Tables of integrals and other math data 3rd ed h dwight

When an integral contains the logarithm of a certain quantity, integration should not be carried from a negative to a positive value of that quantity.. If the quantity is negative, the

A Series of Mathematical Texts

NEW YORK CHICAGO DALLAS dTI,ANTh BAN FRANCISCO LONDON M*NILA BRETT-MACMILLAN LTD

TORONTO

f

i

i

TABLES OF INTEGRALS AND OTHER MATHEMATICAL DATA

Professor of Electrical Machinery

Massachusetts Insta’tute of Technology

THIRD EDITION

.New York

Third Edition @ The Macmillan Company 1957 All rights reserved-no part of this book may be reproduced in any form without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in magazine or newspaper Printed in the United States of America

Firsl Printing

Previous editions copyright, 1934, 1947,

by The Marmillan Company

Library of Congress catalog card number: 57-7909

PREFACE TO THE FIRST EDITION The first study of any portion of mathematics should not be done from a synopsis of compact results, such as this collection The references, although they are far from complete, will be helpful, it is hoped, in showing where the derivation of the results

is given or where further similar results may be found A list

of numbered references is given at the end of the book These are referred to in the text as “Ref 7, p 32,” etc., the page num- ber being that of the publication to which reference is made Letters are considered to represent real quantities unless other- wise stated Where the square root of a quantity is indicated, the positive value is to be taken, unless otherwise indicated Two vertical lines enclosing a quantity represent the absolute or numerical value of that quantity, that is, the modulus of the quantity The absolute value is a positive quantity Thus, log I- 31 = log 3

The constant of integration is to be understood after each integral The integrals may usually be checked by differentiat- ing

In algebraic expressions, the symbol log represents natural

or Napierian logarithms, that is, logarithms to the base e When any other base is intended, it will be indicated in the usual ~ manner When an integral contains the logarithm of a certain

quantity, integration should not be carried from a negative to a positive value of that quantity If the quantity is negative, the logarithm of the absolute value of the quantity may be used, since log (- 1) = (2k + 1) ?ri will be part of the constant of integration (see 409.03) Accordingly, in many cases, the loga- rithm of an absolute value is shown, in giving an integral, so as

to indicate that it applies to real values, both positive and negative

Inverse trigonometric functions are to be understood as refer- ring to the principal values

Suggestions and criticisms as to the material of this book and

as to errors that may be in it, will be welcomed

v

PREFACE TO THE SECOND EDITION

A considerable number of items have been added, including groups of integrals involving

(ax2 + 62~ + fP2, r+kx and a + l cos 2 )

also additional material on inverse functions of complex quanti- ties and on Bessel functions A probability integral table (No 1045) has been included

It is desired to express appreciation for valuable suggestions from Professor Wm R Smythe of California Institute of Tech- nology and for the continued help and interest of Professor Philip Franklin of the Department of Mathematics, Massachusetts In- stitute of Technology

CAMBRIDGE, MASS

PREFACE TO THE THIRD EDITION

In this edition, items 59.1 and 59.2 on determinants have been added The group (No 512) of derivatives of inverse trigo- nometric functions has been made more complete On page 271 material is given, suggested by Dr Rose M Ring, which extends the tables of ez and e-z considerably, and is convenient when a calculating machine is used

Tables 1015 and 1016 of trigonometric functions of hundredths

of degrees are given in this edition on pages 220 to 257 When calculating machines are used, the angles of a problem are

PREFACE vii usually given in decimals A great many trigonometric formulas involve addition of angles or multiplication of them by some quantity, and even when the angles are given in degrees, minutes, and seconds, to change the values to decimals of a degree gives the advantages that are always afforded by a decimal system compared with older and more awkward units In such cases, the tables in hundredths of degrees are advantageous

