SCHAUM'S OVTLmE SERIES

MATHEMATICAL HANDBOOK

of

Formulas and Tables

BY

MURRAY R. SPIEGEL, Ph.D.

Professor of Mathematics Rensselaer Polytechnic Institute

SCHAUM'iS OIJTLIAE SERIES

McGRAW-HILL BOOK COMPANY

New York, St. Louis, ^um^rancis£a»Soronlo, Sydney

.- 'f ■ ^

Copyright © 1968 by McGraw-Hill. Inc. All Rights Reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

60224 234567890 MHUN 72 10 698

Tvpoaravhv by Signs and Symbola. Inc., New York, N. Y.

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 mclude 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 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 mclude 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 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, AgHcultural 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

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 ^1

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 ^

14. Indefinite Integrals 57

15. Definite Integrals ^

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 HO

21. Bernoulli and Kuler Numbers 114

22. Formulas from Vector Analysis H®

23. Fourier Series 131

24. Bessel Functions 13^

25. Legendre Functions 14®

26. Associated Legendre Functions 149

27. Hermite Polynomials 1^1

28. Laguerre Polynomials 1^

29. Associated Laguerre Polynomials 1^5

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. Cotx {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 lege a; or In a; 224

16. Exponential functions c* 226

17. Exponential functions e"'' 227

18a. Hyperbolic functions sinh x 228

18b. Hyperbolic functions cosh x 230

CONTENTS

Page

19. Factorial n 234

20. Gamma Function 234

21. Binomial Coefficients 236

22. Squares, Cubes, Roots and Reciprocals 238

23. Compound Amount: (1 + r)" 240

24. Present Value of an Amount: (1 + r)"" 241

25. Amount of an Annuity: (^+^)"'~ ^ 242

r

26. Present Value of an Annuity: ^^'^'^^

r 243

27. Bessel functions Jo{x) 244

28. Bessel functions Ji{x) 244

29. Bessel functions Yq{x) 245

30. Bessel functions Yi{x) 245

31. Bessel functions Iq{x) 246

32. Bessel functions Ii{x) 246

33. Bessel functions Ko{x) 247

34. Bessel functions Ki{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 Pr.{cos 6) 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

Part I

FORMULAS

THE GREEK ALPHABET

1.2 e = 2.71828 18284 59045 23536 0287... = lim f 1 +-)"

= natural base of logarithms «-» \ n/

1.3 V2 =: 1.4142135623 73095 0488...

1.4 ^/3 := 1.73205 08075 68877 2935... IS v^ ^ 2.23606 79774 99789 6964...

3 _

1.6 V2 = 1.25992 1050...

1.7 ^/S = 1.44224 9570...

1.8 y/2 = 1.14869 8355...

1.9 ^ = 1.24573 0940...

1.10 e"" = 23.14069 26327 79269 006...

1.11 n-e = 22.45915 77183 61045 47342 715...

1.12 e^ = 15.15426 22414 79264 190...

1.13 log,o2 = 0.30102 99956 639811952137389...

1.14 logioS ^ 0.47712 12547 19662 43729 50279... 1-15 logio e = 0.43429 44819 0325182765...

1.16 log,o,r = 0.49714 98726 94133 85435 12683...

1.17 log^ 10 ^ InlO = 2.30258 50929 94045 684017991...

1.18 ]og^2 = ln2 = 0.69314 71805 59946 309417232...

1.19 log, 3 = In 3 = 1.0986122886 68109 69139 5245... y = 0.5772156649 01532 86060 6512... = Euler's constant

1.23

1.24 1.25

1.20

1.21 ey ^ 1.78107 24179 90197 9852... [see 1.20]

1.22 ^fe = 1.64872 12707 00128 1468...

V^ = r(^) = 1.77245 38509 05516 02729 8167... where r is the gamma function [see pages 101-1021.

r(^) = 2.67893 85347 07748...

r(^} = 3.62560 99082 21908...

1.26 1 radian = l%0°h ~ 57.29577 95130 8232...°

1.27 1° ^ ,r/180 radians ^ 0.01745 32925 19943 2957... radians

2.2 {X - j/)2 = x2 - 2xy + y^

2.3 (a: + j/)3 = x3 + 3x2]/ + Sxy2 + j,3

2.4 (x - ^)3 = x3 - 3a:2i/ + 3x1/2 - j/3

2.5 (X + y)* = X* + iX^y + 6x2j/2 + 4a;y3 + j^

2.6 (» - y)-* = X*- Ax^y + 6x2]/2 - 4xy3 + y^

2.7 (X + y)5 = x5 + 5r»j/ + 10x3i/2 + 10x2y3 + ^y;yi + j^5

2.8 {x - j/)5 = x5 - 5x^1/ + 10x3^2 _ I0x2y3 + 5a:y4 _ yS

2.9 (x + y)^ = x« + 6x5i/ + 15x4i/2 + 20xV + I5a:2y4 + gxyS + y6

2.10 (x - i/)fl = x8 - 6x5y + 15»*i/2 - 20x3tf3 + 15x2y4 - 6x^/5 + y^

The results 2.1 to 2.10 above are special cases of the binomial formula [see page 3].

2.11 a:2- y2 = (a; - y)(a; + y)

2.12 x3-|/3 = (X - 1/)(X2 + xy + 1/2)

2.13 X3 + y3 = (;;e + y)(a;2_a;y + y2)

2.14 z*~y* = (a: - y)(x + y)(x2 4-y2)

2.15 x5 - yS = (x-y)(x4 + a;3y + x2y2 + a:y3 + y4)

2.16 x5 + y5 = (x + y)(x* — x3y + x2y2 - xy3 + y*)

2.17 x6-y6 = {x-y)ix + y)(x^ + xy-hy^){x^-xy + yi)

2.18 X4 + X2y2 + y4 = (a;2 + :cy + y2)(a;2 _ a;y + y2)

2.19 x* + 43/4 = (a.2 + 2xy+2y2)(x2-2xj/ + 2y2)

Some generalizations of the above are given by the following results where m is a positive integer.

2.20 x2" + l - y2n + i = (a- - y)(a;2" + x2n-ly + x2"-2y2 + . - . + y2n)

2xy cos

= (x - y) X

x2 — 2xy cos

2n + 1 2nir

2n+l

+ y2 )( sc2 — 2xy cos

+ y2

Air

2n + l

+ y2

2.21 x2"+l + y2n+l = (x + y)(x2n - x2''-ly + x2n-2y2 _ . . . + y2n)

2

= (x + y) { x2 + 2xy cos^—x~r + y^ )( a^ + 2xy cos _ x2 + 2xy cos ^— -^ + y2

^.vO

2.22

r2n _ ,,2n —

— (* ~ y)(3^ + y)(a:"~' + x'*~2y + 3.n-3y2 + .. .)(a.n-i _ x^^^y + a;"~3y:i _ ...)

— (sJ ~ J/){^ 4- y) ( x2 — 2xy cos— + y2 )[ x2 — 2xy eos-^ + y^ ■ ~ 2xy cos ^" ~ ^^"' + y2

2^3

+ y2n - ( x2 + 2xy cos^ + y2 J( a;2 + 2xy cos |^ + y

2n

2n

•} I n (2n — l)jr , n

x'- + 2a:y cos -z + y2

The BINOMIAL FORMULA and BINOMIAL COEFFICIENTS

If 71 — 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

mwf^m fm^im mB^

If 71 = 1,2,3, ... then 3.3 {x + vr = a;" + nx^^^y + ^^^^^f^ a:"-2y2 + "^"^ ~ g^"" ^^ a:"^3y3 + ... + y«

This is called the binomta? formula. It can be extended to other values of n and then is an infinite series [see Binomial Series, page 110].

BINOMIAL COEFFICIENTS

imiiiiiiiiiiiiiiii^

The reaxilt 3.3 can also be written

3.4 (x + j/)" = x^ + ('^Jx'^-^y + f 2Jx"-2y2 + [^ )x^-3y^ +

where the coeflicients, called binomial coefficients, are given by

3.5

n\ ^ n{n - l){n - 2)- • -{n- k + 1) ^ n\

kj k\ k\(n-k)\

'n

n n ~ k

THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS

PROPERTIES OF BINOMIAL COEFFICIENTS

3.6

+

« + 1

k-{- 1

k / V A; + 1 This leads to Paacal's triangle [see page 236].

3.7

n\ in

+

ly + UJ +

+

= 2"

3.8

(;)-(:)-(;)-•■•<-(:)

= 0

3.9

C)

■nl \ n I \ n I \ n

« + »i + 1 n + 1

3.10

C) - (:

+ (4) +

= %v.-\

3.11

vm:^-^"'-

= 2"-*

3.12

+

+

+ ■■■ +

3.13

:)(:)-(:)(.-.-

m + m P

3.14

/n

(Di^lJ + (2)1^2; + (3}(^3J + ■

+ («1

■= n2"-'

/?i

3.15 (l)(^i^ - (2)(^2y + (3)(^3

,(„!)«+ !(„)( ^ ) = 0

MULTINOMIAL FORMULA

3.16

(ici + a:^ + ■ • ■ + iCp)" = 5

w! _n, „n« _n,

where the sum, denoted by 5, is taken over all nonnegative integers n^tU-i,

., n^ for which

GEOMETRIC FORMULAS

RECTANGLE OF LENGTH 6 AND WIDTH o

4.1 Area = ab

4.2 Perimeter = 2a + 2b

b Fig. 4-1

PARALLELOGRAM OF ALTITUDE h AND BASE h

4.3 Area = bk = ab sin e

4.4 Perimeter = 2a + 26

b

Fig. 4-2

4.5

RIANGLE OF ALTITUDE h AND BASE 5

Area = ^hh = ^ab sin e

where s^ ^{a + b + c) = semiperimeter 4.6 Perimeter = a + b + c

4.7

4.8

TRAPEZOID OP ALTITUDE h AND PARALLEL SIDES a A

N

Area - ^k{a + b)

Perimeter = a + b -i- h ( —L -\ ^

\sinff sin^

= a + b + /i(csc e + CSC -p)

b Pig. 4-4

GEOMETRIC FORMULAS

4.9 4.10

4.15

REGULAR POLYGON OF n SIDES EACH OF LENGTH b

Area = ^nb^ cot— = Inb^^

cos JTr/n) sin (ir/n)

Perimeter = nb

CIRCLE OF RADIUS r

4.11 Area = trr^

4.12 Perimeter = 2irr

SECTOR OF CIRCLE OF RADIUS r

4.13 Area - ]^r^$ [ff in radians]

4.14 Arc length 8 = re

RADIUS OF CIRCLE INSCRIBED IN A TRIANG

\/8(« — a)(s — 6)(« — c)

where s = ^(a + b + c) — semiperimeter

Fig. 4-5

Fig. 4-6

RADIUS OF CIRCLE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c

4.16

R =

abc

4^/s{8 — a)(s — 6)(s — c) where s = ^(a+ b + e) — semiperimeter

Fig. 4-9

GEOMETRIC FORMULAS

REGULAR POLYGON OF n SIDES INSCRIBED IN CIRCLE OF RADIUS r

4.17 Area = ^nr^ sin ~ = ^nr^ sin ^^

4.18 Perimeter = 2nrsin— = 2nr sin

n ft

n SIDES CIRCUi

REGULAR POLYG

4.19 Area = nr^ tan- = nr^ t&n^^

n n

4.20 Perimeter = 2nr tan — = 2nr tan ^^

71 n

IE6MENT OF CIRCLE OF RADIUS r

4.21 Area of shaded part = ^+-2 {$ — sin tf)

Fig. 4-11

Fig. 4-12

ELLIPSE

nkmmmMMimkimmm^

4.22 Area = vob

r/2 y/l —

k^ sin2 $ de

— 2v y/^{a^ + b^) [approximately] where k = Va^ — by a. See page 254 for numerical tables.

fMililiJB^ AXIS b

Fig. 4-13

lEGMENT OF A PARABO

4.24 Area = %ab

, 1,2 /4a + y/b^'+lG^

4.25 Arc length ABC = A VP+Jg^ + ~ In

GEOMETRIC FORMULAS

RECTANGULAR PARALLELEPIPED OF LENGTH a, HEIGHT I, WIDTH

4.26 Volume = ahc

4.27 Surface area = 2(o6 + tie + 6c)

a Fig. 4-15

PARALLELEPIPED OF CROSS-SECTIONAL AREA ^ AND HEIGHT h

4.28 Volume = Ah — ahc si

sin e

Fig. 4-16

SPHERE OF RADIUS r

4.29 Volume := ^.^

4.30 Surface area = Attt'^

Fig. 4-17 RIGHT CIRCULAR CYLINDER OF RADIUS r AND HEIGHT h

4.31 Volume = irr^h

4.32 Lateral surface area = 'i.Trrh

Fig.4-18 CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT I

4.33 Volume ~ tttH = —. —■ = trr'^h esc B

sin e

4.34 Lateral surface area = %^tI — ~-— = 2?rr^ esc B

sin B

Fig. 4-19

GEOMETRIC FORMULAS

CYLINDER OF CROSS-SECTIONAl AREA A AND SLANT HEIGHT I

4.35 Volume = Al = — — = Ah esc e

sine

4.36 Lateral surface area = vl = -?■ — ■= ■ph cse B

sme

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 = lir-r^h

4.38 Lateral surface area - vr v'r2"4~P = wrl

Fig. 4-21

PYRAMID OF BASE AREA A AND HEIGHT h

4.39 Volume = ^Ah

SPHERICAL CAP OF RADIUS r AND HEIGHT h

Fig.4-22

4.40 Volume {shaded in figure) = ^7r/t2{3r - h)

4.41 Surface area = 2irrh

Fig. 4-23 FRUSTRUM OF RiGHT CIRCULAR CONE OF RADII a.b AND HEIGHT h

4.42 Volume ^ ^^h{a^-\-ah-\-h^)

4.43 Lateral surface area = Tr{a + h) yjh^ + (6 - a)2

Fig.4-24

GEOMETRIC FORMULAS

PHERICAL TRIANGLE OF ANGLES A,B,C ON SPHERE OF RADIUS r

4.44 Area of triangle ABC = {A + B + C - jr)r'^

Fig. 4-25

TORUS OF INNER RADIUS a AND OUTER RADIUS b

4.45 Volume - iTrHa + b)(b - a)2

4.46 Surface area = Jr2(b2 - ^2)

Pig.4-26

;OID OF SEMI-AXES a.b^c

4.47 Volume — jvabc

Fig.4-27

PARABOLOID OF REVOLUTION

4.48 Volume = j^irb^a

Fig. 4-28

DEFINITION O F TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE

J

Triangle ABC has a right angle (90°) at C and sides of length a, 6, c. The trigonometric functions of angle A are defined as follows.

5.1 S.2 5.3 5.4 BS 5.6

. . a, opposite

»i7te of A = sin A = = .„„„■„„.. — c hypotenuse

. . A ^ adjacent

coaiTie ot A = cos A = — —

c hypotenuse

, . ^ . O' opposite

tangent ot A - tan A - ^ - ^j^^^

cotangent of A = cot A

& _ adjacent a opposite

, . . c hypotenuse

secant of A = sec A = -r = — j-- t—

b adjacent

_. . . A ^ hypotenuse

cosecant of A = esc A = — = -—~— tt —

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 y is positive along py and negative along OY'. The distance from origin O to point P is positive and denoted by r — y/x^ + y^ . 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 y axis respectively.

The various quadrants are denoted by I, II, III and IV called the first, second, third and fourth 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.

TRIGONOMETRIC FUNCTIONS

For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5-/ sin A = y/r

5.8

5.9

5.10

5.11

5.12

cos A = xfr

tan A = y/x

cot A = x/y

sec A = r/x

CSC A = r/y

RELATIONSHIP BETWEEN DEGREES AND RADIANS

A radian is that angle 6 subtended at center O of a circle by an arc MN equal to the radius r.

Since 27r radians = 360° we have

5.13 1 radian ^ 180°/;r = 57.f9577 95130 8232... °

5.14 1° = B-/180 radians ^ 0.1745 32925 19943 2957.. .radians

Fig.5-4

5.15

5.18

RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS

tan A

sin A

5.16 cotA =

5.17 sec A =

CSC A = —:

sin A

5.19 sin2A + cos^A = 1

5.20 sec2A - Un2A = 1

5.21 csc2A - cot2A - 1

"AMErVARIATIONS OF TRIGONOMETRIC FUNCTIONS

TRIGONOMETRIC FUNCTIONS

13

EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES

For tables involving other angles see pages 206-211 and 212-215.

»■ —MT"

14

TRIGONOMETRIC FUNCTIONS

GRAPHS OF TRIGONOMETRIC FUNCTIONS

In each graph x is in radians. 5.22 y = sin a:

y

Fig. 5-5

5.24 y = tanx

Fig. 5-7

5.26

y = sec a; J/

0

Fig. 5-9

5.23

y — cos X

Fig. 5-6

Fig. 5-8

5.27

y = CSC X

O

Fig.5-10

FUNCTIONS OF NEGATIVE ANGLES

2:r

5.28 sin (—A) = —sin A 5.31 CSC (—A) = — c3Ci4

5.29 cos (—/4) = cosA 5.32 sec (—A) = sec A

5.30 tan(-A) 5.33 cot(-A)

— tan A

— cot A

TRIGONOMETRIC FUNCTIONS

16

ION formul:

5.34 5.35

5.36 5.37

sin (A^B) =

cos (A ± B) =

tan (A±B) =

cot (A^B) =

sin A cos B ± cos A sin B cos A cos B m: sin A sin fi

tan A ± tan g

1 ^ tan A tan B

cot A cot B ^ I cot A ± cotB

FUNCTIONS OF ANGLES IN AIL QUADRANTS IN TERMS OF THOSE IN QUADRANT I

RELATIONSHIPS AMONG FUNCTIONS OF ANGLES IN QUADRANT I

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

TRIGONOMETRIC FUNCTIONS

5.38 5.39

5.40

sin2A cos2A

tan2A

DOUBLE ANGLE FORMULAS

2 sin A cos A

cos^A — sin^A

2 tan A 1 - tan2 A

= 1 - 2 sin2 A = 2 cos^ A

r

5.41

5.42

5.43

. A sin-

COSg

* A tan-

HALF ANGLE FORMULAS

V

1 — cos A

1 + COS A 2

1 — cos A 1 + cos A

sin A

+ 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 1 — cos A

1 + cos A

sin A

— 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

sinSA cosSA

tanSA

sin4A cos4A

tan 4A

sin&A cosSA

tan5A

See also formulas 5.68 and 5.69.

3 sin A — 4 sin^ A

4 cos^ A — 3 cos A

3 tan A - tan^A 1-3 tan2A

4 sin A cos A — 8 sin^ A cos A 8 cos-* A - Scos^A + 1

4 tan A — 4 tan'* A 1-6 tan2 A + tan^ A

5 sin A - 20 sin^ A + 16 sin^ A 16 cos^ A — 20 cos^ A + 5 cos A

tanS A - 10 tan^ A + 5 tan A 1 - 10tan2 A + Stan^A

POWERS OF TRIGONOMETRIC FUNCTIONS

5.53 sin2A ^ ^-^cos2A

5.54 cos2A = | + ^cos2A

5.55 sin^A — £ sin A — ^ sinSA

5.56 cos'^fA — £cosA + icosSA See also formulas 5.70 through 5.73.

^ — ^ COS 2A + ^ cos 4A f + ^ cos 2A + ^ cos 4A f sin A — ^ sin 3A + ^ sin 6A ; 3A + ^ cos 5A

§ cos-

^cos;

TRIGONOMETRIC FUNCTIONS

17

M, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIO:4S

5.61 sin A + sin B - 2 sin ^(-4 + B) cos ^(A - B)

5.62 sin A - sin B = 2 cos ^(A + F) sin ^{A - B)

5.63 cos A + cos F = 2 cos ^(A + B) cos ^{A-^ B)

5.64 cos A - cos B = 2 3in^(A + B) sin i(B - A)

5.65 sin A sin B = ^{cos (A - B) - cos (A + B)}

5.66 cos A cos B = |{cos (A - B) + cos (A + B)}

5.67 sinAcosB = ^{sin (A - B) + sin (A + B)}

GENERAL FORMULAS

5.68 sinTiA ^ sinA ■j(2cosA)"-i - f " ^ ) (2 cos A)"-3 + f '^ 2 j (2 cos A)"-5 - ■ • ■ I

5.69 cosnA ^ |-^(2cosA)" - y{2cosA)"-2 + |-("~^j(2cosA)''-4

n In — 4

3 V 2

(2cosA)''-6 +

5.70 sin^n-iA =

5.71 cos2"-iA =

<g^|sin(2n-l)A - (^"^ ^ ] sin (2n - 3)A +

4)'

^-^^-^^t'-l'^^^"^

^A_|eos(2n-l)A + (^"^ ^)cos(2n-3)A + ••• + (^^_ ^^) cosaI

5.72 sin2nA =

5.73 cos2"A =

= 2^(1") +|2;r^|cos2nA -(2«)cos(2n-2)A + ■•- ("D-^ (^^^j )cos2A h^l) + 2^f<'«2nA + (2n^ cos (2«-2)A +

+ ( ^ J* ^ ) cos 2A

If a: — sin J/ then y — sin~' x, 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.

TRIGONOMETRIC FUNCTIONS

NCiPAL VALUES FOR rNVERSE TRIGONOMETRIC FUNCTIO

REt^TtON$ BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS

In all cases it is assumed tbat principal values are used.

5.74 sin-'x + cos-'a: = 7r/2

5.75 tan-'x + cot-'ar ^ 7r/2

5.76 aec-'a: + esc-'a; = 57/2

5.77 CSC"' X — sin-' (1/a:)

5.78 sec-'3; — cos-'(1/a;)

5.79 cot-' X = tan-' (1/a;)

INVERSE TRIGONOMETRIC

In each graph y is in radians. Solid portions of curves correspond to principal values.

5.86 y — sin-'x

5.87 y ~ cos-'a;

5.88 y = tan-'x

S-/2-

-W2 ■

Fig.5-11

O :

Fig. 5-12

Fig. 5-13

TRIGONOMETRIC FUNCTIONS

19

5.89 y = cot-^a;

Fig. 5-14

5.90 y = sec~'x

y

t/2

-1

-ir/2

O i\

5.91

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, c and angles A, B, C.

5.92 Law of Sines

5.95

sin A sin B sin C

5.93 Law of Cosines

c2 = o2 + 62 _ 2a6 COS C with similar relations involving the other sides and angles.

5.94 Law of Tangents , ,,^10*

a + h _ tan IJA + B)

a- b ~ tan ^(A - B) with similar relations involving the other sides and angles.

sin A = T- y/a{8 — a)(8 — b){a — c)

Fig. 5-17

where s = |(a +b + c) is the semiperimeter of the triangle. Similar relations involving angles B and C 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,c [which are arcs of great circles] are measured by their angles subtended at center O of the sphere. A, B, C are the angles opposite sides a,b,c respectively. Then the following results hold.

5.96 Law of Sines

sin a sin b

sin A BinB sinC 5.97 Law of Cosines

cos o = cos b cos c + sin 6 sin c cos A cos A = — cos B cos C + sin B sin C cos a with similar results involving other sides and angles.

Fig. 5-18

]

TRIGONOMETRIC FUNCTIONS

5.98 Law of Tangents

tan ^(A + B) _ tan ^(a + b) tan ^{A-B) ~ tan |(o - b) with similar results involving other sides and angles.

5.99 cos— = ^I sin8sin(i

2 V sin 6 sii

where s = ^(a + 6 + c). Similar results hold for other sides and angles.

5.100

(s-c) sin c

cos

a _ j cosjS-B) cos (S-C) 2 V sin 5 sin C

where S ~ ^(A + B + C). Similar results hold for other sides and angles. See also formula 4.44, page 10.

NAPIER^-S^ RULES FOR RIGHT ANGIED 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 Pig. 5-20 where we attach the prefix co [indicating coviplement] to hypotenuse c 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 b tan (co-B) or sin o — tan b cot B

sin (co-A) = cos a cos (co-B) or cos A = cos a sin B These can of course be obtained also from the results 5.97 on page 19.

DEFINITIONS INVOLVING COMPLEX NUMBER

A complex number is generally written as a + bi where o and b are real numbers and i, called the imaginary unit, has the property that v^ - ~1. The real numbers a and 6 are called the real and imaginary parts oi a + bi respectively.

The complex numbers a + bi and a — bi are called complex conjugates of each other.

EQUALITY OF COMPLEX NUMBEI

6.1

a + bi — c + di if and only if a = c and b = d

6.2

{a+bii-i- ic + di) = {a+c) + ib + 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) = {ae - bd) + {ad + bc)i

DIVISION OF COMPLEX NUMBE

6.5

a + bi _ a + bi ^ c — di _ ac + bd f be — a,d\ . c + di ~ c + di' c- di " c^ + d^ \ c'i + d? ' *

Note that the above operations are obtained by using the ordinary rules of algebra and replacing i^ by —1 wherever it occurs.

COMPLEX NUMBERS

GRAPH OF A COMPLEX NUMBER

A complex number a + bi can be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane. For example in Fig. 6-1 P represents the complex number —3 + 4i.

A complex number can also be interpreted as a vector from O to P.

POLAR FORM OF A COMPLEX NUMBER

In Fig. 6-2 point P with coordinates (x, y) represents the complex number x + iy. Point P can also be represented by pola/r coordinates (r, e). Since x = r cos 6, y = r sin 6 we have

6.6

X -\- iy =^ r(cos 9 -V i sin e)

called the polar form of the complex number. We often call r = s/x^ + y^ the modulus and $ the amplitude of a; + iy.

Fig. 6-2

lUlTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POU^R FORM

6.7 6.8

[ri(cos tf 1 + 1 sin e{i\ [r2(cos $2 + * sin «2)] — »'i'"2[cos (*i + tfg) + ^ sin (ffj + e^i] ri(cos ffi + 1 sin ffx) '"i

—; :—: T — COS (81 — So) + * 31" (*1 ~ *2)l

r2(cos e-i + I sin 62) rg ^ ^ ^ ^ ^ ■"

DE MOIVRE'S THEOREMi|H

If p is any real number, De Moivre's theorem states that 6.9 [r(cos 9 A- i sin ff)]" = r''(cos pS + i sin p$)

ROOTS OF COMPLEX NUMBERS

6.10

If p = 1/n where n is any positive integer, 6.9 can be written

9 -I- 2fcff

[r(cos ff + t sin ff)]^■''' = r^'

+ i sin

* + 2k

^]

where k is any integer. From this the n nth roots of a complex number can be obtained by putting k = 0,1,2, . ..,n-l.

EXPONENTIAL and LOGARITHMIC FUNCTIONS

LAWS OF EXPONENTS

In the following p, q are real numbers and m, n are positive integers. In all cases division by zero is excluded.

7.1 aP' ai = aP + 'i

7.4 qO = 1, o ?^ 0

7.7 v^ - oi/«

In aP, p is called the exponent, a is the base and op is called the pth power of a. The function y = a^ is called an exponential function.

LOGARITHMS AND ANTILOGARITHMS

If aP = N where a # 0 or 1, then p = logo N is called the logarithm ot N to the base a. The number N = a'' is called the antilogarithm of p to the base a, written antilogy p.

Example: Since 3^ = 9 we have logs 9 — 2, antilog3 2 = 9. The function y = log„x is called a logarithmic function.

LAWS OF LOGARITHMS

m^: i>m-i-l;^-mU-7''liiikV.9Ui v*p,1H^KS*«

7.10 7.11 7.12

log„ MA' ^ log„ M + log„ N

M

logo -^ logo M - log„ A^

iogfl M^ = p logn 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 logigN or briefly logN. For tables of common logarithms and antilogarithms, see pages 202-205. For illustrations using these tables see pages 194-196.

23

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

NATURAL LOGARITHMS AND ANTIIOGARITHMS

Natural logarithms and antilogarithms falso called Napierian] are those in which the base a= e = 2.71828 18... [see page 1]. The natural logarithm of N is denoted by logg 'N or In A'. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving e^ for values of ic] 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 6 is given by

log.iV

7.13

In particular, 7.14

7.15

logoiV =

logb a

loggiV = InN = 2.30258 50929 ...logioN logjoN = logiV =: 0.43429 44819 ...log^iV

i

RELATIONSHIP BETWEEN EX

TIAL AND TRIGONOMETRIC FUNCT IONS

7.16 e^ = cos 9 + isins, e~'^ = cos ff — i sin ff

These are called Euler's identities. Here i is the imaginary unit [see page 21].

7.17 7.18 7.19 '

7.20 7.21 7.22

e« -f e-'«

,w _ «-«

PERIODICITY OF EXPONENTIAL FUNCTIONS

7.23 e«»+2feTJ = fiW k = integer

From this it is seen that e* has period 2irx.

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] *'*■* X + iy = r(cos a -\- i sin e) — re'S

OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM

Formulas 6.7 through 6.10 on page 22 are equivalent to the following.

7.26

7.27

7.28

r2e'"2 r2

(rei«)p = r^e'pfl [De Moivre's theorem]

LOGARITHM OF A COMPLEX NUMBER

7.29

In (re») = \nr + ie + 2kwi k = integer

DEFINITION OF HYPERBOLrc FUNCtlONS

8.1

8.2 8.3 8.4 8.5 8.6

Hyperbolic sine of a; = sinh x =

Hyperbolic cosine of a; = cosh x =

Hyperbolic tangent of z = tanh z —

Hyperbolic cotangent of x = coth x —

Hyperbolic secant of a; = sech x =

Hyperbolic cosecant of x = csch x =

dI _ P--I

RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS

8.7 8.8 8.9

8.10

8.11 8.12 8.13

sinh a;

FUNCTIONS OF NEGATIVE ARGUMENTS

8.14 sinh (—a:) — — sinh a: 8.17 csch (—ic) = —csch a:

8.15 cosh(—«) = cosh a: 8.18 aech(-a;) = sech a;

8.16 tanh (—a:) = —tanha; 8.19 coth (-a;) - -cotha;

HYPERBOLIC FUNCTIONS

27

8.20 8.21 8.22

8.23

sinh {x ± y) cosh {x ± y)

tanh (x ± y)

DITION FORMUiAS

sinh X cosh y ± cosh x sinh y

cosh X cosh v ± sinh x sinh i/

tanh X ± tanh |/ 1 ± tanh a; tanh j/

coth ix±y) =

coth z coth y ± 1 coth v ± coth X

DOUBLE ANGLE FORMULAS

= 2 cosh2 X - 1 = 1 + 2 sinh2 x

8.27 8.28 8.29

HALF ANGLE FORMULAS

sinh I

cosh:

tanh I

cosh X — 1

cosh X + 1

cosh X — 1

[+ if X > 0, - if X < 0]

\ cosh X + 1 sinh X cosh x — 1

[+ if X > 0, - if X < 0]

cosh X + 1

sinhx

8.30 8.31

8.32

8.33 8.34

8.35

sinh 3x =

cosh 3x —

tanh 3x =

sinh 4x =

cosh 4x =

tanh 4x —

3 sinh X + 4 sinh^ x

4 cosh^ X — 3 cosh x

3 tanh x + tanh^ x 1 + 3 tanh2 x

8 sinh^ X cosh x + 4 sinh x cosh x

8 cosh'' X — 8 eosh2 x + 1

4 tanh x + 4 tanh-^ x 1 + 6 tanh2 x + tanh* x

HYPERBOLIC FUNCTIONS

8.36 8.37 8.38 8.39 8.40 8.41

8.42 8.43 8.44 8.45 8.46 8.47 8.48

SUM, DIFFERENCE AND PRODUCT OF HYPEl

sinh X + sinh i/ = 2 sinh ^{x + y) cosh ^(x — y)

sinh X - sinh y = 2 cosh ^(x + y) sinh ^(x - y)

coshx + coshy = 2 cosh ^(x + y) cosh ^(x — j/)

cosh X ~ cosh y = 2 sinh ^{x + y) sinh ^(x — y)

sinh X sinh y = ■J{cosh ix + y) - cosh {x - y))

cosh X cosh y — ^{cosh (x + y) + cosh (x — y))

sinh X cosh y — ^{sinh {x + y) + sinh (x — y)}

XPRESSION OF HYPERBCtIC 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.

