
3.32
3.03
2.80
2.64
2.50
2.40
2.32
2.24
2.18
2.13
2.08
2.04
2.00
1.96
1.91
1.86
1.82
1.78
1.76
1.66
1.60
1.56
1.53
1.51
1.48
1.44
1.42
1.38
1.32
100
253.0
19.49
8.56
5.66
4.40
3.71
3.28
2.98
2.76
2.59
2.45
2.35
2.26
2.19
2.12
2.07
2.02
1.98
1.94
1.90
1.84
1.80
1.76
1.72
1.69
1.59
1.52
1.48
1.45
1.42
1.39
1.34
1.32
1.28
1.24
CO
254.3
19.50
8.53
5.63
4.36
3.67
3.23
2.93
2.71
2.54
2.40
2.30
2.21
2.13
2.07
2.01
1.96
1.92
1.88
1.84
1.78
1.73
1.69
1.65
1.62
1.51
1.44
1.39
1.35
1.32
1.28
1.22
1.19
1.13
1.00
Source: G. W. Snedecor and W. G. Cochran, Statistical Methods Fth edition, 1967), Iowa
State University Press, Ames, Iowa, by permission of the authors and publisher.
260

TABLE
51
99th PERCENTILE VALUES FOR
THE F DISTRIBUTION
«i = degrees of freedom for numerator
»2 = degrees of freedom for denominator
(shaded area - .99)
щ\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
40
50
60
70
80
100
150
200
400
CO
1
4052
98.49
34.12
21.20
16.26
13.74
12.25
11.26
10.56
10.04
9.05
9.33
9.07
8.86
8.68
8.53
8.40
8.28
8.18
8.10
7.94
7.82
7.72
7.64
7.56
7.31
7.17
7.08
7.01
6.96
6.90
6.81
6.76
6.70
6.64
2
4999
99.01
30.81
18.00
13.27
10.92
9.55
8.65
8.02
7.56
7.20
6.93
6.70
6.51
6.36
6.23
6.11
6.01
5.93
5.85
5.72
5.61
5.53
5.45
5.39
5.18
5.06
4.98
4.92
4.88
4.82
4.75
4.71
4.66
4.60
3
5403
99.17
29.46
16.69
12.06
9.78
8.45
7.59
6.99
6.55
6.22
5.95
5.74
5.56
5.42
5.29
5.18
5.09
5.01
4.94
4.82
4.72
4.64
4.57
4.51
4.31
4.20
4.13
4.08
4.04
3.98
3.91
3.88
3.83
3.78
4
5625
99.25
28.71
15.98
11.39
9.15
7.85
7.01
6.42
5.99
5.67
5.41
5.20
5.03
4.89
4.77
4.67
4.58
4.50
4.43
4.31
4.22
4.14
4.07
4.02
3.83
3.72
3.65
3.60
3.56
3.51
3.44
3.41
3.36
3.32
5
5764
99.30
28.24
15.52
10.97
8.75
7.46
6.63
6.06
5.64
5.32
5.06
4.86
4.69
4.56
4.44
4.34
4.25
4.17
4.10
3.99
3.90
3.82
3.76
3.70
3.51
3.41
3.34
3.29
3.25
3.20
3.14
3.11
3.06
3.02
6
5859
99.33
27.41
15.21
10.67
8.47
7.19
6.37
5.80
5.39
5.07
4.82
4.62
4.46
4.32
4.20
4.10
4.01
3.94
3.87
3.76
3.67
3.59
3.53
3.47
3.29
3.18
3.12
3.07
3.04
2.99
2.92
2.90
2.85
2.80
8
5981
99.36
27.49
14.80
10.27
8.10
6.84
6.03
5.47
5.06
4.74
4.50
4.30
4.14
4.00
3.89
3.79
3.71
3.63
3.56
3.45
3.36
3.29
3.23
3.17
2.99
2.88
2.82
2.77
2.74
2.69
2.62
2.60
2.55
2.51
12
6106
99.42
27.05
14.37
9.89
7.72
6.47
5.67
5.11
4.71
4.40
4.16
3.96
3.80
3.67
3.55
3.45
3.37
3.30
3.23
3.12
3.03
2.96
2.90
2.84
2.66
2.56
2.50
2.45
2.41
2.36
2.30
2.28
2.23
2.18
16
6169
99.44
28.63
14.15
9.68
7.52
6.27
5.48
4.92
4.52
4.21
3.98
3.78
3.62
3.48
3.37
3.27
3.19
3.12
3.05
2.94
2.85
2.77
2.71
2.66
2.49
2.39
2.32
2.28
2.24
2.19
2.12
2.09
2.04
1.99
20
6208
99.45
26.69
14.02
9.55
7.39
6.15
5.36
4.80
4.41
4.10
3.86
3.67
3.51
3.36
3.25
3.16
3.07
3.00
2.94
2.83
2.74
2.66
2.60
2.55
2.37
2.26
2.20
2.15
2.11
2.06
2.00
1.97
1.92
1.87
30
6258
99.47
26.50