LEXINGTON, MASS

38 RATIONAL ALGEBRAIC FUNCTIONS

CONTENTS

1 ALGEBRAIC FUNCTIONS 1

60 Algebraic Functions-Derivatives 14

80 Rational Algebraic Functions-Integrals 16

180 Irrational Algebraic Functions-Integrals 39

400 TRIGONOMETRIC FUNCTIONS 73

427 Trigonometric Functions-Derivatives 87

429 Trigonometric Functions-Integrals 87

500 INVERSE TRIGONOMETRIC FUNCTIONS 112

512 Inverse Trigonometric Functions-Derivatives 115

515 Inverse Trigonometric Functions-Integrals 116

550 EXPONENTIAL FUNCTIONS 125

563 Exponential Functions-Derivatives 126

565 Exponential Functions-Integrals 126

585 PROBABILITY INTEGRALS 129

600 LOGARITHMIC FUNCTIONS 130

610 Logarithmic Functions-Integrals 133

650 HYPERBOLIC FUNCTIONS 143

667 Hyperbolic Functions-Derivatives 146

670 Hyperbolic Functions-Integrals 147

700 INVERSE HYPERBOLIC FUNCTIONS 156

728 Inverse Hyperbolic Functions-Derivatives 160

730 Inverse Hyperbolic Functions-Integrals 160

750 ELLIPTIC FUNCTIONS 168

768 Elliptic Functions-Derivatives 170

770 Elliptic Functions-Integrals 170

800 BESSEL FUNCTIONS 174

835 Bessel Functions-Integrals 191

840 SURFACE ZONAL HARMONICS 192

850 DEFINITE INTEGRALS 194

890 DIFFERENTIAL EQUATIONS 204

ix

CONTENTS

APPENDIX

PAGE

A TABLES OF NUMERICAL VALUES 209

TABLE 1000 Values of du2 + b2/a 210

1005 Gamma Function 212

1010 Trigonometric Functions (Degrees and Minutes) 213

1011 Degrees, Minutes, and Seconds to Radians 218

1012 Radians to Degrees, Minutes, and Seconds 219

1015 Trigonometric Functions: Sin and Cos of Hundredths of Degrees 220

1016 Trigonometric Functions: Tan and Cot of Hundredths of Degrees 238

1020 Logarithms to Base 10 258

1025 Natural Logarithms 260

1030 Exponential and Hyperbolic Functions 264

1040 Complete Elliptic Integrals of the First Kind 272

1041 Complete Elliptic Integrals of the Second Kind 274

1045 Normal Probability Integral 275

1050 Bessel Functions 276

1060 Some Numerical Constants 283

1070 Greek Alphabet 283

B REFERENCES 284

INDEX 287

TABLES OF INTEGRALS

2 The coefficient of z+ in No 1 is denoted by : or ,C,

0 Values are given in the following table

TABLE OF BINOMIAL COEFFICIENTS

,C;: Values of n in left column; values of T in top row

1 cent numbers cent numbers in same row is in ame row is

4 6 equal to number just below

3 (1 - x)* = 1 - nx + -2r n(n 1) x2 - n(n - l)(n - 2) ~

3!

+ + ( l)r(n f&!” + -**- [See Table 2 and note under No 1.1

4 (a&x)“=al(l*gn

1

* + (- 1)’ @, _ l)ir! xr + -**> [x2 < 11 (1 _ x)-?a = 1 + nx + n(n2t 1) ,&2 + n(n + ‘,‘1” + 2) $3

+ +(n+T’-l)!xr+

(72 - l)! r! [x2 < 1-J (a f x)d = u-n (1 f y, [x2 < a2]

SERIES AND FORMULAS

+ 4.8.12.16 1.5.9.13 tiT

For a large table see Ref 59, v 1, second section, pp 58-68

11 ,limm nne nll n = d PI n!