HYPERBOLIC FUNCTIONS

29

)F HYPERBO

8.49

Fig. 8-1

8.50

y = cosh X

O

Fig.8-2

8.5 i

y = tanh x

Fig. 8-3

8.52 y = coth X

y

Fig. 8-4

8.54 y = csch X

y\

Fig. 8-5

Pig. 8-6

JNVERSE HYPERBOLIC FUNCTIONS

I^ X = sinhj/, then y = sinh~'x 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

sinh->a; = In (x + y/x^ + 1)

cosh""! X — In (a; + ^/x^ — 1)

tanh ^ X = — In ( :;

2 \1 — X

coth * X = - in — --

2 \x — 1

c«:h->x = In(^^+ -^^+1

—» < « < =0

x^l [cosh^^x > 0 is principal value] -1 < X < 1

X > 1 or X < —1

0 < X S 1

x^ 0

[sech~' X > 0 is principal value]

HYPERBOLIC FUNCTIONS

8.61 8.62 8.63 8.64 8.65 8.66 8.67

RELATIONS BETWEEN INVERSE HYPE^G^fcFUNCTIONS

esch-'a; = sinh~^(l/a:)

sech-^a: = cosh^i(l/a:)

coth-ix = tanh"' (1/a;)

sinh~i {—x) = — sinh-i x

tanh~*(— a;) = — tanh^'a;

coth^i {—x) = — coth~' X

csch^* (—ar) — — cseh~' x

RAPHS OF INVERSE HYPERBOLIC FUNCTIONS

8.69 y = cosh-la:

Fig. 8-8

8.72 y - sech-ia:

V

O

II

Fig. 8-10

Fig. 8-11

Fig. 8-J 2

HYPERBOLIC FUNCTIONS

31

RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS

8.74 sin (ix) = i sinh z

8.77 CSC {ix) = — i csch x

8.80 sinh [ix) — i sin x

8,83 csch (ix) = — i CSC x

8.75 cos [ix) — cosh x

8.78 sec (ix) = sech x

8.81 cosh (ia;) — cos a;

8.84 sech (ix) — sec x

8.76 tan (ix) — i tanh x

8.79 cotii'x) — — icothx

8.82 tanh {ix) = i tan x

8.85 coth (ia:) = —i cot a:

PERIODICITY OF HYPERBOLIC FUNCTIONS

In the following k is any integer.

8.86 smh(x4-2ijrt") = sinhx 8.87 cosh(x +Zkin) ^ cosh a: 8.88 tanh{x + kvi) = tanh a:

8.89 csch (a; + 2kiri) ^ csch x 8.90 sech {x + 2kTri) = sech x 8.91 coth (a; + kiri) = coth a;

RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS

8.92 sin~i(ix) = i sinh"'a;

8.94 cos ~^ X =■ ± i cosh~' x

8.96 tan-i(iar) = i tanh-'a

8.98 cot"''(ta;) — — ieoth~'a:

8.100 sec~'x = ±isech~^a;

8.102 csc~' {ix) ^ ~i csch~^ x

8,93 sinh~i {ix) = i sin~' x

8.95 cosh"'a: = ±icos~^x

8.97 tanh-'(ix) = itan-'a:

8.99 coth-'(ia:) = -icot-ix

8.101 sech~^ X = ±i sec~^ x

9.1

Solutions:

-6 ± \/b^ - iac

2a

9.2

If a, b, c are real and if D = b^ — 4a« 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 Z? < 0

If x^, X2 are the roots, then Xi + arg — —b/a and a;,a-2 — c/a.

Ee

CUBIC EQUATION; x'^ + a,x^-\-a-iZ + a^ = 0

L^

Q =

Za^ ~ (i\

R =

9aja2 — 27a3 — 2ai

9 ' " 54

a:, = S + r - lai 9.3 Solutions: -j ajg = -^(S + T) - ^Oi + ^i-/3 (S - T)

3^3 = -i(S + r) - !^a, - \iVZ(S - T)

If ttj, 03,03 are real and if D = Q'^ + R'^ 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 Z> = 0 (iii) all roots are real and unequal if D < 0.

If D < 0, computation is simplified by use of trigonometry.

9.4

93

Solutions if Z> < 0:

Xi - 2\/—Q cos (J*)

Xz = 2-/^ cos (^« +120°)

23 = 2^/^cos(^tf+ 240°)

where cos S = —R/yJ—Q^

where Xi,xi,,x^ are the three roots.

= -a-i

32

SOLUTIONS OF ALGEBRAIC EQUATIONS

33

* B^ BM I I ■ iiii^ii l i I i llllllM^^BTCT i l ll ll il llll fcii

QUARTIC EQUATION: x* + a,*^ + azX^ + CsX + a4 = 0

Let yi be a real root of the cubic equation 9.6 y3 ~ 02^2 -I- (a^ag - 404)1/ + (40304 -al- ofo^) - 0

9.7 Solutions: The 4 roots of z^ + ^{a, ± y/a\ - ia^ + Ay^ )z + ^{j/j ± y/y\ - 404 } = 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

Kj + a^a + ajg + 3:4 = —Oi

XiX2 + X2ir3 + X^X^ + X^Xi + X^X^ + X2X^ = a2 XiX2X^ + ^2X3X4 + 9;j!E2-'^4 "I" ^l^Z-'^i ^^ —*^3 3:1X2X3X4 = 04

where Xj, Xg, Xg, X4 are the four roots.

FORMULAS from PLANE ANALYTIC GEOMETRY

DISTANCE d BETWEEN TWO POINTS Piixx,yi) AND ^2(2^2,^2)

10.1

d = V(3^2-Xi)2 + (j/2-1/1)2

:iL

P,(a;i,!/i)

l(3/2-!/i)

1^2-X,)

-1

Xi

Fiff. 10-1

HOPE m OF LINE JOINING TWO POINTS Piixi,yi) AND P2{x.z,yz)

10.2

V2 - Vi m = = tariff

X2 - «!

UATION OF LINE JOINING TWO POINTS Pi{Xi,yi) AND Pz{x2,y2)

10.3 10.4

y-y\ _ vz- vi

X — Xi x^— Xi

m or y — Vi = «*(« — a^i)

y = mx + 6

XzVi — Xiifz where 0 =^ y.— mx, = is the intercept on the y axis, i.e. the y intercept.

X2 - i'l

lEQUATION OF LINE IN TERMS OF x INTERCEPT a^O AND y INTERCEPT b ¥= 0

10.5

- + ^ - 1 a b

Fig. 10-2

34

FORMULAS FROM PLANE ANALYTIC GEOMETRY

35

10.6 xcosa + ysina — p

where p = perpendicular distance from origin O to line

and a = angle of inclination of perpendicular with

positive X axis.

Fig.10-3

GENERAL EQ

10.7

Ax + By + C = 0

DISTANCE FROM POINT {xi,yi) TO LINE Ax + By + C = 0

10.8

Axj + Byi + C

where the sign is chosen so that the distance is nonnegative.

LNGLE ^ BETWEEN TWO LINES HAVING SLOPES mi AND Mz

10.9

tan^ =

Wl2 ~ Wi

1 + mim2

Lines are parallel or coincident if and only if wii = m2. Lines are perpendicular if and only if m2 =■ —iMi-

Fig. 10-4

slope Bii slope nij

AREA OF TRIANGLE WITH VERTICES AT {a;i,?/i), (:r2,3/2), (a^a.^s)

10.10 Area = ±

Xi Vi 1

X2 i/2 1

X3 Va 1

ix2, j/a)

where the sign is chosen so that the area is nonnegative. If the area is zero the points all lie on a line.

(*i.Vi)

Fig.10-5

FORMULAS FROM PLANE ANALYTIC GEOMETRY

rRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLA

10.11

X = x' + X(i

y - y' + vq

JX' = X ~ Xq

{y' = y - Vq

where {x, y) are old coordinates [i.e. coordinates relative to xy system], {x',y') are new coordinates [relative to x'y' system] and (aro.tfo) are the coordinates of the new origin O' relative to the old zy coordinate system.

O

y'

ixa-Vii)

O'

Fig. 10-6

TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION

x'

10.12

ic — a; cos a — 1/ sin a (x' — x cosa + y nma

_ , , or -^

y — X sma + y'cosa yy' = ?/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.

.y'

0^

\

Fig.10-7

TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION

10.13

X = x'cos a — y'sin a + xq V = x' sin a + y' cos a + y^

x' = (x — Xq) cos a + iy — yo) sin a

y' = [y — Vo) cos a — (x — x^) sin a

where the new origin O' of x'y' coordinate system has coordinates (xQ,yQ) relative to the old xy coordinate system and the x' axis makes an angle a with the positive x axis.

-^

\

\

o \

\ \

Fig. 10-8

x'

A point P can be located by rectangular coordinates (x,^) or polar coordinates (r,»). The transformation between these coordinates

Is

10.14

X = r cos * y — r sin $

r = V*^ + y^

0 = tan-^ (y/x)

O

Fig.10-9

FORMULAS FROM PLANE ANALYTIC GEOMETRY

37

EQUATION OF CIRCLE OF RADIUS R, CENTER AT [x^yo)

10.15

(a!-Xo)2+(wo)2 = fl2

Fig. 10-10

IQUATION OF CIRCLE OF RADIUS R PASSING THROUGH ORI6I

10.16 r = 2Rcos(s-a)

where (r, e) 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 diflFerent angles].

If the focus is chosen at origin O the equation of a conic in polar coordinates (r, e) is, if OQ = p and LM = D, [see Fig. 10-12]

10.17

P

tD

1 — e cos e

The conic is

(i) an ellipse if « < 1 (ii) a parabola if « = 1 (iii) a hyperbola if e > 1.

1 — e cos 8

Directrix

Fig.10-12

FORMULAS FROM PLANE ANALYTIC GEOMETRY

ELLIPSE WITH CENTER C(Xo, 1/0) AND MAJOR AXIS PARALLEL TO x

10.18 Length of major axis A'A — 2a

10.19 Length of minor axis F'B = 26

10.20 Distance from center C to focus F or F' is

= Va2 - 62

10.21 Eccentricity ^

v^airp

O

10.22 Equation in rectangular coordinates: {a: - Xo)g {y - Vp)^

n "r 10 — 1

62

10.23

Fig. 10-13

Equation in polar coordinates if C is at O: r^ = -r —

a2 sin2 9 + 0* cos2 6

10.24 Equation in polar coordinates if C is on « axis and F' is at O: r — -^ —^

1 — « cos e

10.25 If P is any point on the ellipse, PF + PF' - 2a

If the major axis is parallel to the y axis, interchange x and j/ in the above or replace e by iir

90° - e].

e [or

PARABOLA WrTH AXIS PARALLEL TO X AXIS

If vertex is at A{xQ,yff) and the distance from A to focus F is a > 0, the equation of the parabola is

10.26 iy — yo)^ = ^a{z — XQ) if parabola opens to right [Fig. 10-14]

10.27 {y~yo)^ = —4a{ic-a;o) if parabola opens to left [Fig. 10-15] If focus is at the origin [Fig. 10-16] the equation in polar coordinates is

10.28

2a

1 — cos 9

Fig. 10-14 Fig. 10-15 Fig. 10-16

In case the axis is parallel to the y axis, interchange x and y or replace 8 hy ^ir — 8 [or 90° — e].

FORMULAS FROM PLANE ANALYTIC GEOMETRY

PERBOIA WITH CENTERX^o^yo) AND^AJ OR AJOSPll^rLilTOT^

«'n.

^H

Fig. 10-17

10.29 Length of major axis A'A = 2a

10.30 Length of minor axis B'B = 26

10.31 Distance from center C to focus F or F' = e = yaf+b^

C V^2T62

10.32 Eccentricity e = — =

a a

10.33 Equation in rectangular coordinates:

(g - iCoP {y - yg)^ a2 62

= 1

10.34 Slopes of asymptotes G'H and Gff' = ±~

10.35 Equation in polar coordinates if C is at O: r^ =

a262

62 cos^ tf — a2 sin2 g

10.36 Equation in polar coordinates if C is on X^ axis and F' is at O: r = - °^ ~ —ii-

1 — E cos S

10.37 If P is any point on the hyperbola, PF ~ PF' = ±2a [depending on branch]

If the major axis is parallel to the y axis, interchange x and y in the above or replace e by i^r — e for 90° - el

LSMNfSCATE

11.1 Kquation in polar coordinates:

7-2 — Or cos 2tf

11.2 Equation in rectangular coordinates:

11.3 Angle between AB' or A'B and x axis ~ 45''

11.4 Area of one loop — a^

Fig.I1-1

CYCLOID

11.5 Equations in parametric form:

X — a(0 — sin 0) — a{l — COS0)

11.6 Area of one arch = Swa^

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:

n .9 Equations in parametric form:

X — a cos** 8 y = a sin'^ 8

11.10 Area bounded by curve = ■f"''*^

11.11 Arc length of entire curve — 6a

This is a curve described by a point P on a circle of radiiis a/4 as it rolls on the inside of a circle of radius a.

Fig.11-3

SPECIAL PLANE CURVES

41

11.12 Equation: r — a(l + cos tf)

1 1.13 Area bounded by curve = j^a^

1 1.14 Arc length of curve = 8a

This is the curve described by a point P of & 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].

lATENARY

11.15 Equation: y = -{e^^'^+e^'^") = acosh-

2 o,

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 3«

The equation r = a sin 3s is a similar curve obtained by rotating the curve of Fig. 11-6 counterclockwise through 30^^ or Tr/6 radians.

In general r — a cos ns or r = a sin n8 has n leaves if n is odd.

Fig. n-4

Fig.ll-«

11.17 Equation: r = a cos 2tf

The equation r = a sin 2e is a similar curve obtained by rotating the curve of Fig. 11-7 counterclockwise through 45° or w/4 radians.

In general r — a cos ntf or r ~ a sin ne has 2n leaves if n is even.

Fig. 11-7

SPECIAL PLANE CURVES

11.18 Parametric equations:

K = (a + 6) COS 8 — b cos ( — -. — ) 0

y = (a + 6) sin tf — 6 sin ( —;— I e

This is the curve described by a point P on a circle of radius 6 as it rolls on the outside of a circle of radius a.

The cardioid [Fig. 11-4] is a special case of an epicycloid.

Fig.11-8

11.19 Parametric equations:

X = (a—b) cos0 + 6 cos [ — z — )*

y = {a—b) sin^ — 6 sin [ —j— ]^

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 a.

If 6 = a/4, the curve is that of Fig. 11-3.

Fig.11-9

TROCHOID

11.20 Parametric equations:

X = atp — b sin <p y = a — b cos 0

This is the curve described by a point P at distance 6 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 b > a, the curve is as shown in Fig. ll-H and is called a prolate cycloid.

If & = a, the curve is the cycloid of Pig. 11-2.

Fig. 11-10

Fig.U-U

SPECIAL PLANE CURVES

43

11.21 Parametric equations:

a; — a In (cot ^^ — cos ^) y — 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

8aS

11.22 Equation in rectangular coordinates: y = ■ „ -l 4„2

11.23 Parametric equations:

X — 2a cot e

y — a{\ — cos 2fl)

In Fig. 11-13 the variable line OA intersects y = 2a and the circle of radius a with center (0, a) at A and B respectively. Any point P on the "witch" is located by constructing lines parallel to the x and y axes through B and A respectively and determining the point F of intersection.

Fig.11-13

FOLIUM OF DESCARTES

11.24 Equation in rectangular coordinates;

3[;3 4- ^3 = Zaxy

11.25 Parametric equations:

V =

Sat 1 + (3

3at^ 1 + (3

11.26 Area of loop

= ^nZ

1 1.27 Equation of asymptote: x + y + a = 0

Fig. 11-14

INVOLUTE OF A CIRCLE

11.28 Parametric equations:

x = a(cos 0 + 0 sin 0)

y = a(sin 0 — 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

SPECIAL PLANE CURVES

Equation in rectangular coordinates:

(oa;)2/3 + (6y)2/3 = (a2 _ 62)2/3

11.29

11.30 Parametric equations:

ax = (a^ ~ 62) cogS g

by = {a2 - 62) ginS g

This curve is the envelope of the normals to the ellipse xya^ + y^/b^ = 1 shown dashed in Fig. 11-16.

Fig.11-16

11.31 Polar equation: r* + a* - 2aV2 cos 29 = 6*

This is the curve described by a point P such that the product of its distances from two fixed points [distance 2a apart] is a constant b^.

The curve is as in Fig. 11-17 or Fig. 11-18 according as 6 < a or b > a respectively.

If 6 = o, the curve is a lemniaeate [Fig. 11-1].

^^i

^

Fig.11-17

Fig. 11-18

:ON OF PASCAl

11.32 Polar equation; r = b ■¥ a cos 8

Let OQ be a line joining origin O 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 6 > a or 6 < a respectively. If b = a, the curve is a cardioid [Fig. 11-4].

y

Fig.11-19

Fig. n-20

SPECIAL PLANE CURVES

45

CISSOID OF DIOCIES

11.33 Equation in rectangular coordinates:

y2 =

2a — X

11.34 Parametric equations:

fx — 2a sin^ 9

V -

2a sin^ 9 cos 9

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 — a9

Fig. 11-22

FORMULAS from SOLID ANALYTIC GEOMETRY

DISTANCE d BETWEEN TWO POINTS Pi{xi,yuZi) AND P%{x2,%2,Z2)

12.1 d = \f(x^ - a;,)2 + (ya - y,)2 + (z^ - ^^)2

Pz(^2.y2'^2)

FifT. 12-1

DIRECTION COSINES OF LINE JOINING POINTS P^{xx,yi,z,) AND >2(i2,1/2,^2)

12.2

i — COS a = -t , m = cos/3 = , n — cos y —

add

where a,/?,y are the angles which line PjPa 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 ct + cos^ j8 + cos2 7 = 1 or I- + nfi -\- v?- = \

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

y/L^ + M^ + m \/lJ+~M^Tm' ^/UVmTm

46

FORMULAS FROM SOLID ANALYTIC GEOMETRY

47

EQUATIONS OF LINE JOINING Pi{xuyuZi) AND ^2(^2, ?/2,22) IN STANDARD TORM

12.5

X - xi y - j/i z - z, X- Xi y - y\ z — z^

or

These are also valid if l,m,n are replaced by L,M,N respectively.

EQUATIONS OF LINE JOINING Pi{xi,yuZi) AND ^2(^2, ?/3.22) IN PARAMETRIC FORM

12.6 X = X] -^ It, y — ]/] + mi, z — zj + wt

These are also valid if t,m,n are replaced by L,M,N respectively.

ANGLE 0 BETWEEN TWO LINES WITH DIRECTION COSINES U.muni AND ;2,m2,«2

12.7

cos 0 — ^1^2 + Wimg + »liW.2

GENERAL EQUATION OF A PLANE

12.8

Ax + By + Cz + D = 0

[A,B,C,D are constants]

EQUATION OF PLANE PASSING THROUGH POINTS {Xi,yi,Zi), (3^2,3/2,22), (3:3,1/3,23)

12.9

12.10

X — xi y - ifi z - zi

= 0

or

^2 - yi ^2 - 2l

(x-x^) +

Zg ^ Zi Xa — Xi H ~ ^1 ^3 ^ a^i

iy-yi) +

X2 - xi vz- y-i X3 - Xi yg - yi

(z - zi) = 0

EQUATION OF PLANE IN INTERCEPT i^ORM

12.11

a 6 c where a,b,c are the intercepts on the x,y,z axes respectively.

Fig. 12-2

FORMULAS FROM SOLID ANALYTIC GEOMETRY

12.12

EQUATIONS OF LINE THROUGH (xo, ?/o, So) AND PERPENDICULAR TO PLANE Ax + By -\-Cz + D = 0

^ — a^o y - yo z~ ^0

or X — Xq + At, y = Va + Bt, z = Zq + 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,z,y) TO PLANE Ax + By -^ Cz -\- D = 0

12.13

Axo + Byo + Czo + D

where the sign is chosen so that the distance is nonnegative.

12.14

NORMAL FORM FOR EQUATION OF PLANE

X COS a + y cos j3 + 2 cos y = p

where p = perpendicular distance from 0 to plane at P and a,p,y are angles between OP and positive x,y,z axes.

Fig. 12-3

TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION

12.15

X = x' + Xq

y - y' + Vo

Z - Z' -\- Zq

or

= z — Za

where (x, y, z) are old coordinates [i.e. coordinates relative to xyz system!, {x',y',z') are new coordinates [relative to x'y'z' system] and (kqi l/o-'^o) ^^^ ^^^ coordinates of the new origin O' relative to the old xyz coordinate system.

A

\z'

Jjfo^o^^o)^ _ ,

Fig.12-4

FORMULAS FROM SOLID ANALYTIC GEOMETRY

49

12.16

TRANSFORMATION OF COORDINATES rNVOLVING PURE ROTATION

X = l^x' + I2V' + I3Z'

y = Ttiix' + m^y' + m^z' z = n,r' + n2V' + TI32'

x' = lix + m-iy + riiz

y' - I2X + m^y + 7132 s' - l^x + m^y + Mgz

where the origins of the xyz and x'y'z' systems are the same and /i,7ni,?i,; i2.»«2>"2'> '3.'«3."3 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

= hx' + l^y' + hz' + aro — «,^'-J-™_„'-f ^^2'

+ ngz' + Zo

a; = l^x' + /gy' + l^z' + 1/ = Wia;' + m2y' + wigz

+ I/O

y' = ya: - Xq) + m2{y - yg) + n^iz - «o) ^ 2' = ^3(3; - »(,) + ?n3(y - yo) + 713(2 - Zq)

where the origin O' of the x'y'z' system has coordinates (a^o. Vd' ^o) relative to the xyz system and ly, m^, jj,; ^2>*"2."2! h''^S'i^z ^^^ the direction cosines of the x',y',z' axes relative to the x,y,z axes respectively.

/ /

T

CYIINDRICAL COORDINATES {r,e,z)

Fig.12-6

A point P can be located by cylindrical coordinates (r, 9, z) [see Fig. 12-7] as well as rectangular coordinates (ir.y, 2).

The transformation between these coordinates is

12.18

X = r cos

y = r sin (

z = z

r — y/x^ + j/2 e — tan-1 {yfx) z = z

(^.tf.z)

Fig. 12-7

m

FORMULAS FROM SOLID ANALYTIC GEOMETRY

SPHERICAL COORDINATES {r,e,,p)

A point P can be located by spherical coordinates (r,e, [see Fig. 12-8] as well as rectangular coordinates (x, y, z).

The transformation between those coordinates is

12.19

<ar

X = r sm e cos <f> y = r sin e sin 0 z — r cos e

'r = V^2T^2 + ^ ■^ 0 = tan^i (j//ar)

EQUATION OF SPHERE IN RECTANGULAR COORDINATES

12.20 (X - xo)2 +{v- vo)^ + (z- zo)2 = R^

where the sphere has center {xq, i/q, zq) and radius R.

Fig.12-9

EQUATION OF SPHERE IN CYLINDRICAL COORDINATES

12.21 7-2 _ 2rorcos{o-9Q) + r^ + (z-Zq)^ = R^

where the sphere has center (r(,,tfo,«o) in cylindrical coordinates and radius R. If the center is at the origin the equation is

12.22 r^ + z^ = iJ2

EQUATION OF SPHERE IN SPHERICAL COORDINATES

12.23 r^ + To — Zr^r sin 9 sin So cos (^ - <Pq) = R~

where the sphere has center {7*o, 6q, <Pq) in spherical coordinates and radius R. If the center is at the origin the equation is

FORMULAS FROM SOLID ANALYTIC GEOMETRY

51

EQUATION OF ELLIPSOID WITH CENTER (xo,yo,2o) AND SEMI-AXES a,b,C

a^ b^ c^

Fig. 12-10

ELLIPTIC CYLINDER WITH AXIS AS z AXIS

12.26

where a, 6 are semi-axes of elliptic cross section.

If h — a it becomes a circular cylinder of radius a.

12.27

Fig. 12-n

riC CONE WITH AXIS AS z AXIS

a2 ^ 62

HYPERBOLOID OF ONE SHEET

Fig. 12-12

12.28

^ A. y^ - ^ - 1

a2 "^ 62 c2

Fig. 12-13

FORMULAS FKOM SOLID ANALYTIC GEOMETRY

PERBOLOID OF TWO SHEET

12.29

a2 62 c2 ~

Note orientation of axes in Fig, 12-14.

Fig.12-14

ELLIPTIC PARABOLOI

12.30

x2 ^ a2 "^ 62

Fig. 12-15

12.31 4-tf = -

Note orientation of axes in Pig. 12-16.

Fig. 12-16

DEFINITION OF A DERIVATIVE

If 1/ - f{x), the derivative of y or f(x) with respect to 2 is defined as

13 1 ^ = lim /<" + fc)-^<'> = lim /(^ + ^^)-/(^)

da; ji-o ft Ax-*o Az

where h = ^x. 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 m is the natural logarithm of u [i.e. the logarithm to the base e] where it is assumed that a > 0 and all angles are in radians.

13.2 £(c) = 0

13.3 4-i<^=^) = « ax

13.4 ^-ica;") = mcx"-> dx

13.5 13.6 13.7 13.8 13.9

d , ^ ^ ^ V du . dv dw

da; ' dx dx dx

d ^ . du

(cw) = e-y-

dx

' dx

d , , dv ^ du

dx'

dx

d , , dw , dv . du

-;—{uVw) ~ UV-T' + UW-i r VVJ-j—

dx dx dx dx

d_fu

dx \v

v(du/dx) — u(dv/dx)

13.11 ^ = ^^ (Chain rule) dx du dx

13.12 ^ =^ jV dx dx/du

1313 — = ^^^^^ dx dx/du

53

DERIVATIVES

DERIVATIVES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNC

fi lA d . du

Ii».l4 ^—sinjt = cosM^—

dx ax

13.15 -T—cosu = —sinw-;— dx dx

13.16 -J—tanw — sec^M^— dx dx

13.17 3—COtU = — CSC^M-r-

dx dx

13.18 j--secM — secMtanu-j—

13.19 3—cscM = — cse M cot It-r-

dx dx

13.20 ^sin-i« =

1 du

13.21 -I— cos ' u =

dx

:-l,i —

13.22 Atan-.t ^ T^f^ (13: 1 + 1*2 f^x

~2< sin '«< 2

-T [0 < cos J u < ■n\

-- < tan-i w < -

13.23 £eot-„ = ^1 10<cot-.«<.]

13.24 ^sec-.« =

du

±1 dw

13.25

dx

csc~^u =

\u\ Vm^-I t'* mVm2^ '^^

—1 dw q:l du

\u\ y/u^— 1

da;

tVw^ -1

da:

+ if 0 < sec-iM < jr/2

- if n-/2 < sec-^M < TT

- if 0 < csc^u < tt/Z + if -7r/2 < csc-^u < 0

DERIVATIVES 01^

PanW

OGARIfHMIC FUNCTIONS

,**, d , ^ogg^ dtt

13.26 -3—log„M = -3-

dx u dx

13.27 -r- In M = -j- log. m = ,

dx dx ^^ u dx

1 du

13.28 :^a" = aMna^ da: rfa;

13.29 :^c" - e"^ da; da;

13.30 ^u^ =. A ewIn« = e^^^^^[v\n u]

dx

dx

,du, , dv dx dx

»IRIVATIVES of HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIOI

13.31 -^— sinh u dx

, dv,

cosh u -J—

da;

1 3.34 -=- coth u — dx

, ,, du

— csch^ M 3-

dx

13.32 :^coshM da;

. , du

sinh u 3-

aa;

13.35 3- sechw = dx

~ sech M tanh u 3— da;

13.33 3— tanh « da:

, , du da;

13.36 -— cschM — da;

— csch u coth tt 3— dx

DERIVATIVES

55

13.37 Asinh-iM ax

13.38 3—cosh~^M dx

13.39 -^t&nh-^u ax

13.40 ~coth-iu dx

13.41 ^sech-iw ax

13.42 -^csch-iu da;

1 du

±1 dw 1 f^"

1 - m2 dK

1 du l--u:idx

^1 du uVT^^^^ -1 du

r+ if cosh-^ M > 0, u > 1"| [- if cosh-iM< 0, M> ij

[-l<u< 1]

+1 dit

Imj y/TT^ ^ MvTT^ '^

[m > 1 or u < -1]

r- if sech-i w > 0, 0 < u < 1"| L+ if sech-i M < 0, 0 < w < ij

[- if M > 0, + if u > 0]

HIGHER DERIVATIVES

The second, third and higher derivatives are defined as follows.

13.43 Second derivative = ±(^^ = ^^ = fix) = y"

13.44 Third derivative - ^(0) = ^ = /'"(*> = V

jr /jn—1.,

13.45 nth derivative

- dx \dx^-i dx« ^ ^''* ^

= -. ^fl. ^'Va^.t-

k. irfiB iV •^^ -t-fc TTff T 'tf^^^^t'\— iT^T^^l^ WT ^^ .^^^

LEIBNITZ'S RULE FOR HIGHER DERIVATIVES OF PRODUCTS

Let Z?" stand for the operator 3—5- so that D'm = -^-5 = the pth derivative of u. Then

ax'^ ax

13.46

■where ( 1, ( J, -.- are the binomial coefficients [page 3].

As special cases we have 13.47

^ I \ d^ , „ dudv , d^u

13.48

dx^

d^v

du d^v . „ d^u dv . d?u

Jz—( \ — _u 9 . ^ 4- Q . A-

d«3

dat^ rfa; da;^ dx^ dx di^

DIFFERENTIALS

Let y = f{x) and ^y = f{x +&.x) - f{x). Then 13.49

Ax iz ^ dx

where e -* 0 as ix -• 0. Thus

13.50 Ay 33 /'(x)Ax + *Ax

If we call Ax = dx the differential of x, then we define the differential of y to be

RULES FOR DIFFERENTIALS

The rules for differentials are exactly analogous to those for derivatives. As examples we observe that '3-52 d{u±v±iv± ■••) = du :t dv ± dw ± ■■■

'3-53 d{uv) ~ udv + vdu

1 3.54 d f?i\ - ^^dw — udv

13.55 d{M") = nw-idu

' 3-56 d(sin u) = cos« du

'3.57 rf(cosM) = — sin udu

PARTIAL DERIVATIVES

Let fix, y) be a function of the two variables x and y. Then we define the partial derivative of f(x, y) with respect to x, keeping y constant, to be

13.58 ^ ^ lim /(»+ Ax, y) - fix, y)

dZ Al-*0 Ax

Similarly the partial derivative of fix,y) with respect to y, keeping x constant, is defined to be

13.59 il = li^ /(x, y + Aj/) - fix,v)

^y Ay-.0 Ay

Partial derivatives of higher order can be defined as follows.