13.83
9.38
7.23
5.98
5.20
4.64
4.25
3.94
3.70
3.51
3.34
3.20
3.10
3.00
2.91
2.84
2.77
2.67
2.58
2.50
2.44
2.38
2.20
2.10
2.03
1.98
1.94
1.89
1.83
1.79
1.74
1.69
40
6286
99.48
26.41
13.74
9.29
7.14
5.90
5.11
4.56
4.17
3.86
3.61
3.42
3.26
3.12
3.01
2.92
2.83
2.76
2.69
2.58
2.49
2.41
2.35
2.29
2.11
2.00
1.93
1.88
1.84
1.79
1.72
1.69
1.64
1.59
50
6302
99.48
26.35
13.69
9.24
7.09
5.85
5.06
4.51
4.12
3.80
3.56
3.37
3.21
3.07
2.96
2.86
2.78
2.70
2.63
2.53
2.44
2.36
2.30
2.24
2.05
1.94
1.87
1.82
1.78
1.73
1.66
1.62
1.57
1.52
100
6334
99.49
26.23
13.57
9.13
6.99
5.75
4.96
4.41
4.01
3.70
3.46
3.27
3.11
2.97
2.86
2.76
2.68
2.60
2.53
2.42
2.33
2.25
2.18
2.13
1.94
1.82
1.74
1.69
1.65
1.59
1.51
1.48
1.42
1.36
6366
99.50
26.12
13.46
9.02
6.88
5.65
4.86
4.31
3.91
3.60
3.36
3.16
3.00
2.87
2.75
2.65
2.57
2.49
2.42
2.31
2.21
2.13
2.06
2.01
1.81
1.68
1.60
1.53
1.49
1.43
1.33
1.28
1.19
1.00
Source: G. W. Snedecor and W. G. Cochran, Statistical Methods Fth edition, 1967), Iowa
State University Press, Ames, Iowa, by permission of the authors and publisher.
261

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54398
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53298
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52078
75797
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97810
09591
85762
48236
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06568
25424
45406
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25708
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16530
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21387
55870
86707
85659
55189
41889
85418
16835
28195
76552
51817
90985
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11785
92843
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15815
30763
03878
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36081
00745
25439
68829
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36732
28868
09908
55261
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92486
07516
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10863
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17169
50884
65253
88036
06652
71590
47152
24819
72484
99431
36574
59009
91341
99247
85915
54083
95715
19640
97453
93507
88116
14070
11822
24034
41982
16159
35683
52984
94923
50995
72139
38714
84821
46149
19219
23631
02526
¦ 87056
90581
94271
42187
74950
15804
67283
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14676
47280
76168
75936
20507
70185
38723
63886
03229
45943
05825
33537
у /
Ci
А
262

Index of Special Symbols and Notations
The following list shows special symbols and notations used in this book together with pages on which
they are defined or first appear. Cases where a symbol has more than one meaning will be clear from
the context.