This gives approximate values of n! for large values of n When n = 12 the value given by the formula is O.O07(n!) too large and when n = 20 it is O.O04(n!) too large [Ref 21, p 74 See also 851.4 and 850.4.1

c-9 + a4x + a3x2 + a2x3 + ax4 + x6

G (a” - x”>/(a -x) = (a3+x3)(u2+ux+x2)

a - 2 = (a2 - ~“)/(a + 2)

a2 - ax + x2 = (2 + $)/(a + x)

a3 - da: + ax2 - x3 = (G4 - xJ)/(a + x)

= (a” + x”)(a - x)*

a4 - a32 + a2x2 - a2 + x4 = (a” + x”)/(a + x)

a6 - cb4x + a3x2 - a22 + ati - 2

E (a” - x”)/(a + 2) E (a” - xy(a2 - ax + 9) a4 + a2x2 j- x4 = (a6 - x6)/(a2 - x2)

= (a” + ux + x”)(a2 - ux + xya a4 - a2x2 + ti = (d + x6)/(a2 + x2)

a4 + x4 = (a2 + x2)2 - 2$x2

= (a2 + ax\‘2 + x”)(a” - ax42 + g>

25 Arithmetic Progression of the first order (ikst differences constant), to n terms,

n+(a+d)+(a+2d)+(a+3d)+ +Ia$-(n l)d!

Em+&2 - 1)d

= 3 (1st term + nth term)

26 Geometric Progression, to n terms,

a + ar + ar2 + aA + - - + urn-l = ă1 - r”)/(l - r)

= ẳ - l)/(r - 1) 26.1 If r2 < 1, the limit of the sum of an infinite number of terms is a/(1 - r)

27 The reciprocals of the terms of a series in arithmetic pro- gression of the first order are in Harmonic Progression Thus

6 ALGEBRAIC FUNCTIONS

28.1 The Arithmetic Mean of n quantities is

28.2 The Geometric Mean of n quantities is

(a1 u2 a3 * * - c&p

28.3 Let the Harmonic Mean of n quantities be H Then

28.4 The arithmetic mean of a number of positive quantities

is 5 their geometric mean, which in turn is z their harmonic mean

29 Arithmetic Progression of the kth order (kth differences constant)

Series: UI, uz, ua, 0 + - un

First differences: dl’, dz’, d3’, *

where dl’ = u2 - ul, dz’ = UI - UZ, etc

Second differences: dl”, dz”, da”, 0 0 *

where dl” = dy‘ - dl’, etc

Sum of n terms of the series

SERIES AND FORMULAS 7

Interpolation Coefficients For numerical values of these co- efficients see Ref 44, v 1, pp 102-109 and Ref 45, pp 184185 29.1

omitting terms in no and those that follow

For values of B, B2, - - -, see 45

The above results may be used to find the sum of a series whose nth term is made up of n, n2, n3, etc

30.1 1 + 3 + 5 + 7 + 9 + * * * + (2n - 1) = n2

30.2 1 + 8 + 16 + 24 + 32 + + 8(n - 1) = (2n - 1)“ 33.1 1 + 32 + 5x2 + 7x3 + * * * = l+ 2 *

0 - x)”

33.2 1 + ax + (a + b)z2 + (a + 2b)aa + * * *

=1+ QX + (b - a)x2*

(1 - x)2 33.3 1 + 2zx + 3rx2 + 429 + = ’ + ’

(1 - 2)”

0 l+sb [a, b > 01 35.1

SERIES AND FORMULAS 9

:I 39.1 f(x+h) =f(x> +h$‘(x) +$‘(x) +

+ (nh:-;) ! f’“-“(x) + R,, where, for- a suitable value of 0 between 0 and 1,

+ kngn) f(z + f&h, y + &k) [Ref 5, No 807.1

divisible by 3

42.2 A number is divisible by 9 if the sum of the figures is

divisible by 9

42.3 A number is divisible by 2” if the number consisting of the

last n figures is divisible by 2”

LOGlo E,

0 0.698 9700 1.785 3298 3.1414498 4.703 4719 6.4318083 8.299 6402

The above notation is used in Ref 27 and 34 and in “American Standard

ent notations in use and, as stated in the above report, it is desirable when

47.4 as definitions, or to state explicitly the values of the first few numbers,

-1 SERIES AND FORMULAS 11

E = $ (6a2bd + 3a22 + 14~ - a3e - 2lab++)

F = -& (7a3be + 7a3cd + 84ab3c - a7 - 28a2b2d

- 28a2bc2 - 42b6)

G = -$ (8a”bf + 8a4ce + 4a4d2 + 120a2b3d + 180a2b2C2

+ 132bs - u6g - 36a3b2e - 72a3bcd - 12a3c3 - 330ab4c)

[See Ref 23, p 11, Ref 31, p 116 and Philosophical Magazine, vol 19 (1910), p 366, for additional coefficients.]