13.60 ^ = A. f^] ^ ^ ±(^

dx^ dx \BxJ' dy2 sy [dy

13.61 -^ = A/a/"\ j^ = ±f^

dxdy dx ydyj ' SyBx 3y \Sx

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 fix, y) is defined as

13.62 df = fdx + ^dy

ax dy

where dx = Aa; and dy = Ay.

Extension to functions of more than two variables are exactly analogous.

DEFINITION OF AN INDEFINITE INTEGRA1

li ~-= fix), then y is the function whose derivative is f(x) and is called the anti-derivative of /(ar) dx " -^ /• dy

or the indefinite integral of f{x), denoted by | f{x)dx. Similarly if y - \ f{u) du, then ^ = f{u).

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.

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 u denotes the natural logarithm of u where it is assumed that M > 0 [in general, to extend formulas to cases where w < 0 as well, replace In u by In |m|]; all angles are in radians; all constants of integration are omitted but implied.

14.1 14.2 14.3 14.4

14.5 14.6

14.7 14.8

14.9

■ dx = ax

I af{x) dx = a \ f{x) dx

I (u±v±w:ii ■ ■ •)di£ = I udx ± I vdx ± \ w dx ±

i udv — uv — I vdu [Integration by parts]

For generalized integration by parts, see 14.48.

j f{ax) dx = — \ /(«) du

/w" + 1 Wdu = -^V-T» K^-1 [Forn = -l, see 14.8]

r — - In M if «> 0 or In {-u) if u < 0 J u

= In ImI

du = e"

14.10 i a"-du = I e"'"

1 du

gU In a In a

In a '

a> 0, a¥^l

INDEFINITE INTEGRALS

_ . _, r . « . w sin 2« ,, .

14.21 I sin^K du = o" 2 — — ^(w — sinu cosu)

1 4.22 j cos2 II du = — -\ 2 — — ^(m + sin u cos u)

14.23 Jsec»tan«du = secu

14.24 I CSC II cot u du = — CSC u

14.25 I sinh tt du = cosh u

14.26 I coah udu = sinhu

14.27 I tanhitdw — In cosh m

14.28 I coth It du = In sinh u

14.29 I sechttrfw. — sin-Mtanhw) or 2 tan"* e«

14.31 I sech-w d« — tanh u

14.32 j csch2 urfu = -cothii

x

14.30 i csch u du = In tanh ^ or — coth-'c"

14.33 I tanh^w du - u — 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

I coth2 u du = u — coth u

I sinh^ u du = —r -^ — ^(sinh « cosh u — u)

1 cosh^ udu — \- — — ■|(sinh u cosh M + «)

I sech M tanh « du = — sech u

I csch u coth u du = — csch u

f du _ 1 _,M

J u2 + a2 ~ a ^" a

/du 1 - /u — a\ 1 ^. , " 9^5

-5 ^ = TT 1" ( . ~ ~-coth-i- m2 > o2

u^ — a? 2a \u + aJ a a

/ /

^ oin 1 —

du

. '^" = In (m + Vw2 + a2) or sinh-i -

Vu2 + a2 "

r ^" ^ In {« + Vm2 - a2 )

— , — -sec~M-

wV^^23^ a I a

r dw ^ _l^^f a + ^u^ + a^ J uyju^ -t- a2 » " V tt ^

r du ^ _\^^{°:1S^^Z^

This is called generalized integration by parts.

^'''dx

IMPORTANT TRANSFORMATIONS

Often in practice an integral can be simplilied by using an appropriate transformation or substitution and formula 14.6, page 57. The following list gives some transformations and their eflfects.

14.49 r F{ax +b)dx = - f F(«) du

14.50 r FcVax + b )dx = | J «^(w) du

14.51 r Fi^fax + 6 ) dx = - f w'«-iF(M)du

14.52 f F(Va2 -x'^)dx = o J F(a cosw) cos«

14.53 j F(y/W+~^)dx ^ « ) F(a sec u) see2«

du du

where u = ax+h

where u = •\Jax+ 6

where u = v**3J + ^

where x = a sin u

where x — a tan u

INDEFINITE INTEGRALS

14.54 J F{\Jx^ — a?-) dx = a \ F{a tan u) sec u tan u du where x = a sec m

14.55

fFie^^)d. = If

F(u)

du

1 4.56 J F{ln x)dx - ( F(u) e« du

14.57 fFUin-i^j (te = orF{w)cosw

du

where u = e"^

where u = \nx

where u = sin^^ — a

Similar results apply for other inverse trigonometric functions. 14.58 f Fisinx,co.x)dx - 2Jf(^^,{^\^ where u

= tang

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.

14.59 14.60 14.61 14.62 14.63 14.64 14.65 14.66 14.67 14.68 14.69 14.70 14.71

INTEGRALS INVOLVING &x + b

7~r = - In (ax + b)

ax + 0 a ^ '

Jxdx _ X b r-r =: 5 In (ax + 6) ax + b a a^ ^ '

f x^dx _ {ax + 6)2 2b(ax + 6) , 6^

I r-r ~ ~ — „ ■ ' i—5 H ^ In (ace + 6)

J ax + b 2a3 a^ a^ '

/ /

/

^ (ax + &)3 _ Sb(ax + &)2 ^ BbHax + b) _ ^^^ ,^^ ■ ^

3a*

x^ dx ax + b

dx _ i^ I w

x{ax + b) b \ax + b

dx _ 1 a^ / ax+b

zHax + b) ~ ~bx "^P^^l X

dx

x^{ax + 6)

dx (ax + 6)2

dx

2ax-b o2 262^2 ^ 63 '" V ax + 6

-1

J (ax

/x' (ax

a(ax + 6) 6

+ 6)2 2 da;

9, _,_,, + —5 In (ax + 6) a^ax + 0) o2

ax + b

i3

62

+ 6)2 a3 a3(aa; + 6) a^

x^dx _ (ax + 6)2 Bb(ax + b)

26 , , , ,, 5 In (ax + 6)

(ax + 6)2 dx

J a;(o3; + 6)2

2a^

]

6 (ox

+

63

a*{ax + 6) a

+ ^ In {ax + 6)

! + 6) "^ 62^"(^aa: + 6J

a;2(ax + 6)2 62(aa:+6) 62a: 6^

1 , 2a , / ttic + 6 + TTT In

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

dx ^ {ax + 6)2 3a{ax + b)

xHax + 6)2 26*x2 "^ Mx

_ 3a^ , / ax + b

bHax + b) b* \

dx

-1

2((m; + 6)2 -1

aHax + 6) ^ 2aHax + 6)2 o3{ffla: + 6) 2aHax + 6)2 ^ "^3 '" ^'^^ + "'

362

63

a? a'^{ax + 6) 2a4(ox + 6)2 a

T In iax + 6)

a^x<

2aa 1 , / aa; + 6

263(a» + 6)2 63(a« + 6) 63 I x

S X /

/

/

J

J

J J

If n ^ -1, -2, see 14.62, 14.67.

J

If 71 = -1, -2, -3, see 14.61, 14.68, 14.76.

r r-r- + — \ r^ I a!'"(aa; + 6)" i dx

m + m+1 wi + n + lj '

(ax + 6)3

xdx

{ax + 6)3

a;2(ja: {<u; + 6)3

(ax + 6)3

dx

x{ax + 6)3

dx

xHax + 6)3

rfx

x3(ax + 6)3

(ax + 6)" dx =

_ —<t 2a 1 , 3a . f ax + 6

262(ax + 6)2 63(ax + 6) 63x 6* ^" \^ x

a-'x2 4a3x _ {ax + 6)g _ ^ ,„ / ax + b \

ax + 6)2 65(ax + 6) 265x2 b^ \ x )

265(

(ax + 6)" + i (n+l)a ■

If n = -1, see 14.59.

x2(ax + 6)"dx - io-^ + bY*' _ 26(ax + 6)n-H2 62(ax+6)"-^' (n+3)a3 (Ti + 2)a3 ^ (n + l)a3

14.83

J x'"(ax + 6)"dx

x"'(ax4-6n + * -mb T „ ,, , t. ■,

-^-^ , ,. - 7—; r-rr I «'"~Vax + 6)"dx

(m + n + l)a (m + m + l)a J '

:n+l)6 ^ (n + l)6 J "" (ax+ft) rfx

M.84 J

/

dx

14.86

yax + b

dx ■\/ax + 6

x2 dx Vax+6

INTEGRALS INVOLVING y/axTh Zyjax + 6

2(ax-26) /—TT 3a2 ^°^+^

2(3a2a:2 _ 4a6x + 862) _

-^ -—- -' Vox + 6

15a3

14.87

14.88

J J

dx xyax + 6

1 /v'ax+T — ■/&

■/& \v'ax+6 + /6, 2

tan

dx

x^ax + 6

[ax -6a! 26 J

^/=:6

Vax + 6

6

dx

x-\/ax + 6

[See 14.87]

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

ax + b dx =

I xVax + b dx —

I x^y/ax + b dx = /" y/ax + b

r \/ax-J ^

J' g"* VcuF + 6

/

2V(ax + 6)3 3a

X

■\fax-¥h

dx = dx = dx =

2(15a2a;2 - 12abx + 862) . 2yiax + 6 + 6 f

dx

xvax+6 Vax + 6 a /• d«

2 J a:.

"^ -' ■^•\/ax+ 6

[See 14.87] [See 14.87]

2x'«V«a: + 6 2m6 T a;*"- ^

a J

dz

rfa;

x^yax + 6

ax + 6 dx =

dx = -

dx =

\ x"'V

/■\/ax-¥h

/y/ax + b x^

r (ax + 6)'"/2 dx f x(ax + 6)'«/2(ia: j xHax + 6)'n/2 dx

r^ax+_6r^

f (gg + b)"'^2 J

r dx

J x(ax + 6)"'/2

{2m + l)a {2m + l)a J ^^ + (,

Vaas + 6 _ (2wi - 3)a /" dx

*(m-l)6x'"-i {2m~2)6j a^^-iVo^Tfr

= ,o ^Tox (t^ + &P^' - /o ^^',^ r a;"*" Vttg + 6 dx (27n + 3)a^ ' (2m+3)aJ

V ax + 6

s r

dx

(m-l)x'"-i 2(m-l) J a,m-iV^^T6 -(ax+ 6)3/2 (2m BW r Vox + 6

(m-Dbx""-! (2m

^ 2(ax + &)t'" + 2>/2 a(m + 2)

-5)a f -2)6 J

X"!-!

dx

_ 2(ax+6)tw» + 4V2 ^ 26(ax+6)t'" + 2)/2

a2(m + 4) a2(m + 2)

2(ax + 6)t'"+g>^2 46(ax + 6)<'"+<)/2 262(aa; + 6)<"^+2J/2

a3(m + 6) aH7n + 4) a3{m+2)

dx =

dx =

2(ox + 6)'"/2

J

(aX + 6)<'n-2J/2

+ 6 I i;:::^-^^^^- —dx

m J X

_ (aa; + 6)t"'+2)/2 mo r (ax + 6)"'/a ^^

6x 2

(m - 2)6(ax + 6)('«-2>/2

26

+

IS

dx

x(ox + 6)<"'-2>/2

INTEGRAIS INVOLVING ax-\-h AND px + g

14.105 J

14.106 ^

14.107 f

14.108 r

dx

14.109

J

In

px + q

bp — aq \ ax+ 6

-r-^ i~ln(ax + 6) - ^ In (px + g)

6p — a^ a p

+

In

6p — aq 1 ax + 6 bp — aq V ox + 6

px + q

1

In

ax + 6

bp — aq 1 6p — aqr \px + q

6» 1

o(ax + 6)

+

(ox + 6)2(px + q) (bp - aq)aHax + 6) {bp - aq)^ \ p

^\n{px + q) +

^fc^hn(ax + 6)

INDEFINITE INTEGRALS

14.110

/

dx

-I

dx

(ax + h)^{px + q)" (n - l)(6p - aq) ]{ax + 6)'"^Hp3: + 9)"~^

+ a(m + n-2) J^^^^jj^^p^^^j^^,

14.111 1^ 7—dx = — + ^ „ In (px + q)

J px + q p p2

' rl J(°^+^)"'^' + in-m-2)a C-^^^^^dx]

{n-l){bp-aq)\{px + q)"-^ ' J (px +«)" ^ J

14.112

.) ipx + q)^

dx —

Zl__ji^±^ 4- mibp-aq) f

(n —7n —l)p UpK + g)"^! ^

(px + 7)"

dx

(n-Dpltpx + g)"-! '"''J (px + g)"->'*^

INTEGRALS INVOLVING v/aa; + b AND px + q

14.113 f P^ + ^ dx = 2(apa' + 3ag-2M^/^^:^

-* ^/^.x + h 3a2

14.114

/

dx

'yjp(ax + b) — y/bp —

aq

In

(px + g) y/ax+ b

y/bp — ag Vp V vp("^ + 6) + \bp —

aq

\aq — bp vp

tan

-1 I p(oa; + 6) "^ a? — 6p

14.115

/

\ax + b px + q

dx =

2\ax + b vbp — aq lyJp{ax + 6) — yjbp — aq + =— In

P pVp \yjp{ax + 6) + y/bp — aq^

2yax + b 2y/aq — bp

pVp

tan

_, / p(qg -"Y ag -

+_61 6p

2(px -r qy ■ 'y/ax-ro hp - aq f

ij>:c + qYy/ax + bdx = ^^^^fs)^ + (2« + 3)p J ^/^^^ip^

14.117

r dx_

-^ (px + q)"- V

2(px + g)« + JVax + 6 ^ 6p - ag r ( pg + ?)"

/ax+ 6

(2n - 3)a T

-6p) J

dx = ,„ ■ „ -- h T^ , ».'' ■ I 'T ' "" ■ da;

(px + g)-V^^Tt ~ Tn-l){aq-bp)(px + qY~^ ' 2{n - l)(aq - bp) J (pa; + g)"-> V^^+6

(2n + 3)p Vax + b

^;i^ +

dx

lAllfi r <Pa^ + 9)" ^ _ 2(px + g)"Vgx+6 2n(ag-6p) r (px + g)"-idx ■ J ,/:r7XT ''"^ {2n + l)a + (2^+1)^ J ,/—Tft

Vox+fe

Va^ + 6 (px + g)"

—\ax + b

V ax + 6 dx

,,o r V"3^ -^ o _ -V«'^'^P g r dx

**■"' J(px + a)" "^ ^ («-l)p(p:r + g)'«-i "^ 2(7i-l)pJ (p^ + ^j„-i^/^^:jrb

"~^^ ^ - ^^ UMI I il

INTEGRALS INVOLVING y/ax + b AND v^xTg

14.120

/

dx

■\/(ax + 6)(px + q)

- In (y/a(px 4- g) + y/p{ax + b) )

2

tan-i

'—ap

V —p(ax + 6) a{px + g)

4^f

14.121 r

xdx _ V(aa: + b)(px -f g) fey +

ag

\/(ax + 6)(px + g) **P ^"P '^ ^/(ax + b)(px + q)

J

dx

U-

INDEFINITE INTEGRALS

14.122 14.123 14.124

|V{a-+6){px+,)d. ^ '"p^ \l^ + "^ V(a.+&)(px+g) - ^''^g-;;^^' 1^

da;

V(aa: + b){px + g)

VX+ q . _ Vi'^ + &)fP=g + g) ^ gg-ftp

ax + b

dx

dx

^ 2a

2^/axTb

f

dx

y/{ax + h){px + q)

{px + q) V(aa; + 6)(pa: + g) (ag — 6p) Vpa; + q

CEGRALS INVOLVING a;^ +

- tan-i — a a

= iln{a;2 + a2)

X — a tan~^ — a

y2 /,2

|--|ln(x2 + a2)

2a2^" 1x2+ a3

■ -5 i tan 1 —

2a2x2 2a4 ^"^ l^x^ + a^ a; + J^ tan-i-

2(x2 + a2)

—z , _1_ *an-i —

2(x2 + a2) 2a a

5^ + - In (x2 + a2)

2(x2 + a2) 2 ^ ^

1 + A.i„ ^ ^^

2aHx^ + o2) "^ 2ai \x^ + a^

-^ ^ - -3-tan-i-

a'fx 2a4{x2 + a^) 2a5 a

1_ _ 1 _ J, , / xM

Za*x^ 2aHx^ + a^) o* \^x2 + a2y

rfx (x2 + a2)''-i

2(m-l)a2(ar2 + a2)"-i (2n

^1

2(ri-l)(x2 + a2)n-i

^ 1 + J^ (' dg

2(Ti-l)a2(x2+a2)n-i ^ a2j x(x2 + a2)"-i

r a;'"-2dx _ 2 r x'"-2dg J (x2 + a2)n-i " J (x2 + a2)"

J_ r dx \_ C dx

0.2 J a.m(3;2 + a,2)n-i ^2 J a:">-2(a;2 + tt2)n

INDEFINITE INTEGRALS

65

14.144 14.145 14.146 14.147 14.148 14.149 14.150 14.151 14.152 14.153 14.154 14.155 14.156 14.157 14.158 14.159 14.160 14.161 14.162

r dx

J 3:2 - a2 -

■dx - a?

1 , f x - a \ 1 ., , a:

2a \x + a J a a

pn(x^-a^)

I o, x — a a: + ^ In ,

2 Va: + a

C x^dx J a;2 _ „2 -

r x^dx J a;2 — a2

/ dx iB(a;2 — a2)

C dx _ _1_ , J_, / x-a \

J xHx^ - a2) a^x "^ 2a3 \x + a/

^ x^{x^ — a2)

J (g2-a2)2

f gdg _

J a;2rfa; _ (g2 - a2)2

(x2-a2)2

y + yln{g2-«2)

1 , /g2_(i2

^ -J. In

2a2a:2 2a^ 1x2 - a2

- A: In / '^ - '^

-1

2{g2-a2)

+ -lln/=«-'^

2(a:2-o2) 4a \x + a

~^^ -i- 1 In {g2 - o2)

rfg

2{x2 - (i2) 2

-1 , 1

+ ^In

2a2{a;2 - qS) 2a4 Xx"^ - ti^

- T%ln

a^a; 2a4{x2 - a2) 4a5 \. g + a

+ 4ln

2a^g2 2a4(a;2 - a2) a« Kx-^ - a

-3B

2n-

J g{g2-a2)2

C dx

J g2(;c2-aa)2

r dx

J a;3(a;2_a2)2

r dg

J (g2-a2)n

/ g dg _

{g2 - a2)«

r dg ,

J g(g2 — a2)"

/ g"* dg _

(g2 - a2)n

r dg ^ 1 r dg \^ r

J g'"(g2-a2)n a2j a;'"-2(g2 - a2)n a2 J

2(Ti-l)a2(g2-a2)"-i (2n

-1

t-3 r dx

-2)o2j (a;2-a2)"-l

2(n-l)(g2-a2)"-i

-1 J, r dg

-I)a2(g2_a2)n-1 a2j x(g2-a2)"-i

2(n

J g'"-2dg 2 r g"'-2dg

(g2-a2)n-I ^ " J (g2-a2)n

dg

g'"(g2-a2)"-i

INDEFINITE INTEGRALS

14.163 14.164 14.165 14.166 14.167 14.168 14.169 14.170 14.171 14.172 14.173 14.174 14.175 14.176 14.177 14.178 14.179

14.180 14.181

J a^ — x^ 2a \a —

J a2-

or — tanh~' —

X

dx , a, I a + X

x^ 2 \ a — X

c^ dx

2 2

j^^ = -^-~}n(a^-x^

f dx ^ J_ / gg

J x(a2 - a;2) 2a2 '" \^a2 -

f dx ^ _J^ , J^, /« +

J xHa2 - x2) a2x 2o3 *" \^a -

f '^a: L- , J_ 1 f-^Lj]

J x3(a2-x2) 2a2:c2 ^ 20" \a2 - ara^

r dx ^ _

J (a2-x2)2 2a2(a2-x2) ' Aa^ "\a - x

J x dx _ 1 {a^-x

J (^

(a2 - a;2)Z 2(o2 - x2)

2dx

_ 1 l„/a+x

x2)2 2(a2 - x2) 4a V^a - x

J (a2 - -c2)2 2(a2 - x2) ^ 2 ^ '

+ :=^ln

r dx_

J x(a2 - x2)2 2a2(a2 - x2) ' 2a'' "* \^a2 - x^

/ dx _ —1 , X , _3_ , I a + X

xHa^ - x2)2 a*x 2aHa^ - x2) 4a5 \a - x

r ^g ^ -1 , 1 + J_ In / g^

J x3(a.2 - x2)2 2a4x2 aa^ias - a;2) o* \a? - x2

f dx ^ X 2n - 3 r dx

J (a2-a;2)n 2(n-I)a2(a2 - x2)r.-i "^ {2n - 2)a2 J (a2_a;2)n-i

J xdx {a2 - X

2)« 2(tt-l)(a2-x2)

/

dx

1 , i_ r dx

,-l)a2{a2 —a:2)n-» a2 J a:(a2-x2)"-i

x(a2 - x^)" 2(K

(a2-x2)n "• J (a2_3;2)n J (a^ - x2)n-\ : ■'< -

f dx ^ 1_ r dx , J_ C dx

J «™(a2-a;2)« a2 J a'n(a2-a:2)n-i "•" a? J 3:^-2(^2 _ a-2)n

■»- . r

r ' r>

INDEFINITE INTEGRALS

67

INTEGRALS INVOLVING V*^Ta^

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

dx

f . ""^ = In (a: + VicM^) or sinh" ^ -

cV«^ + o,^ a2

^ V^MT^ 3

xV'x2 + o2

ln(x + VkM^)

1 j a + y/x^ + d^

= - - In

a \ X

/ dx ^ Va=2 + a2

J a;2^a;2 + a2

a^x

Va;2 + o2 1 /a + \/x2Ta2

+ ^;-:i In

2a2a:2 2a3

3;V^2 + a5 0^2

+ ^ In (X + \/x2 + ^2 )

/ da: _

x3\/x2 + a2

I Vx2 + a2 dx =

rx2,/^2T^dx = .(x2 + a2)s;. _ g^vgrg _ 4 ^^ ^^ ^ ^^-.-^^ v' 4 8°

fxaV^T^d^ = (x2 + a2)5/3^^2(:,2 +^2)3/2

./ 5 3

r y/x^ + a2 , „ „ ^ fa + y/x^ + a'^

J a

dx = y/x^ + a2 — a In

Vx^ + a2

J

./ X

dx - -

\/x2 + a2

+ !n (x + ^3:2 + a2)

Vx2 + a2

dx = —

\/x2 + a2 1 /a + \/x2+a

2x2

--^In

J (x2 + a:

X dx —1

f x^dx _ —X

(x2 + a2)3/2 - .^/^5T^ + ln(x + -/^24^)

J (x2 + a2)3/2 ^^2 + a2

J x{x2 + a2

1 /a + Vi^Ta2^ ;;ln

r dx ^ V^' + "■' X

J x2(3;2 + a2)3/2 a^X ^4y^2+^

r dx -1 3

J x3{x2 + o2)3/2

fa + y/^ + a2\

202x2^ + 02 2a4Vx2 + a2

+ 2!b1"

a + Vx2 + a2

INDEFINITE INTEGRALS

14.203 f (a:^ + a2)3/2 da: =

14.204 r x(a;2 + a2)3/2 da;

14.205 ra:2(x2 + a2)3/2da:

14.206 r a:3(3;2 + a2)3/2 da;

,4.207 I i:?i±^>^.x =

_ (^2 + a2)5/2

x(x2 + a2)5/2 a2a;{x2 + a2)3/2 a^^V^M^ „6 . , „ ,

- 6 24 16 - jglnCt + V^^T^)

_ {a:2 + a2)?/2 (l2(a;2 + ^2)5/2

J

(a;2 + a2)3/2

+ a2V»2 + a2 - a3 In

a + V^2T^

(x2 + a2)3/2

14.208 1 '-^ : ^ dx =

a;2

,4.209 I "'^ + f'"' d. =

(3;2 + ^2)3/2 3x-\/xM=^ g

2 + 2a2 1n(x + VxM^)

/

2:r2 +g/x2+a2--aln

INTEGRALS INVOLVING \/a;2 - a^

14.210

14.211 14.212 14.213

14.214

14.215

14.216 14.217

14.218 14.219

14.220

14.221

14.222 14.223

r^^^__ ^ In {x + y/x^ - a2 ), C-^^^ ^ V^^^^-

J ^a:2 - a2 J V^sTT^

f_x2d^_ x\/a;2-o2 ^g ^ ^

dx

= — sec~^

xy/x^ - a2 «

Va:2-a2 ^

xyx'^ — a2 ^2

J

r

•^ x^yjx"^ — a?-C dx •^ a; Va:- - a^

r Va:" - a2 dx = ^

jx^f^^^dx = M^!)!^

I x^yj x^ — d

.7 5 3

C Vx2 - a2

J ^2

C dx

J (a:2-a2)3/2

-^\n{x-¥ y/x^-a^)

a;(a;2 - ^2)3/2 a^x^f x"^ ~ a^ a*, / ^ r^ 2\

2 da; = —^ ;; 1 r- In (a; + \x^ — a^ )

dx = -\/x^ — a2 — a sec~' —

I a I

Vx^ - o2

dx = + In (a; + Va;2 - a^ )

-\/x^ — a2 J

2a:2

X

2a I a I

a2-/a:2 - a2

INDEFINITE INTEGRALS

69

14.224 14.225 14.226 14.227

14.228 14.229

14.230 14.231

14.232 14.233 14.234

14.235

14.236

J x dx _ —1

r dx

C dx

} a:2(a;2 - a2)3/2

r dx

- -3 sec 1

Vx^ - gg _ a;

1 3

)3/2

J(.-„.j3.,, . x(.2 - „2)3. _ Sa^xV^ ^ 1^, ,^ ^^ ^ ^.^.^^

fxHx2-a^)^ndx = (a:^-a2F^^ a2(,2-„2)5/2 t/ 7 5

•^ ^ 3 I a I

r (x2 - a2) 3/2 J X^

(a;2_ 0.2)3/2 3a:\/a;2 - a2 3

-■|a2 1n{x + Va^ - a2)

(g2 _ q2)3/2 3^x2 - gg _ 3

2x2 "^ 2 2

INTEGRALS INVOLVING V^^TT^

14.237 r , '^^ = sin-i^

14.238

J v/q,2 „ 3.2

14.240 14.241 14.242 14.243

/~2 j_ x-Ja^ — x^ „2

Va" - a;2 2 2a

r ^'^tfx _ {a2 _ ^2)3/2 — 5

r rfx ^ _]_ / g + Va2_a.2

J xVa2 - x2 '^ " V 3:

Va^~x2

J 3;Z^tt2 _ a.2 a^x

X rfx x3Va2 — x2

Vo^ - x2 1 /a + Va^-x

7^ In

2a23;2 2a3

fr

r

r Va2 - x^

14.244 I yfd^^x^dx = ^^^^—^ + f sin-'l

14.245 I x^a^-x^dx = _ (5i__^!l!^

I xy/a~ — x^

14.246 J a;2^a2-x2dx = _^i^_-!_ + v_ +__sj„-i_

14.247

14.248

J 0 O

14.249 I-—s dx = sin-'T

14.250 J ^^—^—rfa: = - n^g + ;r: 1° ' ^

2ic2 ^ 2a

^^•^5^ /(^ '^

2^x2)3/2 „2V^rr:i^

14.252 Tt^'^'^

)^'^ V^

14.253 I Y^y 2^372" = , - s'n^-

,4.254 f-V^i^ = V^^:^+^^=

/dx 11, f g + Va^ —a?2 '\

p dx _ -1 3 3_ J / tt +Va

14.257 J ^3(„2_«;2)3/2 - 2a2x2V^3^^ ^ 2a^^^^^^ ^a^ "V

14.258 J (a2-x2)3/2da: = ^ ^ ^ + ^^8 + ga* sin »-

J/-,2 _ 3:2^5/2 X(a2-x2)3/2dx = -IB-^J—

,4.260 J .^(.-.=)«.. = -2(2!:^ + -'-<"'--''"' ^ ^^^f^ + fi^in-.f 14.261 /.»(.-x=)«<£x = i^-=fg^°^'°^--''"'

,4.264 Ji °^-f'" .. = _(£!^^_?VHEE?+|„„(£±3^)

INDEFINITE INTEGRALS

71

14.265

J

dx

ax^ -f 6a; + c

2 , 2aa- + & tan~'

V4ac — 62 yf^a^^^^

In '

V62 - 4ac \ 2aa; + 6 + \/b^ - 4ac,

If 62 = Aae, ax^+hx-¥c = a{x + 6/2a)2 and the results on pages 60-61 can be used. If 6 - 0 use results on page 64. If a or c =^ 0 use results on pages 60-61.

14.266

dx

ax~ + hx ■¥ c

'^"''^ J aa;2+6x + c (jn-l)a a J ax^ + 6x + c a J

dx

ax^ 4- 6x + c x""^^ dx

14.269

J

dx

= i^In

14.270

x(fflx2 + 6x + c) 2c \^ox2 + 6x -I- c

dx 6

2c J

ax2 + 6x + c dx

ax^ + 6x -I- c

KL

dx

az2 + ba; + c

C dx _ _6_, / ax2 + &x + c N _ J_ . ^^ - ^ac T

J x2(ax2 + 6x + c) 2c2 \ x^ ) ex %<^ J

,.„, f dx ^ _ 1 b f dx ^ f —

J x"(ax2 4-bx + c) (n-l)cx"-i cj x'»->(aa;2 + bx +c) cj x"

Id 979 r tfx ^ 2ax + 6 , 2a. f

i*i.i/x J (^3.2 + ^3.4.^)2 (4ac - b2)(ax2 + 6x + c) 4ac - b\}

dx '

-2{ax2 + 6x + c)

dx

14.273 14.274 14.275

/

(ax2 + bx + c)2 (4ac - b2)(ax2 + 6x + c) 4ac - b^ J ax^ + 6x + c

xdx __ bx + 2c b r dx

(ax2 + bx + c)2

2dx

. to

S • -

-bsj

r ^

J (ax2 +

J (ax2 + bx + c)"

{4ac - b2)(ax2 + fcx + c) 4ac (b2-2ac)x + be + 2c T

aic2 + bx + C dx

+ bx + c)2 a(4ac - b2)(ax2 + bx + c) 4ac - b^ J ax^ + bx + c

mdx -''"-' (m—l)c i x'"~2dx

(2n - m - l)a(ax2 + bx + c)

(n —wt)b C x"*~* dx

(m-l)c f

"-I (2n —m-l)aj

(ox2 + 6x4- c)"

.X

14.276

14.277 . j(^2 _j_ bx + c)2 ~ 2c(ox2 + bx + c)

14 978 r ^^ ^ 1 _3g r

1.^/ o J ^2(a^2 + 6a; + c)2 cx(ax2 + 6x 4- c^ c J

14.279 f

{2n-m-l)aJ {ax2 + bx + c)"

2'>-3 dx _ _ ^ f

x2''-2 dx

X x^^-^ dx _ 1 C x2"-3 dx _ c r X

(ax2+bx + c)'' ~ a) {ax2 + 6x + c)"-i a J (ax2

c dx ^ 1 _ J. r '^'^ I ^ f

J x(ax2+bx + c)2 2c(ox2+bx + c) 2c J (aa;2+6x + c)2 cj x(ax2 4-bx + c)

+ bx 4- c)n a J (ax2 + bx 4- c)" dx . 1 r dx

dx

c2(ax2 + bx + c)2 dx

cx(ax2 + 6x 4- c) 1

(Ox2 + bx + c)2

_ 2b f

C J X

dx

(ax2 + bx + c)2

(m + 2to — 3)

x'"(ax2 + bx + c)" (wi - l)cx"'~Hax^ + bx + c)"-i (m- l)c

dx

dx

2(aar2 + bx 4- c)"

(m + n —2)b (m — l)c

J ar"*-'

(ax2 + bx 4- c)"

INDEFINITE INTEGRALS

INTEGRALS INVOtVING \/a^Tb^Tc

In the following resulte if 62 = 4ac, Vax^ + bx + c = yfa{x^ 6/2a) and the results on pages 60-61 can be used. If 6 = 0 use the results on pages 67-70. If a = 0 or c - 0 use the results on pages 61-62.