Symbols
Bern (x), Bein (x) 140
B(m, n) beta function, 103
Bn Bernoulli numbers, 114
C(x) Fresnel cosine integral, 184
Ci(x) cosine integral, 184
e natural base of logarithms, 1
elt e2, e3 unit vectors in curvilinear coordinates, 124
erf (x) error function, 183
erfc (x) complementary error function, 183
E = E(k, тг/2) complete elliptic integral of second kind, 179
E(k, ф) incomplete elliptic integral of second kind, 179
Ei(x) exponential integral, 183
En Euler numbers, 114
F(a, b; c; x) hypergeometric function, 160
F(k, ф) incomplete elliptic integral of first kind, 179
y, y-i Fourier transform and inverse Fourier transform, 175, 176
hlt h2, h3 scale factors in curvilinear coordinates, 124
Hn(x) Hermite polynomials, 151
Han\x), H'l\x) Hankel functions of first and second kind, 138
i imaginary unit, 21
i, j,k unit vectors in rectangular coordinates, 117
/„ (x) modified Bessel function of first kind, 138
Jn(x) Bessel function of first kind, 136
К = F(k, тг/2) complete elliptic integral of first kind, 179
Кегп(ж), Kein(a;) 140
Kn(x) modified Bessel function of second kind, 139
In x or loge x natural logarithm of x, 24
log x or log10 x common logarithm of x, 23
Ln(x) Laguerre polynomials, 153
Ln(x) associated Laguerre polynomials, 155
Jl, .C Laplace transform and inverse Laplace transform, 161
Pn(x) Legendre polynomials, 146
P™(x) associated Legendre functions of first kind, 149
Qn(x) Legendre functions of second kind, 148
Q7n(x) associated Legendre functions of second kind, 150
r cylindrical coordinate, 49
r polar coordinate, 22, 36
r spherical coordinate, 50
S(x) Fresnel sine integral, 184
Si(x) sine integral, 183
Tn(x) Chebyshev polynomials of first kind, 157
Un(x) Chebyshev polynomials of second kind, 158
Yn(x) Bessel function of second kind, 136
263

264
INDEX OF SPECIAL SYMBOLS AND NOTATIONS
Greek Symbols
у Euler's constant, 1
Г(х) gamma function, 1, 101
i;(x) Riemann zeta function, 184
в cylindrical coordinate, 49
в polar coordinate, 22, 36
в spherical coordinate, 50
7Г 1
Ф spherical coordinate, 50
Ф(р) the sum 1 + -r + тг + • •
+ -, Ф@) = 0, 137
Ф(х) probability distribution function, 189
A
A
A
A
A
<B
а В
SB
~B
~B
|A|
w!
/n\
W
etc. 1
Notations
A = В A equals В or A is equal to В
A > В A is greater than В [or В is less than A]
A is less than В [or В is greater than A]
A is greater than or equal to В
A is less than or equal to В
A is approximately equal to В
A is asymptotic to В or A/B approaches 1, 102
A if A S 0
-A if A S 0
factorial n, 3
dP
df_ df_ a2/
dx ' dy ' Bx dy'
t)(x, y, Z)
absolute value of A =
binomial coefficients, 3
derivatives of у or f(x) with respect to x, 53, 55
= -j—j, pth derivative with respect to x, 55
dy differential of y, 55
partial derivatives, 56
Jacobian, 125
a(ttj, tt2, us)
I f(x) dx indefinite integral, 57
f{x) dx definite integral, 94
JA • dr line integral of A along C, 121
с
A • В dot product of A and B, 117
A X В cross product of A and B, 118
V del operator, 119
V2 = V • V Laplacian operator, 120
V4 = V2(V2) biharmonic operator, 120

INDEX
Addition formulas, for Bessel functions, 145
for elliptic functions, 180
for Hermite polynomials, 152
for hyperbolic functions, 27
for trigonometric functions, 15
Agnesi, witch of, 43
Algebraic equations, solutions of, 32, 33
Amplitude, of complex number, 22
of elliptic integral, 179
Analytic geometry, plane [see Plane analytic
geometry]; solid [see Solid analytic geometry]
Angle between lines, in a plane, 35
in space, 47
Annuity, amount of, 201, 242
present value of, 243
Anti-derivative, 57
Antilogarithms, common, 23, 195, 204, 205
natural or Napierian, 24, 226, 227
Archimedes, spiral of, 45
Area integrals, 122
Argand diagram, 22
Arithmetic-geometric series, 107
Arithmetic mean, 185
Arithmetic series, 107
Associated Laguerre polynomials, 155, 156
[see also Laguerre polynomials]
generating function for, 155
orthogonal series for, 156
orthogonality of, 156
recurrence formulas for, 156
special, 155
special results involving, 156
Associated Legendre functions, 149, 150 [see also
Legendre functions]
generating function for, 149
of the first kind, 149
of the second kind, 150
orthogonal series for, 150
orthogonality of, 150
recurrence formulas for, 149
special, 149
Associative law, 117
Asymptotes of hyperbola, 39
Asymptotic expansions or formulas, for Bernoulli
numbers, 115
for Bessel functions, 143
for gamma function, 102
Base of logarithms, 23
change of, 24
Ber and Bei functions, 140, 141
definition of, 140
differential equation for, 141
graphs of, 141
Bernoulli numbers, 98, 107, 114, 115
asymptotic formula for, 115
definition of, 114
relationship to Euler numbers, 115
series involving, 115
table of first few, 114
Bernoulli's differential equation, 104
Bessel functions, 136-145
addition formulas for, 145