12 ALGEBRAIC FUNCTIONS

Powers of S = a + bx + cx2 + dx3 + ex4 + fx5 - - - 51.1 s2 = CL2 + 2abs + (b2 + 2ac)x2 + 2(ad + bC)X3

+ (c2 + 2ae + 2bd)x4 + 2(uf + be + &)x5 * f a

512 ~l,2=al12[1+~~z+(~S ~)z2

+ ;!$;G!c+i!-!-)x3

( + ;;-g.$

(

3 bd 3 c2 1 e + ;?g+g;E; - za ;;!!?+Ek?)~4 ]

The difference of two quantities is inconvenient to compute with pre- cision and in such a case the alternative form& using the numerical sum

of two quantities should be used [Ref 41, p 306.1

SERIES AND FORMULAS 13

55.2 If one root LY has been computed precisely, use

72.1 If a function takes the form 0 X co or co - co, it may,

by an algebraic or other change, be made to take the form

O/Oar w/w

72.2 If a function takes the form O”, w” or l”, it may be made

to take the form 0 X co and therefore O/O or w/co by first

79 General Formula for Integration by Parts

or

u dv = uv - s v du,

= uv - va do

16 RATIONAL ALGEBRAIC FUNCTIONS

The constant of integration is to be understood with all integrals

Integration in this case should not be carried from a negative to a positive value of 2 If z is negative, use log 1x1, since log ( - 1) = (2k + l)lri will be part of the constant of integration [See 409.03.1

FIO 82.1 Graphs of y = l/x and y = log, 1x1, where x is real

[n # 11

INTEGRALS INVOLVING X = a -I: 6% 17 Integrals Involving X = a + bx

s (X - a)“X”dX and expand (X - u)~ by the binomial theorem, when m is a positive integer

85 On integrals of rational algebraic fractions, see the topic partial fractions in text books, e.g., Chapter II, Reference 7

89 General formula for 90 to 95:

the square brackets is

not be carried from a negative to a positive value of X in the case of log 1 X j

If X is negative, use log ] Xi since log (- 1) = (2k + 1)ni will be part of the constant of integration

18 RATIONAL ALGEBRAIC FUNCTIONS

INTEGRALS INVOLVING X = Q + bx 19

92.7 Jq$y-&+$&]

93 S XSdX F = $2 (n &-4 1 E + (n _33”,iXn4

- (n -3;;p-2 + (n _ f)Xn-1 1 ‘

[except where any one of the exponents of X is 0, see 891

20 RATIONAL ALGEBRAIC FUNCTIONS

- (n -5;;x9b-2 + (n -@f)xn-i I p [except where any one of the exponents of X is 0, see 891

+5a41og 1x1 +$]c

95.3 J$$ = ;[$ -F+ 106-x - 1oa310g 1x1

- !g + g2] *

5u4 a5

5x5+im I * 95.8

22 RATIONAL ALGEBRAIC FUNCTIONS

INTEGRALS INVOLVING LINEAR FACTORS 23

ix = -‘+&3-~+E-~log 4ati 2a3x2 a4x I ; I

Integrals Involving Linear Factors

24 RATIONAL ALGEBRAIC FUNCTIONS

-clog Ic+4)

112

s (a + x;c + x)2 = (c - a,“@ + 2)

1 + (c - 42 *w a+x

I cfz’ I 112.1

S

x ax (a+x)(F+x)?= (a-cY(c+x)

121 Integrals of the form

6X3

x3dx x2 a2 -=-

28 RATIONAL ALGEBRAIC FUNCTIONS

Integrals Involving X = a2 + x2 (continued)