14.280

/

dx

14.281 r

14.282 r

yjax"^ + hx + c

xdx ■\/ax^ -f- 6a; -f c

x^dx y/ax^ +bx + c

~^\Tl{2^/a^J ax^ + bx + c + 2ax + 6) y/a

1 .„^i / 2ax-¥ b L V-a \yt2-4acj V^

yjax^ -\-bx + c h

or sinh 1

y/iac — b^

2oJ

dx

y/ax^ + bx + e _ 2ax - 36 j „ 2 , , , , 362- ^^c f

da;

Voic^ -i- 6a; -|- c

14.283

14.284

14.285

/

dx xy/ax- -I- 6a; + c

dx

_ J_ In ( 2y/cy/ax^ +bx + c + bx + 2c yfc

sin'

6a; + 2c

■■^y/ax^+JxTZ

I \^ax^ + bx + c dx =

y~^ \\x\yjb^-4ac

_ y/ax^ +bx + c b

2c J ^

1 • U-. / bx + 2c or — -— sinh ' /

Vc yx\y/4ac-b^

dx

y/ax'^ + bx + c

{Zax + 6) Va«2+"6xTc 4ac — 6^

4a

+

8a

/

dx

y/ax^ + 6a: -f c

14.286 rxVax2 + 6a: + cda: = (ax^ + 6x -H ^)3/2 _ b{2ax + 6) . ^^, ^ ^^ ^ ^

_ 6(4ac - 62) r da;

1^«' J y/ax^+bx + c

14.287 JWart^-hfrx + cdx = ^"'^'J^ ^^^^ + ^'^ + '^>''^^ + ^^'leJ^^ ' f V «=^^ + 6a; + c da;

dz

14.288

14.289

/y/ax^ + bx + c , , h f

^ dx = yjax"^ + 6a; -I- c -h | j

di

-I- c

J

Vaa;2 + &g-l-c , _ yfa^^^hxVc

yjax^ -H 6a; -I- c "^ xyfaifi^Vbx + c

•^ a;i

da; — —

14.290 /^

14.291 r

dx

2(2ax -!- 6)

+

-/

da;

\/a«2 -i- 6a; -f- c ^-^ xy^zM^TxTc

/

dx

ax2 + 6x -H c)3/2 ^^^^ _ j^,^ y^2+6^ + ^

xdz 2(6a; + 2c)

14 292 r ^^'^'^ :^ (26^-4ac)a; + 26c ]. r da;

J („^2 + bx -K c)3/2 ^(4^^ _ ^,^ Vaa;2+6x + c "^ a J ^ ^^2 + 5^ + ,

14.293

J x(

dx

-I-

^ (62 — 4ac) vtta;2 -I- 6x -I- c

_ 1 r dx b_ r dx

iax^+bx + cY'^ ^^^2 + 6^ + ^ "^ cj ^^/^^FT6^T. ^cj (ax^ + 6a;

14294 f ^^^ o3;2 + 26x -t- c 62 - 2ac r dx

J x2(ax2+6x-Fc)3/2 fl23:^ax2-K6x-Fc 2c2 J (a;^2 +6^,4^)3/2

36 r dx

^''^'^ x\/ax2T6xTc

14.295 ({ax-^-^bx + c)^ + ^'^dx = (2ax + 6)(gx2 + 6x-Hc)"+i/2 -^ 4a(m + 1)

+ c)3/2

INDEFINITE INTEGRALS

73

14.296 14.297

14.298

J,

2(2ax + 6)

(0x2 + 6a; + cY +1'2 (2m - l)(4ac - 62)(ax2 + 6a: + cY~^'^

8o(n — 1) r da:

+

/.

dx

(2w-l)(4ac 1

-62) J (aa:

(ax2 + 6x + c)" + >'2 (2n - l)c(aa;2 + 6x + c^-^i^

dx

+

'J

x(ax2+6a; + c)"-i/2

2 + 6x + c)"-i'2

2c J

d:e

(oar2 + 6x + c)«+i/2

INTEGRALS INVOLVING x^ + a^

Note that for formulas involving x^ — a^ replace a by —a.

,4.299 f^^ = ^,„_i£±^ + ^tan-^^

1 , a:2 _ aa, + a2 j _, 2x-a

■T— In — -. —;— 7K 1 —z tan ' p—

6a (x + a)2 ^^ „^

14.301 (' ff^, = |ln(a:3 + a3) J 3;3 + a3 3

14.300

/ xdx _ a;3 + a3

dx

ay o u-v "

;.3 + „3) ,4.302 /^(^ = 3^3ln(^

j_ _ j_ x2 - Qx + g'i _ ^_ ^^^„, 2x^1^ a3x 6a4'" (x + a)2 ^,^ ^^3

ax + a.2 a:2 — fflx + g2

3a4\/3

14.303 J a:2(x3+a3) ^03; oa vx-r or

wsAA r_-^5— g , J_, (g + «)^

14.304 J (j.3 + a3)2 - 3a3(x3 + a3) "^ 9a5 ^" x2 - ax +

lAtn^ r ardx _ a:2 , 1 .^ a:2^axjt^

I4.J03 J (a:3 + a3)2 " 3a3(x3 + a3) ^ 18a* (x + a)2

,--*- r x2dx _ 1

J (x3 + a3)2 3{x3 + a3)

^^•^°^ J x(x3 + a3)2 ^ 3a3(x3 + a3) "*■ 3^ ^" Vx^ + a3 j I^.JUlf J ^2(a^ + q3)2 a^x 3a«(x3 + a^) Sa^ J x^ + a-

14.309 J ^^.p^ = ^^ - "^ J -^T^

J^ —\\ C dx

x"(x3 + a3) = a3(n-l)xn-i " aV x"-3(x3 + aS)

INTEGRALS INVoK^^^^^C?

^ , 2x — a

-^tan-i 3-

Sa^VS aV3

2x -

+ tan"!

[See 14.300]

14.311 14.312

14.313

14.314

x2 + axV2 + o2 a^ + a* ~ 4a3\/2 "' V*^ " «*V2 + ©2

X

C xdx

} x^->r a'^

C x^dx J X* + a*

J x^dx x^ + a*

In

= ::-;; tan » -5-

2a2

1 /x2 — 0x^2 + a2

In

4av^ \a2 + ax\/2 + a^

1 . _. axv2 -tan 1 „ „

2a3V2 * " '^

^ tan- ^1^ 2ay2 a:2 - a^

J In (x* + a*)

INDEFINITE INTEGRALS

14.315

/

dx

^ T^ In

x(x'* + a") 4a4 \x^ + a^

2aG^""'f^

14317 r ^'^ - ^_}: 1

14.318 f-r^ = J_ln/^^^VA

14.3.9 J,!^ = i^I-nfe

M.3ao J,5!i?|^ = -L,„

^ „, axy/ 2

^ tan ' -s 5

V2 * ~ **

tan-i-a

z — a\ , 1 . _, X —;— 4-;;-tan 1— x + aj 2a a

14.324 r

dx

1 , 1 , / 3:2 - a2 + -r^ In

x3(3ri - a*) 2a'»x2 4a6 y ^2 4. ^z

INTEGRALS INVOLVING a;" ± a"

14.325

14.326

14.327

14.328

14.329

14.330

14.331

14.332

14.333

14.334

r dx_

J x{x^ +

+ a")

^ In ^^

na'" a;" + a"

fx"^*dx 1

/pwi— n (£2;

/ x"'dx _ r x"*~"da: _ „ T (x" + a")' ~ J (x" + a")'-! "■" J (x" + a")*-

X dx _ J_ r dx J_ r <ix

x'"(x" + a")'' a*^} x'"{3;" + a")''-* a" J

dx

x"*'~"{x" + a^y

_ 1 / ^/x" + g" - \/a" \

xV^*^" + a" mV'a" \\/x*» + a" 4- yf^J

/ dx _ _1_ - / g" — a" \

x(x" — g") Tia" I X** /

Jxn-irfx 1, ,

—;, ;: = - In (x" — a")

X" — a" n ^ '

/ x^dx _ ^ r x"'-"<?x r x""" dx

{x^'-a^y * J (x"-g")'' J (x''-a")'—1

r dx ^ 1 r dx ?^ f —

J dx

da;

2 . / o"

:cos^

INDEFINITE INTEGRALS

75

14.335

2m

,2m-p

(2fc - l)pir / a: + a COS [(2fc - l)W2m]

2 sin —s— tan^i

fc=i

a sin [{2k - l)jr/2m]

y

where 0 < p ^ 2m.

2ma2m

1.2m-p j^^j^ 2m \ 2m J

14.336

a2m

fepr m

k^

2 cos—^ In [ a;2 — 2(ixcos-^ + a

14.337

14.338

1 '"■^' . fcprr . _, /a: — a cos (fc^/m)

o„-n 2 sin -^— tan ^ :— ., , -:—

^a2m-p ^ ^ 1 a sin (fcirArt)

where 0 < p ^ 2m.

J x2m + 1 4- a2m +1

+ 2^i^t^°^^"''^ + (-l)Mn(x + a)}

2(_i)P-i ™ . 2fcp^ , _i / x + g COS [2kirl{2m +1)]

(2m+l)a2--P + ifc^i "'" 2m + 1 ^^"

(_l)P-i

2kpir

asin[2fc7r/(2m+l)] 2fe7r

-^ 2 cos^^^ In x2 + 2ax cos^^^ + a^

(2m+l)a2'"-p+ifc=i 2m+l \^ 2m + 1

(-l)p-Mn(x + a)

(2m+l)a2'n-p+»

where 0 < p ^ 2m + 1.

j.2m + l _ (i2m+l

-2 ^ _in ^'^/^ tan-i f ^^

(2m+ l)a^'n-P+i ^-f, ="" 2m + 1 1, a sin [2K/(2m + 1)]

_2 ■'• ZkpTT - /''^ ~ a cos [2fcjr/(2m + 1)}

+

(2m + 1)

1 ^ 2fcps- , / 5 „ 2fc:r , „

^ TT 2 cos -= ——■ In a2 _ 2ax cos ■= t—t + or

j(j2m-p+i^"^j 2m + 1 Y 2m+1

+

In {x — a)

(2m + l)a2m-p + i

where 0 < p ^ 2m + 1.

INTEGRALS INVOLVING sin ax

i> ■

14.339 14^40 14.341

sin ax dx — — sin ax dx =

cos gjc a

sinour a2

X cos ox

2x f 2 x^

x^ aiaax dx = -j sin ax +( -3 — — J cos ax

14.342 (n^ainaxdx = (^ " ^) s'"*^ + fe" ^

14.343 14.344 14.345 14.346 14.347

/^

sm ax , (ax)3 , (ax)^

dx = ax — ' '. +

/ sin. a:'

J sin J sir

ax

dx = -

3-3! sin ax

+

X

»J

5'5! cos ax

X

dx [see 14.373]

ax dx

— — In (esc ax — cot ox) = - In tan -^ a a £.

1 j , (ax)3 , 7(ax)5

2(22n-l - l)B„(ax)2n+l

■^ (2n+l)! ^

sin^ ax dx — -^^ —

X sin 2ax

4a

^ M

^':

-S.

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

X sin2 ax dx = ^ - ^ ^^" ^"^ _ ^os ^aa; 4 ia 8a2

sm^axdx = -^osax ^co^ax a 3a

sin^ axdx - ^ - s'" ^"^ , sin Aax 8 4a ^ 32a

dx

/ / / J .f J

J 1 — sin

J

J

sin^ ax

rfa; sin^ax

— cot ax

a

2a sin2 aa: 2a

+ t:- In tan -r-

sin P.C sin 9X dx = ^'" ^P " g>^ _ S'" ^P + ?)» 2(p-9) 2(p-h9)

[If p = ±q, see 14.368.]

^ ltan(f+^ ax a V4 2

i — sin ax a \ 4 2

4lnsin -?-

dx ^ -ltan(^-°^

1 4- sin ax o \ 4 2

J 1 + sin

{X — sin ax)2

r ^

J {X - si

= -^tan(^-«-^ sin ax a V 4 2

dx

_i_ ^ 1 . / f ox H s In sin T + ^r-

„ -tan ^ + ^ 2a \ 4 2

+ f- tan3 [ 5 + ^ 6a \ 4 2

(l-sinax)2 - "2^*^"(4"'2";- 6^**"H4-T

J

dx

p -f g sin ax

a-y/^2Tr^

tan-i

p tan ^ax + q

Vp2l-g2

In

p tan iax + 9 — vg^ — p2

14.361

14.362

14.363

14.364 14.365 14.366 14.367 14.368

If p= ±q see 14.354 and 14.356.

(?x o cos ax . JJ i^' dx

J

Q COS ax

+

(p + 9 sin ax)2 a(p3 - g2)(p ^ ^ gin ax) P^

If p = ^q see X4.358 and 14.359.

+ q sin aa:

Vp^ +

\/p2 _ q2 tan

/dx 1.1 y'p2 4- q,2 tan ax

/

f^rmsinaxdx = ^ ^^ cos ax ^ mx-"-isinax _ m(m-l) C ^rn-2 ^j^^^ g^

J a a^ a^ J

rsinox^^ = _ sinaa^ _^_ _g_ T gosox ^^ [see 14.395]

f sin" axdx = _sin"-iaxcosax ^ »^i_l T ^.^„_,

J an n J

p^ + q'^ sin2 ax

p^ — (p- sin2 ax

Vp^^2 P

^(72 _ p2 t;an ax + p lapyjrp- — p2 Vvg^ —p^ tan ox — p

' 2 ax dx

/ dx _ — cos gx , n — 2 ^ dx

sin" ax a(n — 1) sin"~* ax n— 1 J sin""^^^^.

J xdx _ —z cos ax 1 w — 2 f" x dx

sin"ax a(n —1) sin"""^ ax a2(w,-l)(n — 2) sin"-2ax n — 1 J sin"^^

INDEFINITE INTEGRALS

77

^OTEGRAIS rNVOlVlWs''^^^^

14.369

14.370

14.371

14.372

14.373

14.374

14.375

14.376

14.377

14.378

14.379

14.380

14.381

14.382

14.383

14.384

14.385

14.386

14.387

14.368

14.389

/, _ sin ox cos ax dx — a

/, _ cos ax , X sin ax X cos ax ax — — + a^ a

;2 COS ax dx — —^ COS ax + [^ — -^ ) sin ax

a a-

jx^cwaxdx = (^-±jcosax + [^-—Jsmax

Xcosax . , {ax)2 (ax)* {ax)'^ 4. ...

CS21^dx = -S£l^^ a (^^^^^dx [See 14.3431

J X^ X J X

= "In (sec ax + tan ax) = — In tan t + "^

COS ax

J

xdx ^ J_ Uax)^ (ax)^ 5(oa;)a

cos ax

+i5r+

"^ (2n + 2)(27i)! ■*"

/„ , a; , sin 2aa;

cos^ ax dx = -x +

4a

J- _, x^ , ;c sin 2ax , cos 2ax

C , , sin ax sin^ ax

I cos3 ax dx — r

J a 3a

JZx sin 2ox , sin iax

cos* ax dx - ^ + -^^ + -^2^

J

_ tan ax

cos^ ax dx

r_^ = sinax + 1 intan('^ + ^ -/ cos3ax 2a cos^ ax 2a \4 2

J, sin (a - p)x , sin ( a + p)x cos ax cos px dx = ^ : f-

2(a-p)

2{o + p)

Xr^

dx

1 .ax

= cot -r-

cos ax a ^

h St

dx

cos ax dx

X . ax , 2 , . ax - cot -r- + -5 In sin -^ a 2 a^ 2

1 ^ ax - - tan -^ + cos ax a 2

Jo-/

_JEA^_ ^ ^tan^+ ^.Incosf + cos ax a 2 a^ £

dx 1 ax 1 3 ax

dx

(1 + cos ax)^

1 ^ ax , 1 . „ ax ■^ tan -^ + -F- tan-f -^ 2a 2 6a 2

INDEFINITE INTEGRALS

14.390

14.391 14.392

14.393

14.394 14.395 14.396 14.397 14.398

h

dx

q cos ax

a\p^ —

. tan-^ ^/(p - q)/{p + g) tan^a;

In

tan ^ax + v'(g'+p)7(9--p)

[If p = :tg see 14.384 and 14.386.1

dx

J {p + q cos ax)^ dx

ay/q^ — p2 ^tan ^ax — y/{q + p)li.q — p)l

q sin ax _ p C dx

J

a(g2 _ p2J(p ^ q, (.pg g^jg^

, p tan ox tan"'

+ g cos ax

[If p = ±g see 14.388 and 14.389.

p2 + g2 cos2 aa: apy/^^T^^ y/p^ + q^

J p^ — q2 cos^ ax

dx

1 . 1 P tan ax ■ — tan~i ~

apVp^ — q^ yjp^ ~- g2

p tan ax — yjq^ — p^ 2ap-\/q^ — p2 \p tan ox + ^/q^ — p^

In

x-" cos ax dx = =—»ui_ux ^ »«x — cos ax s—■ I x"" ^ cos ax dx

a a^ o^ ./ _

Tcos^ ^ _^osax _^ C^^dx [Seel4.365]

fcosnaxdx = sin ax COS"-'ax ^ ZL^li C J an tf- J

r do

J cos"

sm ax

ax xdx

a(n — 1) cos'

X sin ax

^ « - 2 r da

"^ ax n — 1 ^ cos"-

cos""2aa! dx

dx

cos" ax a(re - i) cos""' ax aHn — l)(n — 2) cos''-^ oa;

+

K-2 r xd

n — 1 J cos"*-

dx

2 ax

INTEGRAIS INVOLVrNG sin ax AND cos ax

14.399 14.400 14.401 14.402

az cos ax dx =

J sin

fsinpxcosgxdx = COs(p-q)x _ cos {p + q)x J 2{p - q) 2(p + g)

I sin" I cos"

ax cos ax dx =

2a

sin" * ^ ax

{n + l)a

[If n = -1, see 14.440.]

ax sin ax dx = —

cQs"'^^ ax

(n+ l)a

[If n = -1. see 14.429.]

14.403 1 sin^ ax cos^ ax dx = — —

14.404 f-—

J sm a

8 32a

dx

14.405 f^-

J sm

14.406 r^-

J sm

X cos ax a

dx

= - In tan ax

1 , ^ / V , ax \ 1

= - In tan t + -h" ~ r

2 ax cos ax a \i 2 / a sin ax

dx

1 . ^ ax , 1

= i In tan -r- +

14.407

/ii

ax cos^ ox a 2 a cos ax

dx 2 cot2ax

sin^ ax cos^ ax

a

>-^

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

COS ax

dx = -

, 1 , , I ax , IT

J cos sii

I

dx ~

dx

a a 2

cos ax(l ± sin ax)

dx

sinofl:{l ± cos ax)

r dx

.) sin ax ± cos ax

ax dx

f sin aa sin ax ±

J

cos ax cos ax dx

1 + f lntan(^+J

2a{l ± sin ox) 2a \2 4

1 , 1 , . ax

+ ;;- In tan -r-

2a(l±cosax> 2a 2

av^ V2 8

X 1 , , . j_ .

-^ ^ ^ In (sin ax ± cos ax)

sin ax =: cos ax sin ax dx

X 1 = ±— + — In (sin ax ± cos ax)

J p4

I = In {p + g cos ax)

+ q cos ax aq '

cos ox dx

J p + g sin ox aq

C sin ax dx

In (p + g sin ax) 1

(p + g cos ax)" aqi^n — l)(p + g cos Ox)»~ '

C COS ax rfx _ —1

(p + g sin oz)" ag(m - l)(p + g sin ax)"-i

.7 p sin ./ p si

dx

ax+ g cos ax a^jrP■+q^

In tan

ox + tan ' (g/p) \

dx

sin ox + g cos ax + r

a-\]i^ ~ p2 - g2

tan"

^ / p + (r — g) tan (ax/2)

In

p _ Vp2 + q2 _ r2 + (r - g) tan (ax/2) \ a\/p2 + ^2 _ r2 "" Vp + Vp2 + g2 - r2 4- (r - g) tan (ox/2) / If r = g see 14.421. If r2 = p2 -j- ,2 ggg 14.422.

14.421

14.422

14.423

14.424

14.425

/

dx 1 , / , , ax

■• T— TT-T T = — In g + p tan ^

p sin ox + g(l + cos ax) a-p \ 2

dx

p sin ax ■\- q cos ax ± vp^ + g* aVp^ + g^ V

-1 ^J.^ ax^tan-Mg/p) \

J p- SI

dx

_ 1 , _.^ I p tan ox

sin^ ox + g2 cos^ ax apq V g

r dx

J p^ sin^ ox — g^

In

cos- ax 2opg \ p tan ax + q

p tan ax — q

J==

sin"" ax cos" ox dx —

sin"*' ^ ax cos" "^' ax , m

a{m+n) m+

sin"" + ^ ax cos"""' ax

o(m + n)

+

/3in"''"2 ax cos" ax dx

Jsin"^ ax cob''"^ qx dx

INDEFINITE INTEGRALS

J C(

14.426 I ?H^^dx

cos" ax

a(n

sin'"~^aa; _ m — 1 C sin*" ^ ax

— l)cos"~^aa; n — lj cos''~^c[x

(i(re — 1) cos

— sin"* ^ aa; a{m — n) cos"~i oa;

1 gg _ wi — n + 2 r _^

)s"~^ ax n — 1 J cos

sin"" ax

co&"^^ax

dx

+

m — 1 I s]n"'~"2 ax

cos" ax

1 Tsi

— M J I

da;

14.427

/CDS'" aa; , ^ *^ac = sin" ax

— CDS'""^ ax _ m — 1 r cQS*"~^ ax a{n — 1) sin"~i ax n — 1 J sin^~^ ax

— cos'" + ^ax m — 71 + 2 C cos'" ax

a(m—1) sin"-'ax n-1 J sin"-2ax *

— 1 r cos"

— n J sii

COS"" ' ax

,a(m — m) sin"-'ax wt

cos"*-^ ax sin" ox

14.428

/ii

dx

sin"" ax COS" ax

J^ ^ m + 7i- 2 f

"• ax cos"-' ax n — 1 J si

dx

a{n — 1) sin"

^1 _j_ m + ti-2 r

a(«i —1) sin™-'ax cos"-'ox m—1 J sin'"-

sin"" ax cos"--'ax

dx

2 ax cos" ax

IGRALS INVOLVING tai

14.429 I tanoarcZx = Incoaax = -Insecax

J « a

14.430 ftonZaajd* ^ t^EM _ ^

14.431 r tan* ax dx = ^^^ + ^ In cos ax J 2a a,

14.432 rtan"a«sec2axdx - t&n^^^ax J (7i+l)a

J ti

14.433 I I^Si^dx = ^ In tan ax tan ax a

14.434 I 17:7^::: = r In sin ox

tan ax a

M.435 J.tan^.. = ^ 1^ , <^ , ?^ , ... ,

22"(22n-l)B„{ax)2" + i (271 + 1)!

+

14.436 J^dx = ax + Iggli + g<^ + ■■ ■ + ''"<'!" " ^^^.^^'^f""^ +

9 ' 7B

(27t - 1)(2tc) !

14.437 fxtanSaxdx = ^-^^2^^ + ^ In cos ax --^

14.438

/

dx

px

+ ., , —57 In (.Q sin ax + p cos ax)

p + 9 tan ax p2 + ^2 ^ ^(p^ + q^)

14.439 ftan-axdx = tan"-iax _ C^j^n-ZaxOx J (n — l)a J

INDEFINITE INTEGRALS

81

INTEGRALS INVOLVING cot ax

14.440 r

14.441 r

cotaxdx ~ — In sin ax a

cot^azdx = -i^tox _ ^ a

14.442 14.443

J

j cot"

X

14.445 r

cot3 ax da = - S°^.BE - 1 in sin ax

fix C8c2 oa; dx = —

2a a

cot" +' ax

{n + l)a

14.444 I — dx = In cot a*

cot ax a

dx 1 ,

—7 = In COS ax

cot ox a

^AAAJ^ C * J iJ (*«)^ (ax)" 14.446 I X cot ax az = -^<ax — ^—~ Tor"

14.447

/

cot ax , 1 £Ei _ (ttx)^

X ax 3 135

22«F„(ax)2''-(2n + l)!

22«F„(ax)2"->

(2n-l)(2ji)!

14.448 fxcot^oxdx = -?LS^^ + ^insinax -^ J o a* 2

dx „ pg q

14.449

J

p + g cot oar p2 + g2 o(p2 -|- g2j

14.450 rcot«axdx = - cot"-'ax _ rcot"-2axdx J (n - l)a J

In (p sin ax + q cos ax)

INTEGRALS INVOLVING sec as;

14.451 I secaxdx = - In (sec ax 4-tan ox) = — In tan (-—+--J a ^ o V 2 4

J

14.452

sec2 ax dx =

tan ax

14.453 fsecaoxdx = 5e£«fLtanax + 1 j^ (ggc ax + tan ax) J 2a 2a

14.454 fsec-oo^tanaxdx = ^^Sl°± J na

14 455 f_^5_ ^ sin ax J sec ax a

r ,_. _ 1 J(ax)2 , (ax)* , 5(ox)8

/

14.456

sec ax dx =

a2 I 2 "^ 8 "^144 (2n+2)(2Ti)!

■iMJi^-w I sec ax J , ^ (ax)2 ^ 5(ax)^ ^ 61 (ox)" ^ , gt.(ag)'"' ^

14.457 I dx = inx + '-^ + -^ + -^^^ + ■•■ + ^^^^^ +

X

14.458 I X aec2 ax dx = - tan ax -\ s In cos ax

INDEFINITE INTEGRALS

+ q COS ax

14.459 r ^ = a_P f-

J q + p sec ax q q J P

14.460 (sec-ax dx = aeC-'^ax tanax ^ n-2 f ^^^^^ax dx J a{n — l) n — 1 J

'*■»

INTEGRALS INVOLVING cac ax

14.461 I CSC axdx = - In (esc ax - cot ax) = - In tan ^

14.462 fcsc2aa:cte = -S^ax_ J a

14.4«3 /csc=a... = _ esc .. cot „. ^ ^ ,„ ^„ ^

14.464 f CSC" ax cot ox dx = -Hc^L«£ J na

14.465 f— ^5_ - _ COS ax

J CSC ax a

(ttx)3 , 7_(ox)5 , , 2{22"-i-l)B„(ax)2'> + i

■'.! ••»

14.466 J X esc ax dx = ^ ■{ ax + ^-7^ + 'i i ^L 4- • ■ • +

18 1800

(2n +1)!

+

J

14.467 I H^i^dx = _JL + ^+ 7(^_j^ ,,_ ^ ^

2{22n-i-l)F„(ax)2"-i

ax 6 1080

14.468 I X csc2 ax dx = —

14.469

(2n-l)(2n)r

_ X cot ax , 1 ,

~ H 5 In am ax

a a'

J dx — * _ p r q -\- p CSC ax ^ 9 J

dx

P + g sin ax

[See 14.360]

14.470 fcsCoxdx = _cscn ^axcotax ^ n-2 fcsC-^axdx ^ a(n —1) n — 1 J

INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS

14.471 fsin-'-dx = x sin-> - + yJa^-x^ J a a ^

I X sin ' -J a

14.472

''^ - 'T-TJ"^" 'a + 4

14.4/3 I x2 sin-i - dx = -r-sm ' —I ;;

J a Z a V

C sin-Mic/o) rf_ - « . (x/a)3 1»3(x/a)S 1 • 3 ■ 6(x/a)7 , J X a 2-3-3'''2-4-5-5^2'4-6*7*7

f sin-'i J xa

14.476 r^sin-i^j dx = x ( sin-i ^ j - 2a! + 2^52^7^ gi^-i £

14.474

14.475

Jx/a) J _ _sm

in-i j x/a) _ 1 f a + y/a^-x^ a

-f

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

/

J COS"'—da; = a

z cos~^ - dx a

\ x^cos-^-dx a/ a

T cos"' (z/g)

J X

J cos~' {xia) x^

lecos"^- — yJa^ - x^ a

-1 £ _ x-\/a^ — x^ a 4

3 a 9

dx

X

dx

fin. -J _ cos~' (x/a)

sin~i (x/a) , dx

X

[See 14.474]

dx

( tan-i-J a

\ a:tan-> -

_ cos-'(g/g) + 1 In / « + A^a^ - x^ \

X a \ X J

a:fcos-i-j - 2a: - 2y/a^~x^ eos"'-

ztan-i- ^ ^In(x2+a2)

da: = ^(x2 +

o2) tan-i ^ - ^

^ X _ (z/a)3 (a:/a)5 _ (z/g)!

a 32 "^ 52 72 + •■•

a; a 2a \^ a;2 y

xcot-i I + |ln(x2 + a2i

rtan-Ma:/a)^^ ./ X

J tan-Mx/a) ^^

Jcot^'—dx = a

/x cot~^ -dx = a

Jx2 cot~i -dx = g

Jcot-Mx/g)^

^)

-i{x2 + a2}cot-i^ + ^

X3

|ln^- J tan-Mx/a) ^ fS,e 14.486]

_ _ cot-i(x/g) , 1 , fx^ + a^ ~ X + 2^^" -^2-

14.493 j sec-'- dx =

X sec-i a In (x + \/«^ ~ «'

X sec-' - + g In (x + V^^ ~ "'

) 0 <

'x2-a2)

-1 * ^ ^

sec ^ — < n

o 2

n < sec ' < n-<i a

14.494 i xsec-'* dx a/ a

.-. X

, X ^ IT

sec-' — < 7; g 2

0 <

^ < sec-' — 2 a

<

14.495

/

x^ sec -' — dx =

•. 1

x3 a: gxVx^ — a2 g^ , , , /-^ s.

y ^^ a~ — 6 ~ y In (a: + V^^-a^) 0 <

^^ ■=««-! ^ _i. a*Vx2 — a2 , a^ /—

T^'''^ a + 6 "^ yln{x + Va^2_„2)

(in '

I > '

■]•.