asymptotic expansions of, 143
definite integrals involving, 142, 143
generating functions for, 137, 139
graphs of, 141
indefinite integrals involving, 142
infinite products for, 188
integral representations for, 143
modified [see Modified Bessel functions]
of first kind of order n, 136, 137
of order half an odd integer, 138
of second kind of order n, 136, 137
orthogonal series for, 144, 145
recurrence formulas for, 137
tables of, 244-249
zeros of, 250
Bessel's differential equation, 106, 136
general solution of, 106, 137
transformed, 106
Bessel's modified differential equation, 138
general solution of, 139
Beta function, 103
relationship of to gamma function, 103
Biharmonic operator, 120
in curvilinear coordinates, 125
Binomial coefficients, 3
properties of, 4
table of values for, 236, 237
Binomial distribution, 189
Binomial formula, 2
Binomial series, 2, 110
Bipolar coordinates, 128, 129
Laplacian in, 128
Branch, principal, 17
Briggsian logarithms, 23
Cardioid, 41, 42, 44
Cassini, ovals of, 44
Catalan's constant, 181
Catenary, 41
Cauchy or Euler differential equation, 105
Cauchy-Schwarz inequality, 185
for integrals, 186
Cauchy's form of remainder in Taylor series, 110
Chain rule for derivatives, 53
Characteristic, 194
Chebyshev polynomials, 157-159
generating functions for, 157, 158
of first kind, 157
of second kind, 158
orthogonality of, 158, 159
orthogonal series for, 158, 159
recursion formulas for, 158, 159
relationships involving, 159
special, 157, 158
special values of, 157, 159
Chebyshev's differential equation, 157
general solution of, 159
265

266
INDEX
Chebyshev's inequality, 186
Chi square distribution, 189
percentile values for, 259
Circle, area of, 6
equation of, 37
involute of, 43
perimeter of, 6
sector of [see Sector of circle]
segment of [see Segment of circle]
Cissoid of Diodes, 45
Common antilogarithms, 23, 195, 204, 205
sample problems involving, 195
table of, 204, 205
Common logarithms, 23, 194, 202, 203
computations using, 196
sample problems involving, 194
table of, 202, 203
Commutative law, for dot products, 118
for vector addition, 117
Complement, 20
Complementary error function, 183
Complex conjugate, 21
Complex inversion formula, 161
Complex numbers, 21, 22, 25
addition of, 21
amplitude of, 22
conjugate, 21
definitions involving, 21
division of, 21, 25
graphs of, 22
imaginary part of, 21
logarithms of, 25
modulus of, 22
multiplication of, 21, 25
polar form of, 22, 25
real part of, 21
roots of, 22, 25
subtraction of, 21
vector representation of, 22
Components of a vector, 117
Component vectors, 117
Compound amount, table of, 240
Cone, elliptic, 51
right circular [see Right circular cone]
Confocal ellipses, 127
ellipsoidal coordinates, 130
hyperbolas, 127
parabolas, 126
paraboloidal coordinates, 130
Conical coordinates, 129
Laplacian in, 129
Conies, 37 [see also Ellipse, Parabola, Hyperbola]
Conjugate, complex, 21
Constant of integration, 57
Convergence, interval of, 110
of Fourier series, 131
Convergence factors, table of, 192
Coordinate curves, 124
system, 11
Coordinates, curvilinear, 124-130
cylindrical, 49, 126
polar, 22, 36
rectangular, 36, 117
Coordinates, curvilinear (cont.)
rotation of, 36, 49
special orthogonal, 126-130
spherical, 50, 126
transformation of, 36, 48, 49
translation of, 36, 49
Cosine integral, 184
Fresnel, 184
table of values for, 251
Cosines, law of for plane triangles, 19
law of for spherical triangles, 19
Counterclockwise, 11
Cross or vector product, 118
Cube, duplication of, 45
Cube roots, table of, 238, 239
Cubes, table of, 238, 239
Cubic equation, solution of, 32
Curl, 120
in curvilinear coordinates, 125
Curtate cycloid, 42
Curves, coordinate, 124
special plane, 40-45
Curvilinear coordinates, 124, 125
orthogonal, 124-130
Cycloid, 40, 42
curtate, 42
prolate, 42
Cylinder, elliptic, 51
lateral surface area of, 8, 9
volume of, 8, 9
Cylindrical coordinates, 49, 126
Laplacian in, 126
Definite integrals, 94-100
approximate formulas for, 95
definition of, 94
general formulas involving, 94, 95
table of, 95-100
Degrees, 1, 199, 200
conversion of to radians, 199, 200, 223
relationship of to radians, 12, 199, 200
Del operator, 119
miscellaneous formulas involving, 120
Delta function, 170
DeMoivre's theorem, 22, 25
Derivatives, 53-56 [see also Differentiation]
anti-, 57
chain rule for, 53
definition of, 53
higher, 55
of elliptic functions, 181
of exponential and logarithmic functions, 54
of hyperbolic and inverse hyperbolic
functions, 54, 55
of trigonometric and inverse trigonometric
functions, 54
of vectors, 119
partial, 56
Descartes, folium of, 43
Differential equations, solutions of basic, 104-106
Differentials, 55
rules for, 56
Differentiation, 53 [see also Derivatives]

\
INDEX
267
Differentiation (cont.)
general rules for, 53
of integrals, 95
Diocles, cissoid of, 45
Direction cosines, 46, 47
numbers, 46, 48
Directrix, 37
Discriminant, 32
Distance, between two points in a plane, 34
between two points in space, 46
from a point to a line, 35
from a point to a plane, 48
Distributions, probability, 189
Distributive law, 117
for dot products, 118
Divergence, 119
in curvilinear coordinates, 125
Divergence theorem, 123
Dot or scalar product, 117, 118
Double angle formulas, for hyperbolic functions, 27
for trigonometric functions, 16
Double integrals, 122
Duplication formula for gamma functions, 102
Duplication of cube, 45
Eccentricity, definition of, 37
of ellipse, 38
of hyperbola, 39
of parabola, 37
Ellipse, 7, 37, 38
area of, 7
eccentricity of, 38
equation of, 37, 38
evolute of, 44
focus of, 38
perimeter of, 7
semi-major and-minor axes of, 7, 38
Ellipses, confocal, 127
Ellipsoid, equation of, 51
volume of, 10
Elliptic cone, 51
cylinder, 51
paraboloid, 52
Elliptic cylindrical coordinates, 127
Laplacian in, 127
Elliptic functions, 179-182 [see also Elliptic
integrals]
addition formulas for, 180
derivatives of, 181
identities involving, 181
integrals of, 182
Jacobi's, 180
periods of, 181
series expansions for, 181
special values of, 182
Elliptic integrals, 179,180 [see also Elliptic functions]
amplitude of, 179
Landen's transformation for, 180
Legendre's relation for, 182
of the first kind, 179
of the second kind, 179
of the third kind, 179, 180
table of values for, 254, 255
Envelope, 44
Epicycloid, 42
Equation of line, 34
general, 35
in parametric form, 47
in standard form, 47
intercept form for, 34
normal form for, 35
perpendicular to plane, 48
Equation of plane, general, 47
intercept form for, 47
normal form for, 48
passing through three points, 47
Error function, 183
complementary, 183
table of values of, 257
Euler numbers, 114, 115
definition of, 114
relationship of, to Bernoulli numbers, 115
series involving, 115
table of first few, 114
Euler or Cauchy differential equation, 105
Euler-Maclaurin summation formula, 109
Euler's constant, 1
Euler's identities, 24
Evolute of an ellipse, 44
Exact differential equation, 104
Exponential functions, 23-25, 200
periodicity of, 24
relationship of to trigonometric functions, 24
sample problems involving calculation of, 200
series for, 111
table of, 226, 227
Exponential integral, 183
table of values for, 251
Exponents, 23
F distribution, 189
95th and 99th percentile values for, 260, 261
Factorial n, 3
table of values for, 234
Factors, 2
Focus, of conic, 37
of ellipse, 38
of hyperbola, 39
of parabola, 38
Folium of Descartes, 43
Fourier series, 131-135
complex form of, 131
convergence of, 131
definition of, 131
Parseval's identity for, 131
special, 132-135
Fourier transforms, 174-178
convolution theorem for, 175
cosine, 176
definition of, 175
Parseval's identity for, 175
sine, 175
table of, 176-178
Fourier's integral theorem, 174
Fresnel sine and cosine integrals, 184

268
INDEX
Frullani's integral, 100
Frustrum of right circular cone, lateral surface
area of, 9
volume of, 9
Gamma function, 1, 101, 102
asymptotic expansions for, 102
definition of, 101, 102
derivatives of, 102
duplication formula for, 102
for negative values, 101
graph of, 101
infinite product for, 102, 188
recursion formula for, 101
relationship of to beta function, 103
relationships involving, 102
special values for, 101
table of values for, 235
Gaussian plane, 22
Gauss' theorem, 123
Generalized integration by parts, 59
Generating functions, 137, 139, 146, 149, 151, 153,
155, 157, 158
Geometric formulas, 5-10
Geometric mean, 185
Geometric series, 107
arithmetic-, 107
Gradient, 119
in curvilinear coordinates, 125
Green's first and second identities, 124
Green's theorem, 123
Half angle formulas, for hyperbolic functions, 27
for trigonometric functions, 16
Half rectified sine wave function, 172
Hankel functions, 138
Harmonic mean, 185
Heaviside's unit function, 173
Hermite polynomials, 151, 152
addition formulas for, 152
generating function for, 151
orthogonal series for, 152
orthogonality of, 152
recurrence'formulas for, 151
Rodrigue's formula for, 151
special, 151
special results involving, 152
Hermite's differential equation, 151
Higher derivatives, 55
Leibnitz rule for, 55
Holder's inequality, 185
for integrals, 186
Homogeneous differential equation, 104
linear second order, 105
Hyperbola, 37, 39
asymptotes of, 39
eccentricity of, 39
equation of, 37
focus of, 39
length of major and minor axes of, 39
Hyperbolas, confocal, 127
Hyperbolic functions, 26-31
addition formulas for, 27
Hyperbolic functions (cont.)