-1 ^

sec ' -

a

^'i

n < sec ^ —

INDEFINITE INTEGRALS

14.496 f^^^Af^da: = ^ In J X 2

14.497 r^ec Hxia) ^^ J X-

14.498 fcse-i-da; =

14.499 f a; CSC-1-da:

2'""^ + a: + 2.3.3 + 2.4.5.5 + 2T4T

sec-"(3;/a) , \/a;2 — o^ „ ^ .a:

0 < sec-i —

1 • 3(a/x)5 1.3. 5(a/x)? 2.4.5-5 "•" 2.4.6-7-7

(a;/a) ^ \/a;2 - pg ■ ax

a<i

sec"' (x/g) _ yf:

X

x^ ~ a2

ax

X CSC"' - + a In (a: + y/x^ — a^

X cac-^ a In (x + \/^

a:* _, a: , aV^^TT^ 2 a 2

a:^ , a: aylx"^ — a^ — CSC - ^ —i

.3 a 2

■a^]

IT < sec-' - < IT 2 a

0 < csc-i- < ^

a 2

-^ < cse-i - < 0

1 a

0 < csc-i - < ^ a 2

n < CSC-' - < 0 2 a

14.500 r a;2 CSC-' -da; J a

14.501 { esc Mx/g) ^^

-=- CSC ^ h — -—

3 g 6

x^

" ■ ^In(x + Va:2-g2)

^ + -.n

/ CSC 1 (g/g)

K^ , X axyjx^ — a2 a,3 ^

y"^^"'a 6 -|ln(x + V^^^)

I a

0 < <

-1 * ^ iT

CSC ' - < -

a 2

o < csc-1 - < 0 £■ a

<x

^ (g/a;)3 1.3(g/a;)5 1 • 3 • 5(g/g)7 , \

2*3.3 "^ 2.4.5-5 2-4.6.7.7 *"/

14.502 I £^£lM^rf^

/

csc~' (a;/g) _ Va:^ — o-3^ ga:

0 <

_ g'»+^

a;m + l

J

14.505 I x^t&n-^-dx

C5c-^(xla) yjx"^ — a^ X ax

a:"' + i - _, ic 1 C -^ ,

sm 1 7-rr I dx

m + 1 a m+ IJ ^^2 - x^

—-—- cos ' — b r^ I — — dx

m + 1 a m + ij^/^^--^

x^ + '^ . ^. X a C* K^ + i —- tan I —- I -5——^ da;

m + 1 a m + lt'ai^ + a^

, a; . ;r

CSC-* — < 77

a 2

^ < csc-J- < 0 2 a

14.506 f a;- cot-i ^dx = -^ cot"' ^ + -^ (4^ dx J a m+1 gm+ljx2 + g2

14.507 I z^ sec-i^da:

14.508 j a;"'csc-''-da:

a:"' + ' sec-' (g/g) _ a C x*" dx

./;

m + 1 «t + 1 J y^^zr„2

J z"' da; \/a;2 - a2

a;m + i 3ec-i (a;/i

sec-' (a;/o) _| a

m+1 m +

a;*""*"* CSC-* (xla)

* CSC-* (x/a) , g_

m+1 m +

- r^

'da;

\/a2l^

a-m + i

esc ' (x/g) m + 1

g r a;'" dx m+1 J y^^IT^

0 < sec-'-

0^2

o < sec ' — d a

<

- < V

a

0 <

CSC-' - < 5 a 2

K < cse-1 - < 0 2 a

INDEFINITE INTEGRALS

85

IGRALS INVOLVING

14.509

14.510

J J

xe'^dx = — (a:--a \ a

14.511 r»2««d^ = '—(^2^^+^

14.512 { x-efl^dx = ^:i^ ~ ^ C x^-ie<^ J a o. J

dx

= ^f ^n ^ ■»«""' , n(w-l)x"-2

(-l)"rt! a"

14.513

14.514

14.515

14.516

14.517

14.518

14.519

14.520

14.521

14.522

14.523

14.524

^ ^ 1-1! ^ 2-2! ^ 3-3!

f — d* -^°' + ^ f ^"^ ^,

r rfa; X I

J dx _ * I 1 1

dx

pC^ + qe"'^

—— tan->(-/£e<" aVP7 \ V 9

In

if n = positive integer

. 2ay—pq \e^ + yl—qlv> { ^?,mhxdx = ^'^i'^^iribx-bcoBbx)

r e" cos bx dx = ^(a cos bx + b sin &a;) J a2 + 62

r a:e<" sin bx dx = '^^'"(o sin bx - b cos ba) _ c<"{(o2 - 62) sin 6g - 2ab cos 6g}

J a2 + 62 („2 _,_ 52)2

f xe<^ cos bx dx = ^^'^^'^ '^"^ ^^^ + & s'" &a^) _ e^ICa" - 6^) cos 6x + 2ab sin 6a;} J a2 + 62 (a2 + 62)2

r«-l„xd. = e'^lnx _1 fe^^^ J a a J X

J e-sin" 6a. do: = £!^^in^(„ sinfcx - n6 cos6«) + ^^^^^ J e-sin-2 ft^ d^ j'6-cos«6a:d« = ^^ll^^^^^f^bx ^ nb ^r.hx) ^ "^^^^^^ e'^ co^^-ibx dx

INDEFINITE INTEGRALS

INTEGRALS INVOLVING In a:

14.525 14.526 14.527 14.528 14.529 14.530 14.531 14.532 14.533 14.534 14.535 14.536

14.537 14.538 14.539

dx = a; In X — :b

Jin.

J ^2

x\nx dx — -z-{lnx — ^)

Xaim+i / 1 \

X- Inxdx = ;j^^^p^ (^In X - -^j^j^ j [If m = -l see 14.528.]

J X 2

Jinx j^ _ _lil£ _ 1 x2 X a;

j ln2 a; dx = x In^ a; — 2a; In x + 2a;

J in" X dx _ In" + ^ x

n+ 1

[If n = -l see 14.532.]

f-T^ = In(lnx) J X In X ^ '

r^ = ln(lnx) + lnx + ill^ + il^+ ■■■ Jinx 2-2!3'3!

Jinx 2*2! 3'3!

I In^xdx = xln^ic — w | ln"~ia:!dx

\ x^Xn^xdx = ^"'"^^^""'^ ^ rx'«ln"^ixdx

If m = -1 see 14.531.

j ln(x2 + o2)dx = xln(x2 + a2) - 2x + 2atan-i-

rin(x2-o2)dx = « In(x2-o2) - 2x + o In (|-^

rx"»ln(x2^a^)dx = x-^Mn(x^^a^)„ 2 C ^^^ J ' m + l wi+lJx2±o2

INTEGRALS INVOLVING sinh aa;

14.540

J sin

h ax dx =

cosh ax

14.541 fxsinhaxdx = xcoshox _ sinha£ J a a^

14.542 I x2 sinh ax dx = I 1—g ) cosh ax — s ainh 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

J X 3-3! 5-5!

/sinHox , _ sinh ax , T cosh ax , ._ , , „^„,

—^2— "* - ~— "*" " J — X — f 14.565]

J.

x2 "- a:

dx _ 1 , , , OiB

-r — - In tanh -^r

nnax a 2

dx

sinhax a^] 18 1800

sinh^ ax dx

2a 2

_ X sinh 2ax cosh 2ax x^

dx _ coth ax

J sin

rsinh2oxdx = sinh ax cosh ax x J 2a 2

Jxsi

J sinh2ax ~ ^

r sinh ax sinh px dx = sinh (a + p)x _ sinh (a - p)x ^ 2{a + p) 2(a - p)

For a = ±p see 14.547.

(' sinh ax sin px dx = « cosh ax sin px - p sinh ax cos px •^ a-2 + p2

r sinh ax cos px dx = ° cosh ax cos px + p sinh ax sin px J a2 + p2

2(-l)''(23"-l)gJax)2"+i "^ (2n + l)! "^

dx

In

qe

_ *" + p - Vp2 + q2

+ 9 sinh ax aV^TT^ "' V^e^^ + p + ^/^+^

dx — q cosh ax _i. P

(p + g sinh ax)2

+

h

dx

/

da:

a(p2 + g2)(p + g sinh ax) p^ + q^ J p + q sinh ax

^ tan-i Vq^ - p^ tanh ax

ap\j(^ — p2 P

p2 + g2 smh2 aa;

p + Vp2 — gg tanh ax , 2apv'p^ — g2 \^p — \/p2 — qr2 tanh ax

In

/^

dx

p + \/p2 + g2 tanh ax

2 - g2 sinh2 ax 2apyf^^^^ \p - V^^TT^ tanh ax,

J x*" sinh ax dx = x*" cosh ax _ m T j^_i cogj^a. j^j. [ggg 14.585]

J„- i,„„, j„ _ sinh"^i ax coshax n — 1 f . ,,_„ , sinh" ax ax — I sinh'' 2 ^x dx

TsinJ^^^ = -sinh ax ^ _g_ f £2^ ^^ [See 14.587]

J X" (m —1)3!"- > n — 1 i^ x" 1 L J

/ dx — — cosh ttx _ w — 2 r dx

sinh" ax a(n — 1) ainh"^^ ax n — 1 J sinh""^^^;

/ xdx _ — X cosh ax 1 n — 2 r x d;

sinh^ax a(n —1) sinh"-'ax a2(m—l)(n —2) sinh''~2ax n —ij ainh"-

X dx

INDEFINITE INTEGRALS

INTEGRALS INVOLVING cosh aa;

14.562 14.563 14.564 14.565 14.566 14.567 14.568

14.569 14.570 14.571 14.572 14.573 14.574

14.575 14.576 14.577 14.578 14.579 14.580

14.581

14.582

i_ J _ sinh ax cosh ax ax =

X cosh ax dx —

a X sinh ax

cosh ax

x^ cosh ax dx — — cosh ax

2x cosh ax

+ ( — + ^ ) sinh ax

X

cosh ax

x^

d, = in^ + i£^+4«^+i^ +

2*2! dx = -SS^l^ +

4-4! ' 6*6! sinh ax

"J^

dx [See 14.543]

cosh ax a

xdx _ 1 J(ag)

cosh ax

(gg)^ , 5(ax)'>

8 "^ 144 "^

, „ , X , sinh ax cosh ax cosh^ ax dx = -r- + ^

, „ , x2 a; sinh2ax

X cosh^ ax dx = -r + :

4 4a

cosh 2ag 8a2

dx

tanh ax

cosh^ ox

J

J

X

/ J / J J J / J

/, a sinh ax sin px — p co sh ax cospx cosh ax sin px dx = 2 , n

I cosh ax cos px dx =

/ J / J J

J (cosh ax — 1)'^

/ dx .

p + 17 cosh ax

/ dx {p + Q cosh ax)2

(-l)"B„(ax)2« + 2 "^ (2n + 2){27i)! ''■

t. J _ sinh a-p)x . 3mh(o+p)a!

cosh ax cosh px dx = —-rr-^ ^^ 1 n- , „■■ —

2(a —p) 2(a + p)

_ a sinh ax cos px + p cosh ox sin px

a2 + p2

dx

1 . , ax = — tanh-Tp a 2

1 ., ax

= coth -^

a 2

X , ax 2 , , ax

= - tanh ^ - -5 In cosh -5-

a 2 a^ 6

= 2^*^"*^T~6^'^"*' 2 ^ J-coth^-icoth3^

2a

6a

2 ^ _, Qe'^ + P

In

ge'

or + p — '\Jj^ — q^

q sinh ax P r_

a(q2 - p2)(p + g cosh ax) q^ _ p2 J p

dx

+ g cosh ax

INDEFINITE INTEGRALS

89

14.583

J P'-

dx

14.584

14.585 14.586 14.587 14.588 14.589

h

(p- cosh^ ax

dx

In

/ p tanh gg + Vp^ - q^ \ Vp tanh ax — Vp^ ~ 9^/

2 + ^2 cosh^ ax

2apy/f^ — q2 \^p tanh ax — Vp^ ~ 9^ —1 . p tanh ax

rn-n^J

1

V^23^

p tanh ax + Vp^ + g^ 2apVp^ + ^ \p tanh ax - y/p^ + q^.

In

1

Vp2 +

tan-1

j x"" cosh ax dx =

J

ap

x"" sinh ax

p tanh ox \/p2 + g2

_ m /*

cosh-ax dx = cosh"-'axsinhax ^

"-^J

x"-' ainhox dx [See 14.557]

cosh" ^2 da; dx Tcoshox^ ^ -coshax _a_ Tsinhox ^ [See 14.559]

f dx _ sinh ax n — 2 T da

cosh" ax a(n—1) cosh""'ax m— 1 J cosh"

dx

/

cosh" ax X dx

:.flf r.

^',. t.

X sinh ax

+

cosh" ax a{n — 1) cosh""' ox (n — l)(n — 2)a2 cosh'»~2 ^x

n-2 r xdx n — 1 J c03h"~2 ax

INTEGRALS INVOLVING sinh ax AND cosh aa;

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 ax cosh ax dx =

sinh^ ax 2a

J. . V J _ cosh (p + tf)x , cosh (p — g)x

ainh px cosh qx dx = — -/^^ ^ + —57^^=—p-2(p + g) 2(p - g)

C sinh" ox cosh ox dx = 8mh" + 'ax ,jj ^ ^ _j see 14.615.]

. . n LD J _ sinh 4ax x

sinh^ ox cosh^ ax dx = —-rr — -z

32o o

dx

= - In tanh ax

sinh ax cosh ax

^^ = -^Uin-^Binhax-S^h^

sinh^ ax cosh ox a a

r cosh" ax sinh ox dx = cosh" + Ux [if n = -1, see 14.604.]

J (n + l)o ^

J

J sii

J

/ CO si

J

dx

sinh ax cosh^ ax dx

sinh^ ax cosh^ ax

55^*1^ + ilntanh^ o a /

_ 2 coth 2ax a

sinh^ax , sinh ax 1 . _, ._. ._

— : dx = tan ' smh ax

cosh ax a a

=.25h!°£dx = ^2^^^^ + I In tanh ^ sinh ax o a ,2

dx

cosh ax (1 + sinh ax)

1 j„ /I + sinh ax\ + 1 ^^^_, ^^ Za V cosh ax I a

I'.' ■

J. '

INDEFINITE INTEGRALS

14.602 14.603

J si

dx __ J^ ]j^ tanYi — + ^

sinh ax (cosh ax + 1) 2a 2 2a(cosh aa; + 1)

dx

14.604 14.605 14.606 14.607 14.608 14.609 14.610 14.611

14.612 14.613 14.614

J

1_ . ^^ _L a£. _ 1

sinh ax (cosh ax — 1) 2a 2 2a(cosh aa: — 1)

INTEGRALS INVOLVING tanh ax

lanh ax dx = — In cosh ax a

tanh2 ax dx = x —

tanh aa; a

( t^nh^axdx = ilncoshax - ^^"^'^^

2a

f tanh" ax sech2 oa; da: = *^"^''V.°'' ./ (n 4- l)a

X

J

sech^ ax J _ 1 ■ . .

; aa: = - In tanh aa;

tanh ax a

—. = - In sinh ax

tanh ax a

J

r tanh

J

* V . 1 J(ax)^ ((ia:)= , 2(ax)7

tanhaxdx = ^^1^-^+^

2 3

xtanh^axdx = ^ - ?-l5n»L«^ + Xin cosh ax

(-l)n-122«(22n - l)g„(gx)2n + l (27t+l)!

+

ax , (ax)3 , 2(ax)''

-dx = ax--y-+^5-

dx

px

p + qf tanh ox P^ — 9^ a(p2 — q^)

(-l)"->22'"(22n-l)B„(ax)2''-i (2n-l)(2Ti)!

In (q sinh ax + p cosh ax)

+

14.615 14.616 14.617 14.618 14.619 14.620

r tanh" ax dx = tanh" ^ ax ^ T ^^^y^n-z ax dx J a(n-l) J

^^^^^^ ^NTlolBn^NVOmN^otra

/ J

coth ax dx = - In sinh ax a

coth^ ax dx = x —

coth ax

Jcoth»axdx = ilnsinhox - ^°*^^''''=

2a

rcoth«axc8ch8oxdx = -S^^^lll^ J (n + l)a

csch^ ax . _ 1 , ..

— dx = In coth ax

coth ox a

—^— = - In cosh ax coth ax a

INDEFINITE INTEGRALS

91

14.621 I X coth ax dx

14.622 C X coth2aa;dx

\\ax + ^^-^^^

(_l)n-122nBJax)2" + l

225

(2n + 1)!

+

_ X X coth ax

+ —5 In sinh ax

14.623

14.624

{2n-l)(2TO)!

C coth ax ^^ ^ _ ± + ^ _ (gg)^ .... (-l)"22"B„(ga;)2''-i J X ax 3 135 "* (2n-lV2-n.il

+

+ g coth aa; 14.625 rcoth''a«dx =

p2-qz - a(p2 - ^a) ^" (^ »'"*» az + g cosh ax) coth"~* ax

ain — 1}

J

+ I coth" •'^ ax dx

^^^^^^^^^^^^tAtS mvOLVfNG sechaa:

14.626 I sech ax dx = — tan~' e°^ J a

14.627 fsechaaxdx = ^"^°' J a

14.628 Tsech^axdx = sechax tanhax ^ 1 ^_ ./ 2a 2a

14.629 fsech^oxtanhazda; = _ sech^L"^ •^ na

14.630 f—S^^— - sinh gg J sech az a

14.631 fxsechaxdx = A W _ W^. , S(ax)6 (-l)«g„(axpn.2 ^ a2 [ 2 8 ^ 144 ^ (27i + 2)(2m)!

+

14.632 r X sech2 axdx = ^^"^"^ - ^ in cosh ax

14.633 fsechoa ^^ ^ ^^^ _ (^)2 Siax)-* _ 61(ax)« (-l)"g^(ax)2"

J SB 4 96 4320 "^ 2n(2K)! "'"

14.634 r ^ ^ ^_P f dx fS*.pi4KRii

J 9 + P sech ax g Q J p + q cosh ax ^ ^ ^^'^^^^

14.635 fsech" axdx = sechn-2 ax tanh ax ^ ^^ r ^^^^„,, ^^ •-' a(n — li ■» — II

INTEGRALS INVOLVING

14.636

14.637

csch ax dx = - In tanh — a 2

csch^ ax dx = —

coth ax

14.638 r cschs azdx =^ - csch ax coth ax _ ^ i ^^ , ax J 2a 2a 2

14.639 f csch" ox coth ax dx = - ^s^^" '^^ f' na

INDEFINITE INTEGRALS

14.640

J

'^^ = -cosh arc

csch ax

,4.*4, ..csch»x^ = -l„._<^+?^ +

14.642 14.643 14.644 14.645

X csch2 axdx = ~ £cothax ^ 1 j^^j^j^ ^^

/ /

J csch Q

r

J g + P

_1^n/')2n-l_

+

2(-l)''(2

l)B„(aa;)2"+i

(2m +1)!

+

oar 6 "^ 1080

= 5. _ p r

*7 9 J

da;

csch ax Q 9 -' p + g sinh aa;

— csch" ~ 2 ax coth aa;

(-l)"2{22'«-i - l)fi„(ax)2«-i (2«-l){2n)!

[See 14.553]

+

csch" ax dx =

a(n-l)

_ «-2 r

n - 1 J

csch" ~ 2 ax dx

InTEoBS? INVoVviN^mVERSEHYPERBOUcflulmlOl

14.646 14.647 14.648

14.649

/sinh~^ — a

da: =

X sinh~' vxM-a2

a

X sinh~* — dx a

f x2 sinh-» - d3B

t+ti-''-*t-

a: scVk^ + a*

= — sinh"' —h

X ^ (2a2 - a:2) Vx^T^

J

sinh~' (a;/o) j^ *

? _ <g/'>)^ , 1' 3(x/a)5 _ 1' 3 ■ 5{x/ay , ... 2'3'3 2*4'5'5 2*4'6'7*7

a

ln2{2z/a) 2

(g/x)^ , 1' 3(a/x)< 2-2-2 2*4'4-4

1 ■ 3 ■ 5(g/g)» 2*4-6-6'6

+

ln2 (-2g/a) (a/x)a _ 1 ■ 3(g/a!)< 1 ■ 3 ' 5(a/x)« _

2-2-2 2'4'4'4 "*" 2-4-6'"6'6

14.650 fsinh-Mx/g)^^ ^ _

14.651

14.652

14.653 14.654

14.655 14.656 14.657 14.658

J

;

X

J

sinh~' (z/a) _ 11_

a+ \Ax2 + o2

cosh 1 - da; = a

X cosh"^ — dx a

flc2 cosh""' -dx a

X a V X

xcoBh-'(x/o) - Va;2-o2, cosh-Ma:/a) > 0

r co3h-'(x/a) ^^ + if cosh-' (x/a) >

[ X cosh-'(a:/a) + -^x^ - a^, cosh"'(x/a) < 0

ri(2x2-a2) cosh-'(x/a) - ^xVar^ - o2, coah-'(x/a) > 0 [i(2x2-a2) cosh-'(x/a) + ^xy/x^ - a^ , cosh"'(x/a) < 0 r^x3 cosh-'(x/a) - ^(x2 + 2a2) yfx^ - aZ . cosh-Hx/a) > 0 1^x3 cosh-'(x/a) + ^(x8 + 2a2) ^x^ - a^. cosh"'(x/a) < 0

= ^[|ln2(2x/a) + i^^+4:4i^ + UA4<^ +

2.2-2 0, - if cosh-'(x/a) < 0

2.4-4-4 2-4.6-6-6

\x\<a x> a X < —a

cosh~' (x/a) ^ _ _

tanh-' -dx = a

r cosh-'

J J J

_ cosh-'(x/a) _ 1, / a + Vx2+aA h if cosh-'(x/a) > 0, " '" ' ' + if cosh-' (x/a) < 0]

X tanh-i - dx

a

xtonh-'- + ^In(a2-x2)

^ + i(x2-o2)tanh-»f

x^ tanh-' — dx a

= f+ ?'-''-| + f-(-"^-^")

INDEFINITE INTEGRALS

14.659

14.660

14.661

14.662

tanh-' (x/a)

dx =

— ^ a

+ f^/°)^ , (x/a)^

tanh-i jx/a)

J /

J coth-i ^dx = z coth-1 X + I In (ar2 - a2)

X ^ 2a " \ a2 - a;2

a:^

coth-i - dx =

14.663 f x2 coth-i ^ da;

14.664 poth-Mar/g) ^^

14.665 f coth-i (ar/g) ^^ J x^

'' a

_ ax^ , x^ .,, X , a^

32

52

+

14.666

14.667

14.668

14.669

/

sech-i -dx =

/^

sech-i^dx a

f

sech-i (x/a)

dx =

csch-i -dx =

a

.-1 X

14.670 ( X csch -1 - rfa; = J a

14.671

r csch~^ (g,

/g)

dx

14.672 C xrn ainh-^ ^ dx J a

_ cQth'(ar/tt) , 1 ,

a; sech-i (a;/g) + o sin-'(«/«), sech->(3;/a) > 0 a: sech^Mx/g) - a sm~^ [x/a), sech->(ar/g) < 0

J^9;2sech-Ma;/a) - laVa^ - x^, sech-Hx/a) > 0 [ ^x2 sech-i (z/o) + ^gVa2 - ^2, sech-Ma;/a) < 0

^ 2.2-2 2.4-4M

i In {a/x) In (4a/x) + i^^^ + ^ ' ^(x/g)^ ,

I." 2*2-2 ^ 2M'4-4

« cach-i ^ ± a sinh-i | [+ if a: > 0, - if x < OJ

T "^^"^^ ^ - ~^ [+ if X > 0. - if X < 0]

' ^ In (x/g) In (4g/x) + li^l^ _ 1' 3(x/a)^ ^ _ _ 2'2'2 2»4'4'4

^ 2-2.2 ^ 2-4-4-4

_g ^ (g/x)3 _ 1 ■ 3(a/g)5

a; 2*3'3 2'4"5'5 "

- »"''*"^ _. a; J__ r a;m + i

seeh-i (x/a) > 0 sech-i(a;/a) < 0

0 < X < a -g < X < 0

1x1 > a

sinh~i

g m

14.673 j x"* cosh-* - dx =

14.674 I I-" tanh-' ^ dx = •^ g

.4.675 J

m + 1

—- cosh-i

m + 1 a

^I^V^^^^

m

hi

a2

X>n + l

m

x"* coth-i — dx a

14.676 r ajwi sech-' - dx = -^ » ^:;^sech-^-^r^

14.677 C x'"csch-i^dx = -Ell^csch-i^ ± —^^ r_£ •^ a m+1 am+ljr-

+ 1 ^ a^ ~ x^ dx

cosh-i (x/a) > 0 cosh*-i (x/a) < 0

m + 1 m+1 a m + 1J ^^

X

+ 7n

Vo^ - x^

'dx

\/g2 - x2

'dx

\/x2 + a2

sech -1 (x/a) > 0 sech-'(x/o) < 0

[+ if X > 0, - if X < 0]

DEFINITE INTEGRALS

DEFINITION OF A DEFINITE INTEGRAL

Let f(x) be defined in an interval a ^ x S b. Divide the interval into n equal parts of length Aa; = (b — a)/n. Then the definite integral of /(*) between x — a and x = b ia defined as

15.1 f fix)dx = lim {f{a)Ax + f{a + ^x)^x + /(a 4-2Aa;) Ax + ■•■ + /(a + (n - 1) Aa;) Aaj}

The limit will certainly exist if f{x) is piecewise continuous.

If fix) = -raix), then by the fundamental theorem of the integral calculus the above definite integral

ax

can be evaluated by using the result

15.2 J f{x)dx = j i^s{.x)dx =

g{x)

= 9{h) - g{a)

If the interval is infinite or if fix) 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 C f{x)dx = lim f fix)dx

15.4 f fix)dx = lim r f{x)dx

Xb y*b-e

fix) dx = lim 1 fix) dx if 6 is a singular point

15.6 I fix) dx = lim I fix) dx if a is a singular point

OENf^UFBRMfftA^NVofvTN^WBmTinwTfl

15.7 15.8 15.9

^b pi} fh fb

J (fix) ± gix) ± k{x) ± • • •> da: = J fix) dx ± J gix) da; ± J h{x) dx ± ■ ■ •

I c fix) da; = c j f{x) dx where c is any constant

f /(a;)da; = 0

15.10 r /(a;)da: = - C fix)dx

15.n f fix)dx = rf{x)dx+ C fix)dx

a a c

f(x) dx = (b — a) fie) where c is between a and 6

This is called the meayi value theorem for definite integrals and is valid if f{x) is continuous in a^ x^b.

DEFINITE INTEGRALS

96

15.13

J /(x) g(x) dx ^ /(c) J g{x) dx where c is between a and 6

This is a generalization of 15.12 and is valid if f{x) and g{x) are continuous in a-^ x ^ h and p(a;) ^ 0.

15.14

lEIBNITZ'S RUIE FOR DIFFERENTIATION OF INTEGRALS

*'*,ttt> »'*i(a) "" «« aa

^PSOXlWATrft^RMlilAlSoR DEFINITE^»^^^

In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x^, «!. X2, ..., a:„_i, a:„ = 6 and we let y^ =^ fix^), y^ = /(a;,), y^ = /(arg), ..., y„ = /(ic„), /t = (6 - a)/n.

Rectangular formula

15.

r''

IS J /(x) da: « A(yo + yi + y2+ •■■ + y„-i)

Trapezoidal formula

15.16

j;

fix) dx

^iVsi + 2j/, + 2i/2 + • • • + 2|/„_i + ?/„)

Simpson's formula (or parabolic formula) for n even

15.17

J f{x)dx « -g-d/o+ 43/1+ 23/2 +4^3+-■■ +2i/„_2 + 4y„_i + !/„)

IN6 RATIONAL OR IRRATIONAL EXPRCSSIOI

15.18

15.19

15.20

15.21

15.22

15.23

15.24

15.25

X"

dx

x2 + a2

2a

X

sinpiT-

J„ 1 +

J'" x'^dx _ „ a;" + a" " n si

0 < p < 1

sin [(m + IV/n] '

0 < m+1 < n

x"^dx

jr sinmff

+ 2ar cos (i + x^ sin m-rr sin fi

dx

»^o y/cfi - x^ 2

Va2 - a;2 d-c -

I 3;"'(a" - a:")Pda: = j ^*' /

I (^"

x'^dx

vV[{m+\)/n-\- p + 1]

(-l)r-I^ttm + l-nrr[(m + l)/n]

+ a")' nsin[(m + l)Wn](r-l)!r[(7n+l)/m - r + 1]

0 < mi + 1 < Mr

DEFINITE INTEGRALS

DEFINITE INTEGRALS INVOLVING TRIGONOMETR

All letters are considered positive unless otherwise indicated.

0 m,n integers and m ^^ n

m

S ''"

15.26 I sin Tnx sinnx dx =

0 1 jr/2 m, n integers and m = n

0 mi, M integers and m ¥= n 15.27 I cos mx cos nx dx == -s

0 7r/2 m, n integers and m = n

f

15.28 I ainmxcoanxdx —

0 m,n integers and m + n odd

2m/(m'^ ~ n^) m, n integers and m + n even

15.29 I sin2 xdx = i cos^ x dx ^ ^

0

sinS"" x dx

15.31 r sm2^+^xdx = P cosZ'^+ixdx = ^„'^_'^": ^^ . , m = l,2,...

0 ''o

sin^P-ixcos^-i-ixdx = J^P^ ^<'^^ 0 2r(p+(/)

15.33 r ?mp^d« = ^

cos2"'a:da: = ^'■^'.^'." ^'!!~ ^ ^ . m = 1, 2, ...

2 • 4 • 6 ■ • ■ 2mi 2

1 • 3 • 5 ■ ■ ■ 2m + 1

15.32

B-/2 p > 0 0 p = 0 -5r/2 p < 0

15.34

/:

sm px cos qx

x

dx —

I

15.35 f sin pa: sin ga;^^ ^

a^J

0 p > g > 0

:r/2 0 < p < qf jr/4 p = g > 0

7rp/2 0 < p ^ g 7rg/2 p ^ Q > 0

15

sin^ px ^^ _ £p

- cos px .^ _ ffp

x^

da; = ^

.36 r° 15.37 ri

15.38 r"cospx-cosqa; ^ = i„ £

J, « p

15.39 r" cos px - cos qx ^^ _ Trig - p)

V

15.40 CSB^^dx = ^e--J„ x2 + „2 2a

15.41

X°° X sinm x2 + a:

mx ,„ _ 7r„_

dx - -^e-"*"

X x(x2 + a2)

- ^d-^-""")

15.43

15.44

15.45

dx

+ 6 sin X

n a

dx

'o a + 6 cos X

.ff/2

n a

dx

+ 6 cos X yo2 — 62

15.46 r ' ^^ = C Ja; ^ 2r-a J^ (a + 6 sin x)^ J^ {a + b cos a;)2 (a^ - 62)3/2

15.47 T" ^ ^ 2:r 0<a<l

15.48 f g sin a; da; ^ \ (v/a) \n (1 + a) \a\ < 1 J„ l-2acosx + a^ | . In (1 + 1/a) |a| > 1

,5.49 r cos.»xd. ^ _^a™^ ^,^ m-0.1,2..