definition of, 26
double angle formulas for, 27
graphs of, 29
half angle formulas for, 27
inverse [see Inverse hyperbolic functions]
multiple angle formulas for, 27
of negative arguments, 26
periodicity of, 31
powers of, 28
relationship of to trigonometric functions, 31
relationships among, 26, 28
sample problems for calculation of, 200, 201
series for, 112
sum, difference and product of, 28
table of values for, 228-233
Hyperbolic paraboloid, 52
Hyperboloid, of one sheet, 51
of two sheets, 52
Hypergeometric differential equation, 160
distribution, 189
Hypergeometric functions, 160
miscellaneous properties of, 160
special cases of, 160
Hypocycloid, general, 42
with four cusps, 40
Imaginary part of a complex number, 21
Imaginary unit, 21
Improper integrals, 94
Indefinite integrals, 57-93
definition of, 57
table of, 60-93
transformation of, 59, 60
Inequalities, 185, 186
Infinite products, 102, 188
series [see Series]
Initial point of a vector, 116
Integral calculus, fundamental theorem of, 94
Integrals, definite [see Definite integrals]
double, 122
improper, 94
indefinite [see Indefinite integrals]
involving vectors, 121
line [see Line integrals]
multiple, 122, 125
Integration, 57 [see also Integrals]
constants of, 57
general rules of, 57-59
Integration by parts, 57
generalized, 59
Intercepts, 34, 47
Interest, 201, 240-243
Interpolation, 195
Interval of convergence, 110
Inverse hyperbolic functions, 29-31
definition of, 29
expressed in terms of logarithmic functions, 29
graphs of, 30
principal values for, 29
relationship of to inverse trigonometric
functions, 31
relationships between, 30

INDEX
269
Inverse Laplace transforms, 161
Inverse trigonometric functions, 17-19
definition of, 17
graphs of, 18, 19
principal values for, 17
relations between, 18
relationship of to inverse hyperbolic
functions, 31
Involute of a circle, 43
Jacobian, 125
Jacobi's elliptic functions, 180
Ker and Kei functions, 140, 141
definition of, 140
differential equation for, 141
graphs of, 141
Lagrange form of remainder in Taylor series, 110
Laguerre polynomials, 153, 154
associated [see Associated Laguerre polynomials]
generating function for, 153
orthogonal series for, 154
orthogonality of, 154
recurrence formulas for, 153
Rodrigue's formula for, 153
special, 153
Laguerre's associated differential equation, 155
Laguerre's differential equation, 153
Landen's transformation, 180
Laplace transforms, 161-173
complex inversion formula for, 161
definition of, 161
inverse, 161
table of, 162-173
Laplacian, 120
in curvilinear coordinates, 125
Legendre functions, 146-148 [see also Legendre
polynomials]
associated [see Associated Legendre functions]
of the second kind, 148
Legendre polynomials, 146, 147 [see also
Legendre functions]
generating function for, 146
orthogonal series of, 147
orthogonality of, 147
recurrence formulas for, 147
Rodrigue's formula for, 146
special, 146
special results involving, 147
table of values for, 252, 253
Legendre's associated differential equation, 149
general solution of, 150
Legendre's differential equation, 106, 146
general solution of, 148
Legendre's relation for elliptic integrals, 182
Leibnitz's rule, for differentiation of integrals, 95
for higher derivatives of products, 55
Lemniscate, 40, 44
Limacon of Pascal, 41, 44
Line, equation of [see Equation of line]
integrals [see Line integrals]
slope of, 34
Linear first order differential equation, 104
second order differential equation, 105
Line integrals, 121, 122
definition of, 121
independence of path of, 121, 122
properties of, 121
Logarithmic functions, 23-25 [see also Logarithms]
series for, 111
Logarithms, 23 [see also Logarithmic functions]
antilogarithms and [see Antilogarithms]
base of, 23
Briggsian, 23
change of base of, 24
characteristic of, 194
common [see Common logarithms]
mantissa of, 194
natural, 24
of complex numbers, 25
of trigonometric functions, 216-221
Maclaurin series, 110
Mantissa, 194
Mean value theorem, for definite integrals, 94
generalized, 95
Minkowski's inequality, 186
for integrals, 186
Modified Bessel functions, 138, 139
differential equation for, 138
generating function for, 139
graphs of, 141
of order half an odd integer, 140
recurrence formulas for, 139
Modulus, of a complex number, 22
Moments of inertia, special, 190, 191
Multinomial formula, 4
Multiple angle formulas, for hyperbolic
functions, 27
for trigonometric functions, 16
Multiple integrals, 122
transformation of, 125
Napierian logarithms, 24, 196
tables of, 224, 225
Napier's rules, 20
Natural logarithms and antilogarithms, 24, 196
tables of, 224-227
Neumann's function, 136
Nonhomogeneous equation, linear second order, 105
Normal, outward drawn or positive, 123
unit, 122
Normal curve, areas under, 257
ordinates of, 256
Normal distribution, 189
Normal form, equation of line in, 35
equation of plane in, 48
Null function, 170
Null vector, 116
Numbers, complex [see Complex numbers]
Oblate spheroidal coordinates, 128
Laplacian in, 128
Orthogonal curvilinear coordinates, 124-130
formulas involving, 125

270
INDEX
Orthogonality and orthogonal series, 144, 145,
147, 150, 152, 154, 156, 158, 159
Ovals of Cassini, 44
Parabola, 37, 38
eccentricity of, 37
equation of, 37, 38
focus of, 38
segment of [see Segment of parabola]
Parabolas, confocal, 126
Parabolic cylindrical coordinates, 126
Laplacian in, 126
Parabolic formula for definite integrals, 95
Paraboloid elliptic, 52
hyperbolic, 52
Paraboloid of revolution, volume of, 10
Paraboloidal coordinates, 127
Laplacian in, 127
Parallel, condition for lines to be, 35
Parallelepiped, rectangular [see Rectangular
parallelepiped]
volume of, 8
Parallelogram, area of, 5
perimeter of, 5
Parallelogram law for vector addition, 116
Parseval's identity, for Fourier transforms, 175
for Fourier series, 131
Partial derivatives, 56
Partial fraction expansions, 187
Pascal, limacon of, 41, 44
Pascal's triangle, 4, 236
Perpendicular, condition for lines to be, 35
Plane, equation of [see Equation of plane]
Plane analytic geometry, formulas from, 34-39
Plane triangle, area of, 5, 35
law of cosines for, 19
law of sines for, 19
law of tangents for, 19
perimeter of, 5
radius of circle circumscribing, 6
radius of circle inscribed in, 6
relationships between sides and angles of, 19
Poisson distribution, 189
Poisson summation formula, 109
Polar coordinates, 22, 36
transformation from rectangular to, 36
Polar form, expressed as an exponential, 25
multiplication and division in, 22
of a complex number, 22, 25
operations in, 25
Polygon, regular [see Regular polygon]
Power, 23
Power series, 110
reversion of, 113
Present value, of an amount, 241
of an annuity, 243
Principal branch, 17
Principal values, for inverse hyperbolic functions, 29
for inverse trigonometric functions, 17, 18
Probability distributions, 189
Products, infinite, 102, 188
special, 2
Prolate cycloid, 42
Prolate spheroidal coordinates, 128
Laplacian in, 128
Pulse function, 173
Pyramid, volume of, 9
Quadrants, 11
Quadratic equation, solution of, 32
Quartic equation, solution of, 33
Radians, 1, 12, 199, 200
relationship of to degrees, 12, 199, 200
table for conversion of, 222
Random numbers, table of, 262
Real part of a complex number, 21
Reciprocals, table of, 238, 239
Rectangle, area of, 5
perimeter of, 5
Rectangular coordinate system, 117
Rectangular coordinates, transformation of to
polar coordinates, 36
Rectangular formula for definite integrals, 95
Rectangular parallelepiped, volume of, 8
surface area of, 8
Rectified sine wave function, 172
half, 172
Recurrence or recursion formulas, 101, 137, 139,
147, 149, 151, 153, 156, 158, 159
Regular polygon, area of, 6
circumscribing a circle, 7
inscribed in a circle, 7
perimeter of, 6
Reversion of power series, 113
Riemann zeta function, 184
Right circular cone, frustrum of
[see Frustrum of right circular cone]
lateral surface area of, 9
volume of, 9
Right-handed system, 118
Rodrigue's formulas, 146, 151, 153
Roots, of complex numbers, 22, 25
table of square and cube, 238, 239
Rose, three- and four-leaved, 41
Rotation of coordinates, in a plane, 36
in space, 49
Saw tooth wave function, 172
Scalar or dot product, 117, 118
Scalars, 116
Scale factors, 124
Schwarz inequality [see Cauchy-Schwarz inequality]
Sector of circle, arc length of, 6
area of, 6
Segment of circle, area of, 7
Segment of parabola, area of, 7
arc length of, 7
Separation of variables, 104
Series, arithmetic, 107
arithmetic-geometric, 107
binomial, 2, 110
Fourier [see Fourier series]
geometric, 107
of powers of positive integers, 107, 108
of reciprocals of powers of positive integers,
108, 109

INDEX
271
Series, arithmetic (cont.)