^Q 1 — 2a cos a; + a2 1 — a - , ■

15.50 J sinoK^da; - J cos 0^2 da; = \'\[¥'

15.51 I sin aa:" da: = _1^ r(l/M) sin-^, n > 1

15.52 I cosaa:" dx = —Vr l'1/") cos —, n > 1 J^ »mi/" 2t[

15.53 r^inx^^ ^ fsSlSdx = .fl

15.54 r^inx^ ^ „ , ^ :" _, 0<p<l

J(, a:P 2r{p) sin {pn-/2) '^

15.55 r^^^dar = ——J!——-, 0<p<l Jg x» 2r(p) cos (pW2) ^

15.56 I sin ax^ cos26a; da; — — ■\l-^ ( cos sin —

J(, 2 \ 2a \^ o a

15.57 I cosaa!^ cos26a: dx = 7: A /tt" I cos (- sin —

J^ 2 Y 2ay a a

"■58 J^ _^d. = -

15.59 r^il^dx = ^ Jo ^' 3

15.60 r^Hl^rf^ = i J,. X 2

15.61

0

. 1"

dx

.Q . + tan"'x 4

,5.63 /;^H^.. = J,-i+l-i+..

^dx - I In 2

15.64 C ^iill^

15.65 p l-cosx ^^ _ rcosx^^ ^

Jo a; J^ X

15.66 I ( T—I— ■= — cos X 1 — = y

15.67 f^tan-ipx-tan-igx ^^ = £ ]„ £ J„ X 29

DEFINITE INTEGRALS

DEFINITE INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS

15.68 15.69

15.70 15.71 15.72 15.73 15.74

15.75 15.76 15.77 15.78 15.79 15.80

15.81 15.82

15.83

15.84

15.85 15.86

Jg-ar cOs6x dz = 0

e-" sin bx dx =

X X

e-"^'COS6a: da; = i../?e-b« 0 2 \ a

a2+ (,2

&

a2 + 62

e-°-sin6x ^^ = tan-i^ X a

dx = In -

-b'/4a

r"e-""' + f^ + -'da: = i - f^efb^-iacJ/to ^^ _6_ where erfc (p) = — | e-^ dx

C x'^e-'^dx = ^<^+l>

X'

*-4ac}/4a

_^ , _ r[(7n + i)/2]

x^e""* dx =

l'^-'"'*"^'^ =\y[l

gatm + i'/z

/■

xdx _ X-L-L-l-L + JLj-

e' - 1 12 "^ 22 "^ 32 "^ 42 "^

6

For even n this can be summed in terms of Bernoulli numbers [see pages 108-109 and 114-115].

r

xdx ^ 1 I . J^ _ A .

e^ + 1 12 22 32 42

12

For some positive integer values of n the series can be summed [see pages 108-109 and 114-115].

r

sinmx ^^ ^ 1 oth^- J-

g27rl _ X

2 2»i

DEFINITE INTEGRALS

15.87

15.88

/:

r^

X secpx 2 \a^ + p^

dx = tan~i tan~i -

CSC px p V

15.89 r"^-°^a-co5a^)rf^ ^ cot-la- fin (a^+l)

ITE INTEGRALS INVOLVING LOGARITHMIC FUNCTIOI

15.90 { x^(\nx)^dx = , ^ ,^^"'!' w>-1, n^0,l,2,...

If 717^0,1,2,... replace m! by r(n+l). 15.91 17

da: - ^2

+ X

15.92

x 6

J, 1-:

15.93 C'JLiLt^dx = 4 J, 0. 12

15.94 I'lnO^

dx = --

15.95 C lnx\n(l + x)dx = 2-21n2-^

15.96 r lnxln(l-.-'ft

x)dz = 2 - J

15.97

15.98

15.99

pxP_nnx^^ = -irZcBCpjrCOtpTT 0<p<l

— dx — In —r

In X n + 1

I e-'\nx

dx = —y

V^

15.100 r e-^Mnxdx = -■^{7 + 2 1n2)

dx = -T-4

dx = -5 In 2

15.102 I In sina; dx = I In cosx

(ln8inx)2dx = I (lneosx)2dir = |(ln2)2+|^ 15.104 r xlnsinxdx = "T^^^

15.105

XTT/2 sin a; In si

sin X dx ~ In 2 — 1

J»27T /^2Tr

In (a + 6 sin x) dx = I In (a + 6 cos x) dx = 2o- In (a + V^^^) (1 •'n

DEFINITE INTEGRALS

15.107 yin(a+bco^x)dx = ^ln(^£±v5!E^

15.108 j In (o2 - 2a6 C9S a: + 62) da: =

2n- In a, o ^ 6 > 0 2n- In 6, 6 ^ a > 0

15.109

15.110

15.111

I In (1 + tan x) dx - ^ In 2 Jf, 8

r sec a; In ( f^-^-^^^ ) rfa: ^ A{(cos-i a)2 - (cos-^ 6)8}

2.i„|,,. = _,i||i+?i||£H.EnSa^

See also 15.102.

DEFINITE INTEGRALS INVOLVING HYPERBOLIC FUNCTIONS

15.112

4^1;^ dx = ;^tanh^ Sinn ox 26 26

15.113

X

15.114 f

COS ax , _ TT . oir — ; — ; — «* = :rr seen rrr-cosh hx 26 26

sinh aa: 4a2

15.115

fii

a;" da; _ 2"+i - 1 . . .,. | 1 . 1 1

sinhaa; ~ 2"a«+i ^ I" "^-^M x-i + i "^ 2"+' "^ 3» + » "*

If M is an odd positive integer, the series can be summed [see page 108].

15.116 r^^dx = ^C8c^-f I e^x -1-1 26 6 2a

15.117 r"^h«^da: = f-^cot¥ J ebz _ 1 2a 26 6

ISCELLANEOU5 DEFINITE INTEGRAL

M

,5.,,8 J^/W^AMrf^ ^ (/(O)-/{-)} In ^

This is called Frullani's integral. It holds if fix) is continuous and I '^^' ~ '^'^' dx converges.

15.120 r (a + a;)'"-i{a-a:)''-ida; = (2a)'" + n-ilMi:M ^~„ rim + n)

16.2 16.3

DEFINITION OF THE GAMMA FUNCTION T{n) FOR n>0

r(

n) = ( f-^e-

^ n

*dt « > 0

r(n + l) = nr{n)

r{n + l) = nl if n - 0,1,2, ... where 0! - 1

THE GAMMA FUNCTION FOR n<0

For n < 0 the gamma function can be defined by using 16.2, i.e.

16.4

n

GRAPH OF THE GAMMA FUNCTION

Fig. 16-1

16.5 16.6 16.7

SPECIAL VALUES FOR THE GAMMA FUNCTION

rim+i) = l'3>5.^-^(2n.-l) ^ ^.1.2.3,

(_l)m2m^

r(-» + i) = ^^3^^

(2m - 1)

m- 1,2,3,

101

THE GAMMA FUNCTION

RELATIONSHIPS AMONG GAMMA FUNCTION!

16.8 16.9

r(p)r(i-p) =

sin pn-

22*-* r{x) r{x + ^) = V^r(2x) This is called the duplication formula.

16.10

For m = 2 this reduces to 16.9.

OtHErDteFrMlttONS OF THE GAMMA PUNCtrON

16.11

16.12

^^^+^^ = ji"L (.+iK;V2V---t+fe)

fe*

This is an infinite product representation for the ^amma function where y is Euler's constant.

DERIVATIVES OF THE GAMMA FUNCTION

16.13

:"(!) = j e-* In K da: = -y

16.14

r(x) ^ ll =»/ \2 *+ 1

+ ■■- + ^-

n X + n~1

+

16.15

^NS FOR THE Gi

r(« + l) = ■\/2vXX='e-^ ^1 + r^ + ^

139

12x 288a;2 51,840x3

+

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. 71 > 10] is given by Stirling's formula

16.16 n\ ~ \/2^n«e-''

where — is used to indicate that the ratio of the terms on each side approaches 1 as n-* «>.

16.17

|r(tx)|2 =

X sinh vx

DEFINITION OF THE BETA FUNCTION B{m,n)

17.1

B(m,n) = J (^-^l-t)"-i dt m > 0, n > 0

RELATIONSHIP OF BETA FUNCTION TO GAMMA FUNCTION

17.2 B{'m,n) = iMliZii

r(m + n)

Extensions of B(m, n) to m < 0, n < 0 is provided by using 16.4, page 101.

ME IMPORTANT RESULTS!

17.3

B{m, n) = B{n, m)

17.4

B(m

,n) = 2 I

ir/2

gJn2n"-l « cos^"-! 9 rffl

17.5

B(m

'"^ - X (l+V-^«

d«

17.6

BASIC DIFFERENTIAL EQUATIONS and SOLUTIONS

104

BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS

105

OlFFiRENTIAL EQUATION

18.6

y F{xy) dx + X G{xy) dy — 0

!• ^ Linear, homogeneous second order equation

d^y , dy dx^ dx

a, b are real constants.

^ + a— + 61, = 0

18 8 Linear, nonhomogeneous second order equation

a, 6 are real constants.

SOLUTION

■'(

In a: - f ^-^M^1L_ + „ '""^ " J v{Giv)-Fm^'

where v = xy. If G(v) = F(v), the solution is xy = c.

Let mi,m2 be the roots of m^ + am + b = 0. Then there are 3 cases.

Case 1. m,, m^ real and distinct:

y = Cie*"i* + cge""!*

Caae 2. m,, mg real and equal: " ' '

y = c,e"»i* + C2a:e"'i^

Case 3. rtii = p + gi, m2 — p — qv.

y = eP*(ci cos qx + Cj sin qx)

where p = —o/2, g = \/6 — oV4 .

There are 3 cases corresponding to those of entry 18.7 above.

Case 1.

y = Cie*"!* + Cg^"**

mi — 7112 J

WI2 ~ '"i ^ Case 2.

+ see"!* I e~'"i^fi{x)da:

— e*"!* I xe-'^i'^ R(x) dx Cases.

y = ei*E(cj COS qx + e^ sin qx)

+ ^— I e P- «(a:) cos oa; da;

18.9 Euler or Cauchy equation

S + -f + *. = S(.)

Putting X = e', the equation becomes and can then be solved as in entries 18.7 and 18.8 above.

BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS

i^ '

RITHMETIC SERIE!

19.1 a + ia + d) + {a + 2di + ■■■ + {a+ in-l)d} = ^n{2a + (n - l)d} ^ ^n{a-i-l)

where 1 = a+{n — l)d is the last term.

Some special cases are

19.2 19.3

l + 2 + 3+---+n = |n(n +1) 1 + 3 + 6 + •■• + {2w-l) = m2

19.4

EOMETRIC SERIES^

a + ar + wfi + a-fi + • ■ • + ar"""^ where ? = ar«-' is the last term and r t^ 1.

a(l - r")

1 - r

g — rf 1 -r

If -1 < r < 1, then

19.5

a + ar + ar^ + or^ +

1 -r

19.6 a + (a + d)r + (a + 2d)r2 + •-■ + {a+(n-l)d}rn-i = a(l-r«) ^ rd{l - nr"-i + (n-l)r"}

1 - r (1 - r)2

where r ¥^\.

If -1 < r < 1, then 19-7 a + (o + rf)r + (a + 2d)r2 +

r(f

1 - r ' (1 - r)2

SUMS OF POWERS OF POSITIVE INTEGERS

19.8 IP + 2" + 3P + •■■ + „p = ^^ + inp + ^^P" _

P + 1 2 2!

BipW-i B2P(P - 1)(P - 2)nP-3

4!

+

where the series terminates at n^ or n according as p is odd or even, and B^ are the Bemoullx numbers [see page 114].

107

SERIES OF CONSTANTS

Some special cases are

19.9 1+2 + 3+ ■•■+n = "tn+D

2

19.10 12 + 22 + 32 + ... + n2 = Mn + l)i2n + 1>

6

19.11 13 + 23 + 33 + ■•■ + n3 = nHn + l)i ^ (i+ 2 + 3 + • ■ • +n)2

19.12 14 + 24 + 34 + ... + „4 = «(n + l)(2n + 1)(3«2 + 3n - 1)

30

If Sfc — l** + 2"= + Sif + • • • + n'^ where k and n are positive integers, then

'^ + l\« . . /k+l

k

k + \ 1

19.13 C: ^ iSi + [■■;" )Sa +•••+[-;* )Sk = (n+1)^+1 - (n+l)

19.14 19.15 19.16 19.17

19.18 19.19 19.20 19.21 19.22 19.23 19.24 19.25 19.26

19.27 19.28 19.29 19.30 19.31 19.32

2^3 4^5

3^5 7^9

4^7 10 ^ 13

5^9 13 ^ 17

= ln2

^+i 11 14

l_l+i_

2 5 8 J- + J-+J_

12 ^ 22 32

14 ^ 2^ 3«

1+1 + X 16 ^ 2« 3«

1-1 + 1

12 22 32

1-1 + 1

14 2* 3*

1_1 + 1 1« 28 38

1 + 1 + 1

12 ^ 32 52

1 + 1 + 1

1* 3* 5^

1+1+1 + 1 + 16 ^ ga ^ 5« 76 ^

1_1+1_1+

13 33 53 73

1 + 1_1_1 + 13 ^ 33 53 73 ^

1*3 ^ 3-5 ^ 5-7 ^ 7-9

^ + ^ + -^+^ + 1'3 2-4 ^ 3-5 ^ 4-6

rV| , V2 In <1 + V^)

8 "^ 4

6

90

_£l. 945

i^ 12

7^

720

^ 31irB

30,240

8

96

960

zi

32

3ir2\/2

• ■•-■^C J' -'. .'■•■

1 + 1,+ 1,+ 1,+

16 •• -.7 .

_ 1 2 3

4

12.32 ' 32.52 ' 52*72 ' 72*92

16

SERIES OF CONSTANTS

109

19.33

+

+

12,22.32 22-32-42 32-42.52 1 1

+

47^2 - 39

1».34 i 1 ^ ,

a a + d a + 2d a + 3d

+

16 1 ..«-

+

19.35 J-+J^+J^+J_ +

12p ^ 221' 32P ^ 42p ^

19.36 J-+J_+JL+J_ +

12P ^ 32p ^ 52p ^ 72p ^

19.37 i--J_ + J-_J_ +

l2p 22P S^K 42P

X l + «

22P-l^PBp

(2p)!

{22P - l),r2PBp

2(2^)] (22p-i-i)^PBp

du d

{2p)\

19.38

^ + 1

12P+1 32P+I '^ 52P+I 72P+I

+

;r2p + lSp 22p + 2(2p)J

n

r'^A

MISCELLANEOUS SERIES

19.39 l + COSa + C0S2«+ ■-. + COSTla :3: ^^" ^^ ^ ^^^ 2 2 sin (a/2)

19.40 sin« + 3in2a + 8m3a+ ••• +sinn« = sin [^(n +!)]„ sin^n«

sin (a/2)

19.41 1 + rcosa + r2cos2a + r3 cos 3a +

1 — r cos g X — 2r cos a + 9-2 '

|r| < 1

'• .H

19.42 rsina + r2am2a + r«sin3a + ■•• = r_^Tia ,, .

1 -2rcosa +r2' I'^l ^ ■"

19.43 1 + rcosa + r2cos2a + ••■ + r«coan« = r'*'^^ cos na - r" + i co3(n +l)a - r coaa + 1

1 - 2r cos a + r2

19.44 rsina + 7^sin2a + ■•■ + r^sinTta = ^si""-^""^^ sin(n + l)a + r"+2 8inna

1 — 2r cos a + r^

hit

THE EUIER-MACLAURIN SUMMATION FORMULA

19.45

n-l

fc

2 F{k) = f F(k) dk ~ I {F(0) + F(n)}

+ ^{F'(w)-F'(0)} - tIo^^'"**'*"^'"^*'^

A. i

+

+ ... (-l)p-l_^{i^(2p-I)(„) _p(2p-l)(0)) + ...

19.46

THE POISSON SUMMATION FORMULA

2 F(fc) = 2 11 «27Kn«/r(a.) d:c

{/:

TAYiOR SERIES FOR FUNCTIONS OF ONE VARIABLE

2! (n—1)! "

where R„, the remainder after « terms, is given by either of the following forms: 20.2 Lagrange's form ie„ = /^"H$)(a; - a)"

20.3 Cauchy's form

_ /t">a)(x~{)"-Hx-a)

The value $, which may be different in the two forma, lies between o and z. The result holds if f(x) has continuous derivatives of order n at least.

If lim fi„ = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If

a = 0 the series is often called a Maclaurin aeries. 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

20.4

(a+«)" = g" + na^-^x + "t"^~ ^> an-2a.2 + ^(" " IK"" 2) an-3a;3 +

= a" +

Special cases are

a^-ix +

a«-2a:2 +

20.5 (a+a;)2 = a^ + Zaz + x^

20.6 {o + a;)3 - a^ + Sa^x + Saz^ + x^

20.7 (a+ 3:)^ = a4 + 4a3a: + 6a2x2 + 4aa;3 + ar*

20.8 (l + a:)-i = 1-x + x^-x^ + x*----

20.9 (l + a)-2 = 1 — 2a; + 3a;2 - 4x3 + 5a^ - • ••

20.10 (l + a!)-3 ^ 1 - 3x + 6x2 - 10x3 + 153^ _ ..

20.11 (l+x)-/2 = i_l^+|l|,2- 113^^3 +

20.12 (H-x)»/2 = i+±x-^x^ +

1-3 2'4'6

l-4'7

20.13 (1 + x)— . i_l, + ll|,. ___

X3 - -X3 +

20.14 (l + x)./3 = 1-,!, --!_,. + _il^,3_

(in-3a;3 +

-1 < X < 1

-1 < X < 1

-1 < X < 1

-1 < X ^ 1

-1 < X ^ 1

-1 < X ^ 1

-1 < X ^ 1

110

TAYLOR SERIES

111

20,15 e' = i + x+|J + f^+ ...

= ^n + f + T +

-1< X < 1

20.19 Inx = 2U^^] + ^^^ +lf?^^ll

x+l/ '^ 3[x + lj "'■fltTf' "*"

—« < X < *

20.16 a- = e-in« = 1 + « ]„ a + i^il^ + t^J^L^ + ... -« < ar < «

20.17 ln(l + x) = a;-^ + ^-^+... -Kxfil

23 4 i%x = j.

20.20 ,„. = f^)+|(E^y,|(^l-

+

X > 0

x^i

;eries for trigonometric functions

20.21 sinx

20.22 cos X

20.23 tan z ■■

20.24 cot X --

20.25 sec x

20.26 CSC X =

20.27 sin-Ik

20.28 cos~ix

X^ . X^

2! ^4! 6!

1 _ X _ x^ 2x° X 3 45 945

"^ 2 "^ 24 ■*" 720 "^

X 6 360 15,120 "•■

— 00 < a; < oo

— ~ < X < «=

+ 1 —2 +

(2n)!

22nB„K2n-l

+

(2n)!

+

2(22''-'-l)B„x2''-'

(2n)!

= a: +i^+il3xf 1^3^x7 2 3^ 2"4 5 ^ 2-4*6 7

— jr ._, »■ / ,1x3 1-3x5,

20.29 ten-»x =

r 3+5 7 +

U--i +J L +

^2 X 3x3 5a.5 ^

w<|

0 < |xl< »•

w<t

20.30 cot-'x = ^-tan-ix =

x^ . x=

2-(^^-T + T- ■

P" + x-3^+5^-

20.31 sec-i X

20.32 csc-ix

2 V X 2 • 3x3 ^ 2 • 4 • 5x6

+

sin-> (1/x) = - + ^— + ^'^

X 2 • 3x3 ^ 2 • 4 • 5x5

+

0 < |xl < 0-|x| < 1

\x\ < 1

|x| < 1 [+ if X £ 1, - if X ^ -1]

lx| < 1

[p - 0 if X > 1, p - 1 if X < -1]

|xt > 1 Ixl >1

TAYLOR SERIES

SERIES FOR HYPERBOLIC FUNCTIO

X^ , K"

2t ^ dt ^ fit ^

4!

^"T + ir

6! 315

+

ic 3 45 945

x2 . 5^4 61k6

1 - — +

2 ^ 24

+

± f In 12x1 +

20.39 sinh-ia; =

20.40 cosh-' X = ± -(In (2x) -

20.41 tanh->x

20.42 coth-»iB

720

31x5 15,120

„ _ JEL + 1-3x5 2-3 ^ 2-4.5

1 X 7x^ X 6 "^ 360

+

—'^ < X < *

— OS < X < « (_l)n-122n(22n_l)B^a;2t>-I

(2^0!

(-l)''-122«B„x2n-l

(-1)"£„X2" ■ (2«)I ■*" ■■■

(-l)«2(22n-i - l)5„x2n-i

+

1 • 3 • 5x' 2*4'6-7

1'3

+

2*2x2 1

+

(2n)!

1'3'5

+

2 • 4 • 4x4 ^ 2 • 4 • 6 • 6x8

+

1*3 I l'3-6

2-2x2 2-4'4x* 2-4-6-6xfl

^^f^y^y^

a; 3x3 ^ 5a.5 ^ 73.7

■)} [:;

1.1 <i

0 < 1x1 <

w<i

0 < |x| < a-

lx| < L

"+ if X ^ 1 - if X ^ -

if cosh-i X > 0, if cosh-' X < 0,

|x|<l \x\ > 1

.]

X ^ X

:a

MISCELLANEOUS SERIES

20.43 e»- = i+:c+f-f-^+...

20.44 e'^°^ *

, , x2 a:* 31x8

20.45 efn* = 1 +x + ^+ ^ + 3£l+ ...

^ A O

20.46 e-sinx = x + x2+%^-^-^+ ... + 2:!^iliniW4Ix^^ ...

a oU 90 7(,!

20.47 e-cosx = l + x- ^-^+ ■-- + 2"^^ cos (n./4) x» ^ ...

3 D

n!

on ^« 1 I • I lit ^^ a^ 3:8

20.48 Inlamxl = m |xl - - - — - ^835 "

20.49 inlcosxl = -^ - ^^ - g - ^^^^

22"-iB„x2« w(2m)! "^

2a«-i(22«-l)g„g2" 7i(2ti) !

+

on en 11* 1 1 1 1 . «^ . ^ar* , 62x«

20.50 ln|tanx| = in\x\ + ^ +-^ + ^^

+

22'>(22"-i-l)B„x2"

n(2n)!

20

.51 llLikL£) = jc - (l + ^)x2 + (l + ^ + i)x8 - -■-

X "i X

— <* < X < <«

— *o < X < <»

I r ^ "

W<2

— =° < X < •

—«° < X < »

0 < Ixl <

N<^

0 < kl< I

1x1 < 1

TAYLOR SERIES

113

If 20.52

then 20.53

where 20.54

20.55

20.56

20.57

20.58

20.59

y = Cix + c^x^ + Cgx^ + c^^ + CsxS + c^jje + ... a: = Ciw + Czv" + C^y^ + C^y* + €^y^ + Cei/S + • - • c^C^ = 1

c\C^ — Bcic^e^ — 5<^ — eje4

e^Cs = 6<!?C2C4 + 3cfc2 _ cjcj + 14c^ - 21c,c|c3

cJ^Cfl = Tc^cgCs + 84c,c|c3 + Tcf^^^ _ 28c2c2c2 - c^^c^ - 2Sclelc^ - 42c|

TAYIOR SERIES FOR FUNCTIONS OF TWO VARIABLES

20.60 f(x,y) = f{a.b) + (x-a)fMb) + iy-b)f^{a.b)

+ -^(i^- aW^ia. b) + 2(x - a){y - b)f^(a, b) + {,y- 6)»/„(o, 6)} + • • •

where fjfl., b), fy{a, b), ,.. denote partial derivatives with respect to ar, j/, ... evaluated at x = a, y = b.

DEFINITION OF BERNOULLI NUMBERS

The Bernoulli numbers Bj, B^, B^, ... are defined by the series 21.1 '^

e*- 1

= 1 _ ^ 4_ ^1*' ^2^* ^3^:8 2'^ 2\ "IT "^ ~6r

x\ < 27

21.2

, X ^x Bix^ B2X* BgajB

1 — « cot - = —: 1- — 1 =— +

4! ^ 6!

2!

I»l <^

DEFINITION OF EULER NUMBERS

The Elder numbers E^,E^E^, ... are defined by the series 21.3 s«.hx = 1-^ + ^.^ +

2!

4!

6!

W<|

21.4

secx

= 1+^ + ^+^ +

21

4!

6!

W<2

BERNOULLI AND El

llfiRS

BERNOULLI AND EULER NUMBERS

115

JmBwWiW oi» BtU'lffC'yi'tnwB^^HiHffluwWR

2n+ 1 2

2n+ 1

21.5 ( „ ")22Bi - ^^ 4 "J24B2 + [ g ~ )2«B« -

2n+ 1

(_l)n-i(2n + 1)2^B„ = 2n

21.6

2n f/2n - 1

*'•' ^» = ^.=(1^ |(^'"r V ^»-' - ('"3" 0^"- + ('"5-') ^"-3 - • • ■ (-«-

SERIES INVOLVING BERNOULLI AND EULER NUMBERS

21.8 B^

- (2n)!

l + i + a= +

22n-l^nl 22" 32«

21.9 B„ =

2(2?i)!

1+ 1 + 1 +

(22" - l)n^n ] 32" 52"

21.10 B, =

(2n)! __L+J_

21.11 £;„ =

22n+2(2n)! K 1_ 1

y2n+l 1^ 32n + l "•" 52n + l

21.12

B. ~ in^Hirei-^'^Vrn

FORMULAS from VECTOR ANALYSIS

VECTORS 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| A. The tail end of the arrow is called the initial point while the head is called the terminal point.

or

2.

3.

FUNDAMENTAL DEFINITIONS

1. Equality of vectors. Two vectors are equal if they have the same magnitude and direction. Thus A = B in Fig. 22-1.

Multiplication of a vector by a scalar. If m is any real number

(scalar), then mA 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 in = 0, then mA = 0 ia called the zero or null vector.

B.

Fig. 22-1

Sums of vectors. The sum or resultant of A and B is a vector C = A + B formed by placing the initial point of B on the terminal point of A and joining the initial point of A to the terminal point of B [Pig. 22-2(6)]. This definition is equivalent to the parallelogram law for vector addition as indicated in Fig. 22-2(c). The vector A-B is defined as A + {-B).

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 A, B, C and D.

(a)

D

Fig.22-3

(b)

4. Unit vectors. A unit vector is a vector with unit magnitude. If A la a vector, then a unit vector in

the direction of A is a = A/A where A > 0.

LAWS OF VECTOI

tEBRA

If A, B, C are vectors and m, n are scalars, then 22.1 A + B = B + A Commutative law for addition

22.2 A + (B + C) = (A + B) + C

Associative law for addition

22.3 m(7iA) = {mn)A = n(m.A) Associative law for scalar multiplication

22.4 (m + 7i)A = mA + nA Distributive law

22.5 mi{A + B) =^ »iA + rniB 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 = Ax\ + AJ + A^

where Aii.AaJ.Agk are called ctym'poneni vectors of A in the i, j, k directions and A^.A^, A3 are called the components of A.

DOT OR SCALAR PRODUCT

Fig. 22-4

22.7

A • B = AB cos * 0 ^ S S

where e is the angle between A and B.

FORMULAS FROM VECTOR ANALYSIS

Fundamental results are

22.8 A*B = B-A

22.9 A-(B + C) = A-B + A-C

22.10 A-B = AiB^+ A2BZ + A3B3 where A = Aii +A^ + Agk, 3 = 811 + 823 + 8^^.

Commutative law Distributive law

CROSS OR VECTOR PRODUCT

22.11

A X B = AB sin S u

0^ S Sir

where $ ia the angle between A and B and u is a unit vector perpendicular to the plane of A and B 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 B will advance in the direction of u as in Fig. 22-5].

Fundamental results are

22.12 AXB =

i j k

Ai Az A3 Bi Bj ^3

Fig. 22-5

22.13 AXB = -BXA

22.14 AX(B-I-C) = AXB-I-AXC

22.15 |A X B] = area of parallelogram having sides A and B

ISCELtANEOUS FORMULAS INVOLVING DOT AND CROSS PRODUCl

22.16 A'(BXC) =

Aj A2 A3 81 82 8^ Ci C2 C^

= AjfiaCa + Aa^aCi + As^iCg - AgBgCi - AjjBiCg - AjBaCj

22. 17 IA • (B X C) [ = volume of parallelepiped with sides A, B, C

22.18 AX(BXC) = B(A>C) - C(A'B)

22.19 (A X B) X C = B(A • C) - A(B • C)

22.20 (A X B) • (C X D) = (A • C)(B • D) - (A • D)(B • C)

22.21 (A X B) X (C X D) - C{A • (B X D)} - D{A • (B X C)>

= B{A • (C X D)} - A{B • (C X D))

_

FORMULAS FROM VECTOR ANALYSIS

119

DERIVATIVES OF VECTORS

The derivative of a vector function A(m) = A i(u)i + A2(u)j + A^iu^k of the scalar variable u is given by

„-, dA ,. A(M + AM)-A(tt) (Ml dAa dA^

22.22 -J— = hm = -;—i 4- -;—j + -;—k

du iu-*o Au du du du

Partial derivatives of a vector function A(a;, y, z) are similarly defined. We assume that all derivatives exist unless otherwise specified.

RMULAS INVOLVING OERIVATIV£$

22.23 #(A.B) = A.^+^-B du du du

22.24 -^(AXB) = AX^ + ^XB du du du

22.25 £iA.(BXC)> = ^.(BXC) + A. f XC+A.fBxf

22.26 A.^ = A^ du du

22.27 A ■ -r~ = 0 if [A[ is a constant

The operator del is defined by 22.28

THE DEL OPERATOR

V = l~ H ]T h k-r-

dx dy dz

In the results below we assume that U = U{x, y, z), V = V(x, y, z), A = A(«, y, z) and B = B(a:, y, z) have partial derivatives.

THE GRADIENT

22.29 Gradient of U - grad U -

VU = {\^ + \±.+ k^)U '3a: * by dz '

dU. ^ dU. _L 3C7. 5a; dy dz

THE DIVERGENCE

22.30 Divergence of A = divA = V-A =

's+'^ + ''i)-'^'' + ^^ + -*^'"

SAi dAz dAs dx dy dz

FORMULAS FROM VECTOR ANALYSIS

THE CURL

22.31 Curl of A = curl A = V X A

*l; + i^ + »'i)=^<^^*-^^^ + ^3k)

i j k

-1 A A

5a; dy dz

dy

dz

dA2\ fdAi SAgN fdAz dAi

THE LAPLACIAN

22.32 LaplacianofU = V^C/ = V'[VU) = ^+^ + ^

22.33 LaplacianofA = v^A = 0 + 0 + ^

THE BIHARMONIC OPERATE

22.34 Biharmonic operator on C7 = ^*U = V^V^U)

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 V-(t/A) = (VC/)'A+ [/(V'A)

22.39 V X {UA) = iVU)XA + U(V X A)

22.40 V'(AXB) = B'(VXA) - A-(VXB)

22.41 Vx(AXB) = (B"V)A - B(V-A) - (A*V)B + A(V'B)

22.42 V(A-B) = (B*V)A + (A*V)B + BX(VX A) + AX(VXB)

22.43 V X (V IT) = 0, i.e. the curl of the gradient of t7 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

If A(m) = ^^^(w)' *^C" t*^6 indefinite integral of A{u) is

22.46 I A(m) du = B(it) + c c ::= constant vector The definite integral of A(w) from u = a to u = & in this case is given by

22.47 r Aiu)du = B(6) - B(a) The definite integral can be defined as on page 94.

IE INTI

Consider a space curve C joining two points Pi{ai,a2,a^) and ^2(^1. &2. ''3) as in Fig. 22-6. Divide the curve into n parts by points of subdivision {xi.y^.Zi), .. .,(a:„_i,y„_],z^_i). Then the line integral of a vector A(a:, y, z) along C is defined as

22.48 f A-dr = r 'A-dr = lim 1 \{x„,y^,z^)'^r^

where Atp - AKpi + A?/p j + Az^k, AXp = a!p+1 - aip, Ai/p = i/p + j - ^p, A«p = Zp+i — Zp and where it is assumed that as n-» «= the largest of the magnitudes |Arp| approaches zero. The result 22.48 is a generalization of the ordinary definite integral [page 94].

The line integral 22.48 can also be written

22.49 I A-dr = \ Aydx + A^dy + A^dz using A = All + Agj + Agk and dt — dxi + dyj + dzk.

Fig. 22-6

PROPERTIES OF IINE INTEGRALS

22.50

22.51

A • dr = - J A • dr

'-'1 P|

rPi r^s /-pj

I A • dr = ) A • dr + I A • dr

-'p, *^Pj "^Pa

NDEPENDENCE OF THE PATH

In general a line integral has a value which depends on the particular path C joining points Pj and Fj in a region %. However, in case A = V^ or V X A = 0 where <f> and its partial derivatives are continuous in %, the line integral I A • dr is independent of the path. In such case

FORMULAS FROM VECTOR ANALYSIS

where ^(Pi) and 0(^2) denote the values of 0 at Pi and Pa respectively. In particular if C is a closed curve,

X

£

22.53 I A'dr = * A-rfr ^ 0

c *^r

where the circle on the integral sign is used to emphasize that C 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 z and y axes as indicated. Let AAp = Aajp i|/p denote an area of one of these parts. Then the integral of Fix, y) over % is defined as

22.54 r F{x,y)dA =^ lim "2 F(Xp,yj,) AA,

Jio n-*« P=l

provided this limit exists.

In such case the integral can also be written as

I F(«, y) dy dx

i F{x,y)dy\dx

x=a i'^v=f,(x) J

where y = fi(x) and y - fzix) 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

Fig. 22-7

22.56

I f(a:,ff)dicdi/ ^ I 1 I FKX,y)dx\dy

« = C ^1 = 01(1*) •-'u = C L''i = gi(!() J

where x = g^iy), x = ffziy) are the equations of curves HPG and HQG respectively and c and d are the y coordinates of H 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

areaiSp, p=l,2,...,n. Let A(a;p, yp, Zp) = Ap where (Xp, j/p, 2p) Let Np be a unit normal to ASp at P.

is a point P in iSp.

the surface integral of the normal component of A

defined as

Then over S is

22.57

X

A-tidS =

lim 2 Ap

n-*iB p=l

N„AS„

Fig. 22-8

FORMULAS FROM VECTOR ANALYSIS

128

RELATION BETWEEN SURFACE AND DOUBLE INTEGRALS

If ^ is the projection of S on the xy plane, then [see Fig. 22-8] 22.58

Jo J J N-k

%

THE DIVERGENCE THEOREM

Let £r 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 I VA dV = I A'dS

The result is also called Gauss' theorem or Green's theorem.

f V • A dF = r

•-'it ^ e

J

Fig. 22-9

Fig. 22-10

STOKE'S THEOREM

Let S be an open two-sided surface bounded by a closed non-intersecting curve C [simple closed curve] as in Fig. 22-10. Then

22.60 4) A • dr = I {V X A) • dS

where the circle on the integral is used to emphasize that C is closed.

£ A'dr = r {V X A)

GREEN'S THEOREM IN THE PLANE

22.61

£ Pdx + Qdy = C [^ -^\ dxdy

J^ Jj, \3x dyj

where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem or Stoke's theorem.

FORMULAS FROM VECTOR ANALYSIS

22.62

GREEN'S FIRST IDENTITY

J {^W + (v^)-(v^)>dv = ^{^v^)

dS

where p and ^ are scalar functions.

GREEN'S SECOND IDENTITY

22.63

f (0VV - <pV^)dV = f (0V^ - vV0)-dS

22.64

X

V X A dV =

= 1

(fSX A

22.65 f ^dr = J dSXV0

CURVILINSA,R COORDINATES

A point P in space [see Fig. 22-11] can be located by rectangular coordinates {x, y, z) or curvilinear coordinates (Mj, M2» "s) where the transformation equations from one set of coordinates to the other are given by

22.66 X = x{ui,U2,tts)

y = I/{Mi,«2»"8)

If % and Ug are constant, then as u^ varies, the position vector i = xi + yj + zk of F describes a curve called the «i coordinate curve. Similarly we define the 2*2 ^^d ^3 coordinate curves through P. The vectors dr/auj, dt/dv^ dr/du^ represent tangent vectors to the U], 112, U3 coordinate curves. Letting •i.«2. ©3 he unit tangent vectors to these curves, we have

Fig. 22-11

22.67

where 22.68

di . dr

dUi

h, =

dr

are called scale factors. If ene^jCg are mutually perpendicular, the curvilinear coordinate system is called orthogonaL

FORMULAS FROM VECTOR ANALYSIS

1S6

IbRMULAS INVOLVING ORTHOGONAL CURVILINEAR COORDINATES

22.69 (£r — —^ dui + -— du2 + T— d«3 = hi dui e, + hz du^ t^ + h.^ du^ 63

OU| 0M2 "^Z

d«2 = dT'di - h{dul + hldu\ + Aj**"!

22.70

where da is the element of arc length. If dV is the element of volume, then

22.71 dV = I (/t^ei dui) • {^262 ''"2) ^ (^^3*3 ''"a) I — '^i^'^s d«i d«2 dwa

where 22.72

SMi fltt2 flws

3(a:»!/,2)

dwi du2 dug =

3(a:, 1/, z)

a{Wi,U2,W3)

duj dz£2 dug

3(Ui, 1*2. "3)

is called the Jaeobian of the transformation.

Sx/dUi dx/dU2 dx/du^ dy/dUi dy/dii2 dy/du^ dz/dui dz/dll2 dz/du^

TRANSFORMATION OF MUlTIPlE INTEGRALS

The result 22.72 can be used to transform multiple integrals from rectangular to curvilinear coordinates. For example, we have

22.73 (((F{x.y,z)dxdydz = rff G(u,.W2.«3) J^'^'^'^W duiduzdu^

JJJ J J J 3(Mi,M2,M3)

where ^C' is the region into which "3^ is mapped by the transformation and G(ui,W2»W3) ^^ ^h^ value of F(», y, z) corresponding to the transformation.

0RADIENT/DIVER6ENCE, CURL ATID LAPLACIAN

In the following, * is a scalar function and A = A,«i + Agej + AgCs a vector function of orUiogonal curvilinear coordinates Wi, «2> "a-

22 JA Gradient of * = grad * = V* = T^T^+T-|^ + r-T^

22.75 Divergence of A = div A = V-A = i-^ U^daMi) + :S7-(Mi^2) + t?-(M2^8)1

"in-afta L^**i °^2 <'**3 J

hi^i ^262 ^3*3

22.76 Curl of A = curl A = VxA = T-fr- — ~ -^

A1/1.2A3 dUi d«2 o«3

hiAi h2A.2 h^A.^

22.77 Laplacianof* = V^* = ^^^^ ^_ ^_ — j + — (^-^ — j + — (^-^ —^J Note that the biharmonic operator V** = V^iy^) can be obtained from 22.77.

FORMULAS FROM VECTOR ANALYSIS

SPECIAL ORTHOGONAL COORDINATE SYSTEMS

Cylindrical Coordinates (r, 6, z) [See Fig. 22-12]

22.78 22.79

X = rcosff, y = r sin ff, z = z

22.80

V24. ^ ^+1^1+1^+^ 3r2 ^ r Sr ^ r2 5tf2 ^ dz^

Fig. 22-12. Cylindrical coordinates.

Fig. 22-13. Spherical coordinates.

Spherical Coordinates {r,e,^) [See Fig. 22-13]

22.81 22.82

22.83

r sin e cos 0, y = r sin 9 sin <p, z = r cos e hi = 1, k2 = r2, fcg = r2 sin2 9

V24. =

r^ dr\ drJ r^ sin fl 3*

1 324,

Parabolic Cylindrical Coordinates {u,v,z)

22.84

22.85

22.86

— ^{u^ — v^), y = uv, z = z

fe| = u2 + r2, fts = 1

+

32* dz2

The traces of the coordinate surfaces on the zy plane are shown in Fig. 22-14. They are confocal parabolas with a common axis.

Fig. 22-14

FORMULAS PROM VECTOR ANALYSIS

127

Paraboloidal Coordinates {u,v,<i>)

22.87

where 22.88

22.89

X = uv COS 0, J/ — Ml? sin 0, z =■ ^(u^ — v^) u ^ 0, u g 0, 0 g 0 < 2jr

V2* =

^ ^M^i +

1 ALi*u 1-3^

«(w2 + i;2) Am I 5m y v(w2 + v2) flu I 5v / "'^'^ 9*^

Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 22-14 about the X axis which is then relabeled the 2 axis.

Elliptic Cylindrical Coordinates {u,v,z)

22.90

where 22.91

22.92

X = a cosh u cos V, y = a sinh u sin v, z =^ z w ^ 0, 0 ^ V < 27r, —CO < z < » h\ = hi = a2(sinh2M + sin2i;), hi = X

__ 1 /a^ a^\ a^

~ a2(sinh2ii + sin^r) \Su^ &v^) dz^

The traces of the coordinate surfaces on the xy plane are shown in Fig. 22-15. They are con-focal ellipses and hyperbolas.

Vff

u - 2

*1^

u = 1.

Fig. 22-15. Elliptic cylindrical coordinates.

FORMULAS FROM VECTOR ANALYSIS

Prolate Spheroidal Coordinates {$,■>], <i>)

22.93

where 22.94

22.95

X = a sinh i sin t/ cos 0, y = a sinh £ sin 1; sin ^s, z — a cosh £ cos tj

i ^ 0, 0 ^ V ^ ^, O^0<27r

kj=zhl = a2{sinh2f + sin2,j), hi - a2 sinh2 { sin* ij

V2* =

1 5_ / . u,5*

a2(sinh2 ^ + sin^ 7?) sinh $ 5£ T'" a^

+

1

a2(sinh2 { + sin^ i?) gin ij di?

-— sin); ^— 1 +

32*

dv J a2 sinh2 j sin2 i? S4>^

Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 22-15 about the X axis which is relabeled the 2 axis. The third set of coordinate surfaces consists of planes passing through this axis.

Oblate Spheroidal Coordinates i$,ri,4>)

22.96 X — a cosh { cos n cos <p, y = a cosh f cos tj sin <(,, z = a sinh i sin -q

where € - 0, -7r/2 ^ ij S ;r/2, 0 ^ 0 < 2;r

22.97

22.98

V2* =

hi = hi = a2(sinh2£ + sin2)7), /i| - a^ cosh^ £ cos2,

a2(3inh2 £ + sin2 t)) cosh f 5£ l di

+

a2(sinh2 ^ + sin2 ij) cos 1? dr)

-— cos IJ -— +

32*

dv / a^ C03h2 5 cos2 ,, 302

Two sets of coordinate surfaces are obtained by revolving the curves of Fig, 22-15 about the y axis which is relabeled the z axis. The third set of coordinate surfaces are planes passing through this axis.

22.99

where

or

22.100

22.101 22.102

Bipolar Coordinates {u,v,z)

a sinh v

y =

a sin u

cosh V — cos u' " cosh v — cos u ' 0 ^ It < 2jr, —w < V < 09, —« < 2 < »

a;2 + (j/ — a cot w)2 = o2 csc2 u, (x — a coth ^1)2 + j/2 = *(j2 csch^ v, z = z

h-i — h2 —

(cosh V — cos u)2

, /t3 - 1

^2* = (cosh V - COS k)2 /3^ 32*\ 3^

a2 13u2 3^2; "^ 322

The traces of the coordinate surfaces on the xy plane are shown in Pig. 22-16 below.

FORMULAS FROM VECTOR ANALYSIS

129

.e*

(a, 0) or tf =

<\«*

Cv^^

Fig. 22-16. Bipolar coordinates.

Toroidal Coordinates {u,v,^)

22.103

_ a sinh v cos 0 _ a sinh t> sin 0 ^ _

cosh V — cos u cosh V — cos u

a sin u

cosh V — cos w

22.104

' ^ (cosh u — cos u)2 ' ^ (cosh v — cos u)^

,2 a^ sinh^ r . A3 =

22

.105 V2* = (cosh u-cos w)^ d

5*

3u V cosh V — cos u du

(cosh 1; — cos m)3 _a_ / sinh v a*\ , (cosh u — cos u)^ 3f* a^ sinh v Bv\ cosh v — cos u dv J (^ sinh^ v 3^2

The coordinate surfaces are obtained by revolving the curves of Fig. 22-16 about the y axis which is relabeled the z axis.

Conical Coordinates (A, p., v)

22.106

(^2 _ b2)(v2 - &2) 62- a2

22.107

.2 _ , .2 _ \^{i.^ - V^) .2 _ X'(ji

X2(jf2 _ ^)

(^ - a2)(^2 - 62)

FORMULAS PROM VECTOR ANALYSIS

Confocal Ellipsoidal Coordinates (A,^, v)

22.108

+

+

a2-\ ' 62 - X ' c2 - X

3;='

+

yz

+

a^ — /i b^ — fi c^ — fi x^ y2 z2

- 1 X < c2 < 62 < a2

= 1 c2<^<62<a2

= 1 c2 < 62 < P < a2

22.109

fa = (g2-x)(a2-^)(aa-.) (a2 - 62)(a2 - c2)

2 = (62-X)(62-^)(fr2_,) (62-a2){62-c2)

^2 = (C2-X)(c2-^)(C2-.) (c2-a2)(c2_62)

22.110

hi =

(ti - xX" - X)

^2

A3

4(o2-x)(62-X)(c2-X)

(v - fi){\ - /i)

4(a2 - ^)(62 - /x)(c2 - n)

(X - !■)(;. - y)

4(a2-^)(62-^)(c2-^)

Confocal Paraboloidal Coordinates {k,(i,v)

22.111

= z - \ — «'<X<62

- z - fi 62</i<a2

a2 - p 62 -

= Z — V

a^ < p < ^

or

22.112

^ ^ (tl2-X)(g2-;,)(a2-,)

yi =

62-a2

(62-X){62-^)(62-p) a2- 62

X + /I + F - a2 - 62

22.113

A? =

1.2 /12

fe^ =

16(o2 - p)(62 - y)

DEFINITION OF A FOURIER SEtlES

The Fourier series corresponding to a function /(x) defined in the interval c ^ x ^ c + 2L where e and L > 0 are constants, is defined as

23.1

where

23.2

do , S / nnx , , rtirX

y + 2 (a„cos-y- + 6„sin-^

.C + 2L

nirx J coB—=—dx

r 1 /•cfai-

If f(x) and /'(a;) are piecewise continuous and fix) is defined by periodic extension of period 2L, i.e. f(x + 2L) = fix), then the series converges to fix) if a; is a point of continuity and to ^{fix + 0) + fix - 0)} if a; is a point of discontinuity.

COMPLEX FORM OF FOURIER SERIES

Assuming that the series 23.1 converges to f(x), we have

23.3 fix) = 2 c„e'"'^^/^

n = —»

where

23.4 ''•' " xj fix)e-^'^^"^dx = V

^K-i6„) «>0

23.5

tSEVAL'S IDENTITY

lf'ifix))^ax = ^f^^lK^^l)

GENERALIZED PARSEVAL IDENTITY

.C + 2L

23.6 j^ J fix) gix) dx = -g- + ^2 (anC« + K^n)

c

where a„,6„ and c„,d„ are the Fourier coefficients corresponding to fix) and gix) respectively.

FOURIER SERIES

iFtheTrgW^phs

4/sin a; . sin 3a; , sin 5a: ,

23.8 fix) = 1x1 =

X 0 < X < TT

—X —JT < X < 0

£ _ A / cosx , cosSx , cos5x ,

~2t

fix)

Fig.23-1

-2ir —n-

Fig. 23-2

277-

23.9 f(x) =X, -ir<X<Tr

2 I sinx _ sin 2a; , Bin 3x

Fig. 23-3

23.10 f{x) = X, 0 < X < 2,7

— 2 I ^'" "^ 4- sin2x , sin3x ,

23.1 1 /(x) = Isin xl, —TT < X < n-

2 4 /cos 2x , cos4x , cos 6x

TT TT V 1-3 3*5 5-7

-2v —IT

Fig. 23-5

FOURIER SERIES

133

23.12 f(x) =

sin a; 0 < a: < -0 TT < X <2Tr

1 1 . 2/cos 2a: , cos 4a; , cos 6a; ,

- + - sm a; - - (-yT^ +-gTg-+-gT,-+

23.13 f(x) =

cos a; 0 < a; < :r — cos X —77 < X < 0

8 / sin 2x 2 sin 4x 3 sin 6a; 1-3 "^ 3-5 ^ 5-7

23.14 fix) =a;2, -b-< a; < tt

^ _ / cos a; _ cos2g: cos Sa; 3 \ 12 22 "^ 32

23.15 /{a;) = a;(ff-x), 0 < a; < tt

e1 _ Aqs 2a; cos4g , cos 6a; , T ~ ^ 12 22 32

23.16 /(a;) - a;{ir-a;){Tr + a;), -jt < a; < tt

, „ , sin z sin 2a; sin 3g 1-^' ~^s~ 23 33

y^

-2r —Jf

/(a:)

r^ ^

TT 27r

Fig. 23-6

Fig.23-9

Fig.23-10

FOURIER SERIES

23.17 nx)

0 0 < a: < „-a

1 Tr — a<X<7r + a

0 r- + a<X<2n-

a 2/sin a COS a; sin 2a cos 2a;

1 2

, sin 3tt cos 3a:

--)

Pig. 23-11

23.18 f(_x) =

x{ir — X) 0 < X < r-

—a;(jr — a;) — tt < a; < 0

8 /sin a; sin 3a: sin 5a; ,

13

33

53

Fig.23-12

MISCELLANEOUS FOURIER SERIES

23.19 f{x) = ain^, —v<x<v, /i # integer

2 sin /i^ / sinx _ 2 sin 2x 3 sin 3x 12 _ ^2 22 - ^2 32 - ^2

23.20 /(«) = zoBitx, —Tr<x<-tr, y.¥= integer

2^ sin P.-W I 1 , cos a: _ cos 2a; cos 3a:

2^2 12-^2 22 - /i2 ^ 32 _ ^2

23.21 /(a:) = tan-i[(aBinx)/(l - acosx)], -»■ < z < ir, |a| < 1

a sin X + -^ sin 2x + — sin 3x + • • •

23.22 /(x) " ln(l - 2acosx + a2), _^ < 3; < ^^ |a| < 1

—2 ( a cos X + — COS 2x + -^ cos 3x +

23.23 /(x) = \ tan-> [(2a sin x)/(l - a2)], -n- < x < ir, |a| < 1

a sin X + "o" sin 3x + -3- sin 5x +

23.24 f(x) = |tan-'[(2acosa;)/(l-a2)], -ir < x <-^r, \a\ < 1

a cos X — -T- cos 3x + -=- cos 5x — S 0

23.25 fix) = e>^, -^ < x < w

2 sinh /tiT ( \_ , ■^ (—l)"(/t cos nx — n sin na:)

23.26 f{x) = sinh /lar, -b- < a; < n-

2 sinh /in- / sinx _ 2 ain 2x 3 sin Sa:

12 + ^2 22 + u2 "^ 32 + u2

23.27 /(x) = cosh iix, -tt < x < n

2fi sinh fiTT /_1 cos a: cos 2g _ cos 3g ,

TT I 2/x2 12 -|_ ^2 "^ 22 + /i2 32 + ^2

23.28 /(x) - In jsin^x], 0 < a; < n-

-(1„2+ c^+C0|2£^C0|3x^

23.29 fix] = ln|cos^K|, -w < x < v

_ f 1 2 — <^osg COS 2a: _ cos 3x ,

23.30 fix) = ^5r2 - ^irx + ^2:2, 0 ^ a: ^ 2)r

cos a: cos 2a: , cos3g , 12 "*■ 22 32

23.31 /(x) ^ ^X(X - n){x - 2;r), 0 g X S 2;r

sinx sin2x ainSx ,

13 ■•" 23 33

23.32 fix) = ^^^ - ^w^^ + ^7rx3 - ix4, 0 ^ X S 2.

cos a; cos2x cos3x , 14 "*■ 24 3^

^^^^^^^^^^^^^^^^ BESSEl'S DIFFERENTIAl EQUATION

24.1 x^y" + xy' + {x^-nZ)y = 0 n^O

Solutions of this equation are called Bessel functions of order n.

BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n

24.2

24.3

24.4

Jn(x) =

X"

1 -

+

2«r(n + l)l 2(2m + 2) 2 • 4{2n + 2)(2n + 4)

^ "I (-l)''(a;/2)" + 2fe

J-nM = T^z:

1 -

+

X^

2-n r(i _ „) Y 2(2 - 2n) 2 • 4(2 - 2n)(4 - 2n)

^ (-l)fc(g/2)2fe-" k=o /c! r(fc + 1 - ?i)

J_„(x) = i-l)^J„(x) n = 0,1,2....

If n#0,l,2,..., y„(x) and J_„(x) are linearly independent.

If 717^0,1,2,..., J„(a:) is bounded at x =^ 0 while ./_„(x) is unbounded.

For n = 0,1 we have

24.5 Jo{x) — 1 — 22 + 22. 42 ~ 2^ * 4^ * 6^ + "''

_. , - , . _ » _ g^ , ac^ x[

24.0 Ji(a;) - ^ 22 • 4 "*" 22 • 42 • 6 22 • 42 ■ 6^ • 8

24.7 /o'C'^) = -M'^)

BESSEL FUNCTIOI

^KIND OF ORDER n

24.8

i'nla:) =

/„(«;) coswn- — ./-n(a:)

sinTtTT

Jp(x) cospn- - ./-p(a:) lim -i-

M9^0,l,2, ..

n = 0,1,2, ..

This is also called Weber's function or Neumann's function [aUo denoted by iV„(a;)].

BESSEL FUNCTIONS

137

For n = 0,1,2, ..., L'Hospital's rule yields

24.9 Y^ix) = -{ln{x/2) + y}J„{x) -- '^ in-k-l)\ix/2)^k-n

T IT fc=o

--2 (-iH-i>(fe) + f(n+fc)} 7;'^^

^ fc=o kl{n + k)\

where y = .5772156.,. is Euler's constant [page 1] and

24.10 «i.(p) = 1+ 1+1+ ••• +-, <i.(0) = 0 ■ For n = 0,

24.11 Y,{x) = |on(x/2) + r}Jo(x) + f|f-^(l + i) + 2^{l + i + i)

24.12 r_„(x) = (-l)''Y„(x) n = 0,1,2,... For any value ti £ 0, J nix) is bounded at a: = 0 while 3'„(a;) is unbounded.

24.13 y = AJJx) + BJ.^ix)

24.14 y = AJJx) + BY„[x)

n-^ 0,1,2, all n

/ax where A and B are arbitrary constants.

GENERATING FUNCTION FOR 3n{z)

24.16

gX(t

-i/t)/2 = 2 J„(a;)(«

24.17 24.18 24.19 24.20

RECURRENCE FORMULAS FOR BESSEL FUNCTIONS

■'n + l(«) = —•'«(«) - •^n-l(a;)

a;J4{a!) ^ a;/„_j(x) — nJ„{a;) a;J^(a;) = nJ'„(ir) - xJ^^-^{x) d

dx

24.21

24.22

The functions Y„{x) satisfy identical relations.

{x»J„(a:)) = os»J^-i{x)

^{x-«J^{x)} = -x-r'J^^.ix)

BESSEL FUNCTIONS

BESSEt FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER

In this case the functions are expressible in terms of sines and cosines.

24.23 Ju2(x) = \-^sin!t

2 .

24.26 J_,Mx) = -W-^

2 /cos X

WX \ X

+ sin a:

24.24 J_,/2(a:) = V— cos*

vx

24.27 75/2(3

— 1 1 sin X — — cos X

X

24.25 J,„(x) = ylJ(^^-cosx^ 24.28 J.,„M - ^/^jf sinx + (|-1

For further results use the recurrence formula. Results for yi/2(«). Y^/si^), ... are obtained from 24.8

HANKEL FUNCTIONS OF FIRST AND SECOND KINDS OF ORDER n

24.29 Hl,'\x) - J,ix) + iY,{x)

24.30 Hl^'ix) - J«(x) -ir„(x)

BESSEL'S MODIFIED DIFFERENTIAL EQUATION

24.31 . x^" + xy' - {x^ + n^)v = 0 n^O

Solutions of this equation are called modified Bessel functions of order n.

lED BESSEL FUNCTIONS OF Tl

24.32 /„(») = i-^'JJix) = e-^^!^J^(ix)

JT"

1 +

+

xi

2"r(« + l)r 2(2?i + 2) 2 • 4{2n + 2)(2n + 4)

KIND OF ORD1

+ -■ - 2

24.33

/_„(x) = i"J-„(ix) = e^^'^J.^iix)

a;-n

1 +

+

X4

+

2-«ra-n)\ 2{2-2n) 2*4{2-2n)(4-2n)

24.34 I-^{x) = 7„(x) n = 0,l,2,...

If n-/■ 0,1,2, ..., then/„(«) and/_„(a!:) are linearly independent.

For n = 0,1, we have

^ 4. ^ A »°

22 "^ 22>42 "*" 22'42. 62

24.35 /o(x) = 1 + 22 + 22712 + 02.d2.,- +

•9A '«A r / ^ — £ J. "^ 4. g° , x'' ,

^*».-*o 7i(a;j - 2 "*" 22.4 22-42'6 2^ • 42 • 62 • 8

{a:/2)

N + 2h

ft=ofc!r(n + Jfc+l)

= 2

(a:/2)

2fc-«

fc=o k\T{k + l-n)

24.37 /^(x) - /i(x)

BESSEL FUNCTIONS

139

MODIFIED BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n

24.38 K„(x) =

—^— {/_„(x) - /„{x)} « ^ 0,1,2,...

2 sin njT

Urn

{/_„(x)-/»} n = 0,1,2,...

p^n 2 sin Pit '' ^

For n = 0,1,2, ..., L'Hospitars rule yields 24.39 KJ,x) = (-1)"+ Hln (a:/2) + 7>/„(a;) + J- 2 ("DM^ - A: - 1)! (a:/2)2k-«

«-i 2 2

+

2 k^ofc!(« + ft)!

{*{fc) + ^(n + fc)}

where *(p) is given by 24.10. For n = 0,

24.40 ifoCx) = -{In (x/2) + y}/o(«) + ^ + 3^742(1 + *> + gg.^g.ga (1 + i + i) + "' *

24.41 K-J?c) = K^{x) 71 = 0,1,2,...

X*

GENERAL SOLUTION OF BESSEL'S MODIFIED EQUATION

24.42

24.43

24.44

where A and B are arbitrary constants.

GENERATING FUNCTION FOR /n(x)

24.45

ei(t + i/«/2 = 2 /„(a:)t"

RECURRENCE FORMULAS FOR MODIFIED BESSEL FUNCTIONS

0-,

24.46 /,+ ,(x) = I„-i(x) ~—I„(x)

24.47 lUx) = ^{/„„,(x) + /„+i(x)}

24.48 x/;(a!) = xl^-iix) - nl„(x)

24.49 xl'„(x) = xl„^i{x) + nl„(x)

24.50 ^{x^Ux)} - x"/«_i(a;)

277

24.52 K^+.ix) = A-„_i(x) + —K«(ar)

24.53 <(a;) = ^iK,-lix) + K^^t(x)}

24.54 a:<(x) = -xK^.^ix) - nK^ix)

24.55 xK;(x) = ni:„(x) - xK„+i(«)

24.56 ^{x"£:„(x)} = -x-iC„_i(a:)

24.51 ^{a:-n/,(x)} = x-«7„ + i(x)

24.57 ^{x-"iC„{x)} = -x-"K,+ ,(x)

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 /i/2(x) = -\|—sinha;

24.59 /_i/2(a:) = A/4co8ha:

24.61 /_3,,(x) = ^(^sinh.-^

24.62 /5/2(a:) - a/— "1(4+0 sinhx - - cosh a;

24.60 /3/2(x) - A/-±-fcoshx-

sinha;

•\ / — 1 ( ~9 + 1 1 co3^ ^ sinh X >

24.63 /_5/2(x) = ^/^■{(^+11 cosh:

For further results use the recurrence formula 24.46. Results for K^/^i.^), K^/2{x), ... are obtained from 24.38.