orthogonal [see Orthogonality and orthogonal series]
power, 110, 113
Taylor [see Taylor series]
Simple closed curve, 123
Simpson's formula for definite integrals, 95
Sine integral, 183
Fresnel, 184
table of values for, 251
Sines, law of for plane triangle, 19
law of for spherical triangle, 19
Slope of line, 34
Solid analytic geometry, formulas from, 46-52
Solutions of algebraic equations, 32, 33
Sphere, equation of, 50
surface area of, 8
triangle on [see Spherical triangle]
volume of, 8
Spherical cap, surface area of, 9
volume of, 9
Spherical coordinates, 50, 126
Laplacian in, 126
Spherical triangle, area of, 10
Napier's rules for right angled, 20
relationships between sides and angles of, 19, 20
Spiral of Archimedes, 45
Square roots, table of, 238, 239
Square wave function, 172
Squares, table of, 238, 239
Step function, 173
Stirling's asymptotic series, 102
formula, 102
Stoke's theorem, 123
Student's t distribution, 189
percentile values for, 258
Summation formula, Euler-Maclaurin, 109
Poisson, 109
Sums [see Series]
Surface integrals, 122
relation of to double integral, 123
Tangent vectors to curves, 124
Tangents, law of for plane triangle, 19
law of for spherical triangle, 20
Taylor series, 110-113
for functions of one variable, 110
for functions of two variables, 113
Terminal point of a vector, 116
Toroidal coordinates, 129
Laplacian in, 129
Torus, surface area of, 10
volume of, 10
Tractrix, 43
Transformation, Jacobian of, 125
of coordinates, 36, 48, 49, 124
of integrals, 59, 60, 125
Translation of coordinates, in a plane, 36
in space, 49
Trapezoid, area of, 5
perimeter of, 5
Trapezoidal formula for definite integrals, 95
Triangle, plane [see Plane triangle]
spherical [see Spherical triangle]
Triangle inequality, 185
Triangular wave function, 172
Trigonometric functions, 11-20
addition formulas for, 15
definition of, 11
double angle formulas for, 16
exact values of for various angles, 13
for various quadrants in terms of
quadrant I, 15
general formulas involving, 17
graphs of, 14
half angle formulas, 16
inverse [see Inverse trigonometric functions]
multiple angle formulas for, 16
of negative angles, 14
powers of, 16
relationship of to exponential functions, 24
relationship of to hyperbolic functions, 31
relationships among, 12, 15
sample problems involving, 197-199
series for, 111
signs and variations of, 12
sum, difference and product of, 17
table of in degrees and minutes, 206-211
table of in radians, 212-215
table of logarithms of, 216-221
Triple integrals, 122
Trochoid, 42
Unit function, Heaviside's, 173
Unit normal to a surface, 122
Unit vectors, 117
Vector algebra, laws of, 117
Vector analysis, formulas from, 116-130
Vector or cross product, 118
Vectors, 116
addition of, 116, 117
complex numbers as, 22
components of, 117
equality of, 117
fundamental definitions involving, 116, 117
multiplication of by scalars, 117
notation for, 116
null, 116
parallelogram law for, 116
sums of, 116, 117
tangent, 124
unit, 117
Volume integrals, 122
Wallis' product, 188
Weber's function, 136
Witch of Agnesi, 43
x axis, 11
x intercept, 34
у axis, 11
у intercept, 34
Zero vector, 116
Zeros of Bessel functions, 250
Zeta function of Riemann, 184