Ber AND Bei FUNCTIONS

The real and imaginary parts of Jjt (aJe3»ri/4) are denoted by Ber„ (a;) and Bei„ (x) where OAAA x» / ^ -V (a;/2p + « (3n + 2fe)rr

24.65

If « = 0, 24.66

24.67

r, ■ / \ X (a;/2)2fc + " . (3n + 2fc):r

Ber(a:) - 1 " ^72" +-4T2 •'

Bei(x) = (a:/2)2 - -i-g^ + ^,^

Ker AND Kei FUNCTIONS

The real and imaginary parts of e~'^'^^^ KJ^{xe^''^) are denoted by Ker„ (a;) and Kei„ (x) where

24.68

Ker„ (a:) = -{In (a;/2) + y} Ber„ (x) + ^^ Bei„ (a:)

(k - fc - 1)! (^/2)2fc-" (3n + 2A:);r

r-, cos ;;

k\ 4

1 S^ (a;/2)«+2fc

(3n + 2kW 4

24.69

Kei„ (a:) = -{In (x/2) + y) Bei„ {x) - J,r Ber„ (x)

n-l

(TO-A:-l)!(a;/2p-« . (Sti + 2fc)7r

. {3n + 2fc)7r

_^ 1 ^ a;/2« + 2ic , ^, _^, „ . (3n +

and * is given by 24.10, page 137. If 71 3:: 0,

(a;/2)''

(a;/2)8

24.70 Ker (x) = -{In (x/2) + y} Ber (x) 4- jBei (x) + 1 - -^^{1 + ^) + ^^(1 + ^ + ^ + J) -

24.71 Kei (X) = -{In (x/2) + y} Bei (x) - ^ Ber (x) + (x/2)2 - ^^ d + ^ + i) +

BESSEL FUNCTIONS

141

DIFFEI

"fOR Befe^fKer, Kei FUNCTfONS

24.72

kV + xy' ~~ {ix^ + v?)y = 0

The general solution of this equation is 24.73 y = A{Ber„ (a;) + i Bei„ (x)} + B{Ker„ [x) + i Kei„ (a:)}

GRAPHS OF BESSEL rUNCTIONS

Fig. 24-1

Fig.24-2

1—a;

Fig. 24-3

Fig. 24-4

Beia

Kei X

Fig. 24-5

Fig. 24-6

BESSEL FUNCTIONS

INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS

24.74 C xJQ(x)dx = zJi{x)

24.75 C x^Joix) dx = x^Jiix) + xJo(x) - f Joix) dx

24.76 Cx-^jQ{x)dx = x'^Jiix) + (m-l)x'"-^ Jo(x) - (m-l)2 f x'»-2 ^(a;)

dx

M^)

Mx)

)dx

24.77

24 78 f ^ -

J„(x)

dx =

Jy{x)

Joix)

-1 (?n-l)2j

Jai'^)

dx

24.79 r^i(a;)da = -Jo{x)

24.80 ra:yi(a:)da; = -xJo(x)+('jQix)dx

24.81 ra;'"Ji(a:)da: = -x«/„ (*) + *» f ^"""^ *'o(») (^a^

24.82 r^^dx = -J, (x) + J Jo (a') da:

y,(x)

24.83 f-'-l^dx ^ -Jl^^^^r-^dx J x^ 7nx"*~^ m./ a;*" *

24.84 fx^Jn-iix)dx = x^Jnix)

24.85 (x-^J„+i(x)dx = -x-^J„{x) C x'^J„{x)dx = -x'^J„-i(x) + (m + n-1) J z"""'J„_i(x) d«

24.86

24.87

C xJn{ax)Jn(px) dx =

x{a JJPX) Kiax) - /3 J^(aX) J^M)

)S2~ a2

^2 / «2

24.88 Ja;/2(«x)dx = |-V;(««)}2 + y (^1 " ^j(-'n('«=^)P

The above results also hold if we replace J„{x) by Y„{x) or, more generally, A y„(x) + B y„(a!) where

A and £ are constants.

DEFINITE INTEGRAiS INVOLVING BESSEL FUNCTIONS

24.89 Ce-'"Mbx)dx =

24.90 C e-'"J^{bx)dx =

24.91 I cos ax Jo(bx) dx

Va2 + &2

(Vgg + 62 - a)n

6"Va2 + 62

1

Va2-62 0

n > -1

a> 6 a< 6

BESSEL FUNCTIONS

143

24.92 C J„ibz)dx =1 n>~l

24.93 J^ ^^^^ = i n = l,2,3,...

24.94 r e-'^jQ{by/^)dx = ?lU^ Jq a

24.95 I X Jn(ax) J^ifix) dx =

i82-a2

24.96 C xJliax)dx = ^U;(«))2 + ^{l-n2/«2){Jn(«)P J a:yo(o3;)/o(j3fl:) (ix = ^^^

24.97

24.98 yo(a:)

24.99 ^.(x)

24.100 ^(x)

24.101 Yo(x)

24.102 /o{:c)

^2

INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS = - I cos {x sin 9) d9

= — I cos (n* — ac ain tf) d*, n = integer

= _ I COB (a; giQ g\ cos^ tf dtf, Tt > —4

2 r"

= I COS (a: cosh w) du

- I cosh {x sin e) d$ = ~ \ e* ^me d*

ttJ,, 2:rJ„

ASYMPTOTIC EXPANSIONS

24.103 J„(rr)

24.104 Y,{x)

24.105 J,(:r)

24.106 Y^{x)

24.107 /„(»)

24.108 £:„(ie)

2 / na- _

COS ar —s" ~ T jra; \ 2 4

2 . / nr- jr

Sin I X ~

TTX \ 2

'2s^ \2.nJ

2_! ex n-n V 2n

V2^

where z is large

where x is large

where n is large

where n is large where x is large where x is large

BESSEL FUNCTIONS

ORTHOGONAL SERIES OF BESSEL FUNCTIONS

Let Xi, \2>M,--- be the positive roots of R J„(x) + SxJ^(x) = 0, n> -1. Then the following aeries expansions hold under the conditions indicated.

S ~0, R^'O, i.e. Ai, A2, As,... are positive roots of Jn(x) = 0

24.109

where 24.110

fix) = A^J^iXix) + A2Jni\2^) + A^J^iX^x) + 2 C^

In particular if n. = 0,

where 24.112

2 r'

^fc ^ 72777 I a;/(a;)^o(M)d:a'

RIS > -«

24.113

where 24.114

/(jr) = A^J^{\^x) + A2J„(X2a;) + A3J„(X3a:) +

A^ =

J */(a:).^n('^fca;)dx

In particular if m = 0, where

R/S = -n

24.117

where

24.118

24.119

where

24.120

fix) = AqX» + AiJ^{\ix) + AzJniH'^) + ••• Aq = 2(»+l) r x^+^fix)dx

"^^ " j''(x)-j ^xw —;r; J <^m'^niK^)dx

^ •'n(Xfc) — J„_i(Xfc) J„ + i{Xk) -'o

In particular if n = 0 so that i2 = 0 [i.e. Xi, Xg, X3, ... are the positive roots of ^i (a;) = 0], fix) = Ao + AiJoiXiX) + AzJoiXzx) + ••■

Ao = 2J xf(x)

dx

A^ =

4iw

J X fix) J^iX^x) dz

BESSEL FUNCTIONS

145

R/S < -n

In this case there are two pure imaginary roota —iKq as well as the positive roots \i, \2> H' and we have

24.121

where

24.122

fix) = Ao/„{Xox) + AjJ„(Xia:) + AsJniM^) + •"

j X fix) J„(\kX) dx

A. =

Jni\k) - Jn-liK)Jn-niW ^0

M^^UANEOUS^^^^

24.123 cos (a; sin«) = Jo{x) 4- 2Ji{x)cos2e + 2Jt{x)cos4e + ■•■

24.124 sin{a;sinfl) = 2Ji(x)ame + 2J3(x)sin3ff + 2J5(x)sin5tf + ■••

24.125 Jni^ + y) = 2 Ji,ix)Jr,-k{y) n = 0, ±1,^2,...

fc= -"

This is called the addition formula for Bessel functions.

24.126 1 = Jo{x) + 2J2{x) + ■■• + 2/2«(«) + ■"

24.127 X = 2{Mx) + ZJsix) + 6J5(x) + •■■ + (2n+l)J2„+i(x) + ••■)

24.128 a:2 = 2{4J2ix) + ieJ^(x) + 36Je(a:) + •■■ + (2n)2/2„(a;) + •••} X JI (x)

24.129

= J^ix) - 2Mx) + SJ^ix) - ••■

24.130 1 = 4(x) + 2Jlix) + 2JI(x) + 2j|(ar) + •••

24.131 J'^ix) = i{J„-2(x) - 2J„{x) + ^„ + 2(«)>

24.132 J'^'ix) = iU„-8(«) - 3J„_x(ac) + 3J„ + j(x) - Jn + 3(a:)} Formulas 24.131 and 24.132 can be generalized.

24.133 K{x)j^^ix)-JL,JM = ^''""'

vx

24.134 y„(x)J_„ + i(x) + y-„(a:)J„_i(x) = ^ "" "'^

24.135 J„^,{x)Y„{x) - JAx)Y,^tix) = -

24.136 sin a; = 2{J,(x) - J^ix) + ^{x) - •■•}

24.137 cosx = Jo(x) - 2J2(a=) + 2^4(2:) - ••■

24. 1 38 sinh x =^ 2{/,(x) + h{x) + h{x) + • ■ ■}

24.139 cosh x = /©(x) + 2{/2(x) + I^ix) + h(x) + • • •>

LEGENDRE'S DIFFERENTIAL EQUATION

25.1 (i^j^2)y" _ 2xy' + n(n + l)y = 0

Solutions of this equation are called Legendre functions of order n.

LEGENDRE POLYNOMIALS

If m — 0,1, 2, ..., solutions of 25.1 are Legendre polynomials Pn(a:) given by Rodrigue'a formula

25.2

^'^(=^> = 24!£^(^^-^>"

^

25.3 P^ix)

25.4 P,{x)

25.5 P^{x)

25.6 P^ix)

^ 1

1(3x2-1) ^(5x3 - 3x)

SPECIAL LEGENDRE POLYNOMIAL

25.7 P^{x) = ^(35x^-30x2+3)

25.8 P^ix) = ^(63x5 - 70x3 + 15a;)

25.9 Ps(x) = Jg{231a:8-315x-» + 105x2-5)

25.10 Pj(x) = J^(429x7-693x5 +315x3-35x)

LEGENDRE POLYNOMIALS IN TERMS Of ^ WHER!

25.11 Fo(cos#)

25.12 P,(cos(?)

25.13 /»2(costf)

25.17 Pe(coss)

25.18 PyCcoss)

1 25.14 P3(cos«) = ^(3 cos tf + 5 cos 3?)

costf 25.15 P4(costf) = JL(9 + 20cos2ff + 35 cos4ff)

J(l + 3 cos 29) 25.16 Pstcos 8) = i|8(30 cos « + 35 cos 3a + 63 cos 59)

_ 1

~ ^(50 + 105 cos2ff 4- 126 cos4ff + 231 cos 6ff)

- 15^(1 "^B cos 9 + 189 cos 3a + 231 cos 5# + 429 cos 7«)

25.19

GENERATING FUNCTION FOR LEGENDRE POLYNOMI

= 2 Pn(^)t"

Vl - 2tx + (2 n=0

LEGENDRE FUNCTIONS

147

25.20 25.21 25.22 25.23 25.24

URRENCE FORMULAS FOR LEGENDRE POLYNOMI

(n + l)P„+,(a!) - (27i+l)xi>„(x) + nP^_,{x)

F; + i(x) - xP:(x) = (n+l)P„(x)

xP;(x) - P'„-iix) = nP„(x)

K + A') - P'n-iix) = (2n+l)P„ix)

(x^~l)P;,{x) = nxP^(x) ~ nP^_,(x)

^ 0

25.25

25.26

OLYNOMI

P^{x)P^{x)dx = 0 m^n

r,

J_ {Pnix)}

^dx =

2n+ 1 Because of 25.25, Fm(«) and P„ix) are called orthogonal in -1 ^ a; ^ 1.

25.27

where 25.28

jOGONAl SERIES OF LEGENDRE POLYNO

/(x) ^ A^P^ix) + A,P,(x) + A^Pzix) + ■■-2k+1 C^

A^ -

J mP^(x)dz

iSPECIAL RESULTS INVOLVING LEGENDRE POLYNOMI

25.29 P„(i) = I 25.32

25.33

25.34 25.35 25.36

25.30 P„(-i) = (-i)t 0

PniO) = i

25.31 P„(-x) = (-l)"P„(x) n odd

(-1)

n/2

1 ■ 3 ■ 5 • ■ • (71 - 1)

n even

1 /"^

''nCa;) = - I (« + Vk^-I cos 0)" d^

/^n(a:

)da; =

^«+i{»)-P„-i(x)

2n + l

-"W = 2^£#^''^

where C is a simple closed curve having x as interior point

LEGENDRE FUNCTIONS

GENERAL SOLUTION OF LEGGNDRE'S EQUATION

The general solution of Legendre's equation 13 25-37 y = A[7„(x) + BV^{x)

where

25.38 U ix) = 1 - ^^^^±11x2 + "(»-2)(^ + l)(^ + 3) _ ^ ,

n\ / 2! 4!

25.39 VM = X- t"-l)(^ + 2) (n-l)(n-3)(n + 2)(« + 4)

n\ / 3! SI *

These series converge for —1 < x < 1.

LEGENDRE FUNCTIONS OF THE SECOND KIND

25.40

where 25.41

If n = 0,1,2, ... one of the series 25.38, 25.39 terminates. In such cases,

C/„(x)/f;„(l) n = 0,2,4,... 'f'n(«)/^n(l) »i-1,3,5,... n2 /

C;„(l) = (-l)"/2 2n

ir-

n\

n = 0,2,4,...

25.42

^n{l) = (-l)<''-l)/2 2"-»

n-1

nl n= 1,3,5,...

The nonterminating series in such case with a suitable multiplicative constant is denoted by Q„(x) and is called Legendre's furiction of the second kind of order n. We define

25.43

Qnix) =

' U„{l)V„ix) n = 0,2,4,... ~VM)U„ix) M-1,3,5,...

25.44

25.45

25.46

lAl LEGENDRE FUNCTIONS OF THE SECOND KIND

Q.(x) = |ln(i±f

<?'<^> = p4\H^-^

nu\ - 3x2-1 / l + x \ 3x

25.47

i^\ - 5a3-3x, f l + x \ 5a:2 , 2

The functions Q„(fl:) 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

y = AP^ix) + BQ^ix)

LEGENORE'S ASSOCIATED DIFFERENTIAL EQUATION

(1 - x'^)y" - 2xy' + ^ n(n + 1) -

l-a;2

y ^ 0

Solutions of this equation are called associated Legendre functions. We restrict ourselves to the important case where m, n are nonnegative integers.

■GENDRE FUNCTIONS Qd

26.2

rim P^(X) = (1-0:2)^/2 ^P„(X) =

(1 _ x2)m/2 (fm+n

where Pni^) are Legendre polynomials [page 146]. We have 26.3 Plix) = PJx)

-(a.2-1).

26.4

P^{x) = 0 if m > n

SPECIAL ASSOCIATED LEGENDRE FUNCTIONS OF THI F|RST KIND

26.5 Plix) = (l-a;2)i/2

26.6 PUx) = 3a;(l-x2)i/2

26.7 PUx) = 3(1-ir2)

26.8 Plix)

26.9 Plix)

26.10 Plix)

|(5a;2-l)(l-a;2)i/2

15a;(l-x2) 15(l-a;2)3/2

26.11

GENERATING FUNCTIO^O^S

(2m)!(l-gg)"'^gt"

2 i'r(a')i" n=in

26.12 26.13

RECURRENCE FORMUL

{n + l-m)P'^+iix) - (2n + l)xP';:(x) + in + 7n)P^-i{x) = 0 Pr\^)-W^^zPr\^) + (n-m)in + m + l)P:ix) = 0

(1 - a;2)i

149

ASSOCIATED LEGENDRE FUNCTIONS

26.14

26.15

f P™{a;)Pr{x)dx = 0 if n^Z

'•'-1

)}2da! =

2 (m + m)!

2n + 1 (rt - Tn)!

26.16

where 26.17

THOGONAL SERIES

fix) = A^PZix) + A^^.PZ^iix) + A^^^Pl+^M +

A. =

2k + 1 {k-m)]

^ - ~l~(itT^ J_/<^'^'=t^''^^

i£.

ASSOCIATED LEGENDRE FUNCTIONS OF THE SECOND KIND

26.18

(fm

where Q„{x) are Legendre functions of the second kind [page 148].

These functions are unbounded at a: = ±1, whereas P^(x) are bounded at a: — ±1. The functions Q^ix) satisfy the same recurrence relations as P!^{x) [see 26.12 and 26.13].

GENEJ

IF LEGENDRE'S ASSOCJ

kOUATION

26.19

V = AP:^(x) + BQr(a')

HERMrTE POLYNOMIALS

27.1

NTIAI EQUATI

y" — 2xy' + 2ny = 0

If n = 0,1, 2, . . Rodrigue's formula

27,2

RMITE POlYNOMIAtS

then solutions of Hermite's equation are Hermite polynomials H^(x) given by

H^x) = (-i)nex'£L(g-^^j

P6CIAI HERllTri»OlYNOMIALS

27.7 Hi(x) = 16a;-i - 48z^ + 12

27.8 H^{x) = B2z^ - 160x3 + i20x

27.9 H^ix) = 64fl:8 _ 480x4 + 720x2 - 120

27.10 H^(x) = 128x'- 1344x5 + 3360x3 - 1680^

27.11

OENERAtiNG FUNCTION

,2tz-t' ^ -V ^n(a^) *"

nio nl

27.12 27.13

EECURRENCE FORMUL

■«n+i('K) = 2xff„(x) - 2nH„_i(x) Hl,ix) = 2nH„_j{z)

151

HERMITE POLYNOMIALS

27.14

27.15

ORTHOGONALITY OF HERMITE POLYNOMIALS

r 6-==* H^{x) Hnix) dx = 0 m¥=n

27.16

where 27.17

ORTHOGONAL SERIES

fix) = A^H^ix) + A,H,{x) + A^H^ix) +

A^ - ^= f e'^'f{x)Ht,ix)dx

27.18

SPECIAL RESULTS

BJ,) = (2x)n - 2l(ZL^(2.)n-:. + ^(^-l)(^^;2)(--3) (2.)n-4 ,

27.19 H„(-a;) = (-l)''fl„(a:) 27.20 H2„_,(0) = 0

■ff2n(0) = (-1)''2« • 1 • 3 • 5 ■ ■ • (2m,- 1) «„ + ,(«) H„+i{0)

27.21 27.22

27.23

27.24

27.25

J ^«(t)

±f.-^

dt =

2(n + 1) 2{n + 1)

,-!*

dx

{e-'^H^ix)} = -e-^fl„ + i(a;)

J ("e"** Hnixt) dt = Vvnl P„(a;)

27.26 ^ i /n

fc=0

This is called the addition formula for Hermite polynomials.

HJ^x^y) = 2 ^j^(l]H^{xyf2.)H^.^(yyf2)

27.27

" H^{x)H^{y) _ g„+,(ir)H„(y) - H„(x)H^+,iy)

2" + %! (a —1/)

xy" + {X-x)y' + ny = 0

lAOUERRE POLYNOMIALS

Zl7J:\-l:J. '*"'" '°'''**°"' '' Laguerre's equation are Laguerre polynomials L„ix) and

by Rodrigue'a formula. 28.2

are given

rf«

^» = ^:^i^"'-')

da:"

SPECIAL LA6UERRE POLYNOMIAL!

28.6 Lsix) = ~3^ + 9x2 - 18a; + 6

28.7 ^^(a;) ~ x^- 16x3 + 72^2 - 96a; + 24

28.8 Lsix) = -x^ + 25x4 - 200xS + 600x2 - 600« + 120 x6 - 36x5 4- 450x4 _ 2400x3 + 5400x2 - 4320x + 720 -x7 + 49xfl - 882a:5 + 7360x4 - 29.400x3 + 52,920x2 - 36,280x + 5040

28.11

GENERATING FUNCTION

- L„{x) <n

e-xtn~t 1 - t

28.12 28.13 28.14

K+iix) - (2n + 1 - x) L„(x) + m2L„_,(x) = 0

lUx) ~ nLl_i(x) + nL„_^(x) = 0

a!Z*^(x) = nL^ix) - n2L„_i(x)

153

LAGUERRE POLYNOMIALS

28.15

28.16

MlAlS

28.17

where 28.18

ORTHOGONAL SERIES

f(x) = AoL(,(x) + A^Liix) + A^Lzix) +

1 r°*

28.19 L„(0) = m!

28.21

28.22

28.23

28.24

28.25

28.20 J L,{t)dt = L„ix) --^^^^Y

II

2!

I a;Pe-^L„(x) da; := -^

''o [(-l)«(m!

(-!)««!

if p < n )2 if p = n

(n\)^{x~y)

L„(a;) = r «"«*-« Jo (2-/^)

d«.

lAGUERRE'S ASSOCIATED DIFFERENTIAL EQUATION

29.1

xy" + (m + 1 — x)y' + {n — m)y = 0

iSOCIATED LAGUERRE POlYN<

Solutions of 29.1 for nonnegative integers m and n are given by the associated Laguerre polynomials

29.2

where Ln(a;) are Laguerre polynomials [see page 153],

29.3 29.4

Lnix) = Ln(x)

L^{x) = 9 if m > n

ISSOCIATED LAGUERRE POLYNOMIAI.

GENERATING FUNCTION FOR L^j

29.15

(1-«)" + !

- Lnix)

2 -r^***

ASSOCIATED LAGUERRE POLYNOMIALS

ORMULi

29.16

r-i — Ln + i(x) + {x + m — 2ti—l)Lnix) + n^Lrt-iix) =

71 -j- 1

29.17

■^{Ln{x)) = Ln (X)

29.18

■^{x-^e-'^ Ln{x)) = (m~n-l)a:'"-'c-^Ln \x)

29.19

x^{Ln{x)} - ix — m)Ln(x) + {m — n — l)Ln (x)

ORTHOGONALITY

29.20

{c««-* Ln(x)Lpix) dx = 0 p¥^n

A

29.21

X

^ x-e-HLn(xWdx = ^^

ORTHOGONAL SERIES

29.22

where 29.23

f(x) = A^Lm(x) + A^+iLm-nix) + A^ + 2^m+2{x) +

A^ =

{k~

{k\)

x^e~^Lh (x) fix) dx

^SPECIAL RESULTS

29.24

-m, . ,, m! nln — m} , n(n ~ l){n ~ m){n ■-m — 1)

^nix) = (-l)«^^-^|x«-".- -L_^a:«-'«-l + -^ '.^^-^ L-^n-^--

, 3;n—m —2 ^

}

29.25

r" «. + , rrr^'/M^j (2«-Tn + l)(K!)3

CHEBYSHEV'S DIFFERENTIAL EQUATION

30.1

(l-x2)y" ~ xy' + n'^y = 0 n = 0,l,2,...

CHEBYSHEV POLYNOMIALS OF TH£ FIRST KIND

Solutions of 30.1 are given by 30.2 r„(x) = cos (71 cos-i a!) = a:«

- C"^x"-2a-a:2) + /'"'\a;"-»(1-052)2 _

SPECIAL CHEBYSHEV POLYNOMIALS OF THE FIRST KINI

8x* - 8fl;2 + 1

16ar5 - 20x3 + 6a;

32x« - A8x* + 18x2 _ 1

64a:' - 112z^ + 56x3 _ ij^

GENERATING FUNCTION FOR Tn{x)

30.11

I- tx

1-Ztx+ t^

2 r„(x)t«

n=0

SPECIAL VALUE!

30.12 T^{-x) = (-l)T„(a;)

30.13 ^^{l) = 1

30.14 r„C-l) = (-1)"

30.15 r2^(0) = (-1)«

30.16 r2,+,(0) - 0

CHEBYSHEV POLYNOMIALS

Fira^^rFORMuraFo

30.17

T„^i{x) - 2xr„(x) + r„_i(a:) = 0

30.18

30.19

/

I T^{x)T„ix)

J

-1 Vl-3:2

1 {r„(x)}2

da; = 0 m ¥= n

dx =

TT if n = 0 jr/2 if n= 1,2, .

30.20

where 30.21

fix) = \A^T^{x) + A,T^(x) ■\- A^T^ix) + 2 r fix)T^{x}

-IS

-1 \/l-x2

dz

30.22

IHEBYSHEV POIYNOMIALS OF TJ

;OND KIND

Un(x) =

sin {(n + 1) cos~^ a:} sin (cos""^ a;)

n+ 1

a;" —

n + 1

g xn-2Cl-x2) + , 5

m + 1

.11-4/1 —^212 _

x"-4(l-a:2)

SPECIAL CHEBYSHEV POLYNOMIALS OF THE SECOND KIND

FUNCTIOI

30.31

1 - 2tx + t2

2 u^{^)tn

n=0

CHEBYSHEV POLYNOMIALS

169

30.32 U„(-x) = (-l)«t/„(x)

30.33 UJl) = n+1

30.34 I7J-1) = (-l)«(n+l)

30.35 U2n(0) = (-1)«

30.36 [72,,,(0) = 0

30.37

RECURSION FORMULA FOR Un{x)

U^^^ix) - 2xU„(x) + t7„_i(x) = 0

ORTHOGONALITY

30.38

30.39

30.40

where 30.41

30.42 30.43

30.44 30.45

30.46

f Vl-a;2 c;^{iB) C7„(a;) dz = 0 m 9^ n

j Vr^{U^ix)}^dx = I

ORTHOGONAL SERIES

fix) = A^Uoix) + A^U.ix) + A^U^ix) +

A^ - - f' Vl^^fix) U^ix)

dx

RELATIONSHIPS BETWEEN TJx) AND C^J*!

n(x) = U^{x) - xU„^i{x) {l-x^)U^.,{x) = xT^ix) - T^^Ax)

1 r' r^+iMdij

t^™{a:)

_ 1 p ^r.

a:)Vl--y2

r„(x)

1 W VT-^2C7„_,(^)

= -J

dv

GENERAL SOLUTION OF CHEBYSHEV'S DIFFERENTIAL EQUATION

V =

AT„ix) + By/T^^^U n-iix) il n= 1,2,3, ...

A + B sin-i a

if n= 0

HYPERGEOMETRIC DIFFERENTIAL EQUATION

x(l-x)y" + {c-(a+b + l)x)y' - aby = 0

Wl

RGEOMETRIC FUNCTIONS

A solution of 31.1 is given by

310 mn h-.-^\ _ 1 , a.6 ^ a(a + l)&(6 + l) a(a+l)(a +2)6(6 +1)(6+ 2) . ^

31.2 Fia,b,c,x) - l+-^x+ i.2.c{« + l) ^'+ 1.2-3.c(c+l)(c-F2) *' +

If a, 6, c are real, then the series converges for —1 < a; < 1 provided that c — (a + 6) > —1.

31.3 F(-p,l;l;-a:) = (! + «)"

31.4 F{l,l;2;-x) = [ln(l + JB)]/a!

31.5 lim F{l,n;l;x/n) = e*

31.6 F(^,-^;^;sin2a;) = cosx

31.7 F(^,l;l;sin2a;) = secx

SPECIAL CASE!

31.8

31.9 F{^,l;l;-x^) = (tan-iic)/a:

31.10 F{l,p;p;x) = 1/(1 - x)

31.11 F(n + l,-n;l;il-x)/2) = P„(a;)

31.12 F(n.-7i; ^; (1 - a:)/2) = r„(x)

GENERAL SOLUTION OF THE HYPERGEOMETRIC EQUATION

If c, a — 6 and c — a — b are all nonintegers, the general solution valid for ]«! < 1 is 31.13 y = A F{a, b; c; x) + Bx^~<'F(a-c +1, b - c + l;2-c; x)

31.14 31.15 31.16 31.17

MISCELLANEOUS PROPERTIES

F(o, b; c; 1) =

r(o) r(c - g - 6)

T(c - a) r(c - 6)

—Fia, &; c;x) = — F(a + 1, 6 + 1; c + 1; x) dx c ^

F{a,b:c;x) = r(6)r(c-&) J" ^"'Ml-^)'-'''-'(l-ux)"" du

F(a, 6; c; k) = (1 - a)c-a-bF(c - a, c - 6; c; x)

160

DEFINrriON o

^B^ApfAC^ffW^^^oP^fff

32.1

^{Fim = J €-'tF{t)dt = /(8)

In general /(a) will exist for s > a where a is some constant. .C is called the Laplace traneform operator.

DEFINITION OF THE INVERSE LAPLACE TRANSFORM OF f{s)

If ^{F(e)> = /(s), then we aay that F(i) = ^-^{/(s)} is the inverse Laplace transform of /(s). Jl~^ is called the inverse Laplace transform operator.

COMPLEX INVERSION FORMULi

The inverse Laplace transform of /(s) can be found directly by methods of complex variable theory. The result is

32^

F{t)

c + fT

where c is chosen so that all the singular points of /(«) lie to the left of the line Re {s} = c in the complex

s plane.

LAPLACE TRANSFORMS

tE OF GENERAL PROPERTIES OF LAPLACE TRANSFORM

LAPLACE TRANSFORMS

163

LAPLACE TRANSFORMS

IB or

TSPITTCE TRANSFORMS

LAPLACE TRANSFORMS

165

LAPLACE TRANSFORMS

LAPLACE TRANSFORMS

167

LAPLACE TRANSFORMS

LAPLACE TRANSFORMS

169

LAPLACE TRANSFORMS

LAPLACE TRANSFORMS

171

LAPLACE TRANSFORMS

LAPLACE TRANSFORMS

173

32.163

32.164

32.165

32.166

32.167

32.168

f(s)

s See also entry 32.138.

e-os (! — «-")

s{l - 6-"^) See also entry 32.102.

e(l - e-*)''

1- e-

8(1 —re-«) See also entry 32.104.

a2s2 + 7r2

Fit)

Heavislde's unit function V{t — a)

F{t)

1-

Flg. 32-6

Pulse function Fit)

1-

Fig. 32-7

Step function

3-2-1

2a

—I— 3a

Fig. 32-8

Fit) =n2, 71 S t<n+l, 71 = 0,1,2,

Fig.32-9

F(() = r«, n ^ ( < n + 1, n = 0,1, 2,

2 3

Fig. 32-10

Fit) =

sin {Trt/a) 0 ^ t g a

t > a

Fig.32-11

—I— 4a

T '

33.1

where

33.2

FOI

^m

W-iSR^

f{^) = I MW cos aX + B{a) sin ax) da

If" A (a) — ~ I fix) COS ax dx

1 C" B{a) = — I f{x) sin ax dx

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) 1 |/(a;)| dx converges;

(iii) f(x) is replaced by ^{f{x + 0) + f{x — 0)} if a; is a point of discontinuity.

33.3

33.4

EQUIVALENT FORMS OF FOURIER'S INTEGRAL THEOREM

fix

) = 2~ J j ^^^^ *^°^ aix-u)

■^rth^; .A *'*! — M^

du da

a= —30 "11= —

fix) = ^ C e'«* da ( fill) e-*«" du

du da

— 00 — 1

33.5 /(a;) = — I sin ax da I /(w) sin au du where fix) is an odd function {f{—x) = —/(a:)].

33.6 /(a;) = — ( cos aa; da | /(m) cos au du where fix) is an even /uncfcion [f(~x) =f(x)].

FOURIER TRANSFORMS

175

The Fourier transform of f(x) is defined as

33.7 ?"{/(«)> = ^(«) = J /(a;)e-*«*dx Then from 33.7 the inverse Fourier transform of F(a) is

33.8 f-HFia)) = f{^) = ^J ^(«)e*«*d« We call f{x) and F{a) Fourier transform pairs.

CONVOLUTION THEOREM FOR FOURIER TRANSFORMS

If F(a) = Tifix)) and G(a) = Tiffi^)}, then 33.9 ^ f F{a) G{a) e^=^ da = J fiu)g(x-u)

where f*g is called the convolution of / and g. Thus

33.10 7{f*ff} = nnriff}

du = f*g

EVAL'S IDENTITY

If Fia) = nfix)}, then

33.11 J" \fix)\^dx = ^y \F{a)\^da

More generally if F(a) = f{f(x)} and G{a) = ^{^(a!)}, then

33.12 C fix)^)dx = ^ r F{a)GW)da where the bar denotes complex conjugate.

FOURIER SINE TRANSFORMS

The Fourier sine transform of f{x) is defined as

33.13 Fsia) = Tsifix)) = r f{x)amaxdx

Then from 33.13 the inverse Fourier sine transform of Fsia) is

2 r°°

33.14 fix) = frM-PsW} = -\ Fsia)Binaxda

FOURIER TRANSFORMS

dx

The Fourier cosine transform of fix) is defined as

33.15 Fc(a) = TciH^c)) = f f{x)cosax

Then from 33.15 the inverse Fourier cosine transform of Fc(a) is

33.16 fix) = f-HFcM) = Ij Fc(«)cosaxd«

TRANSFORM PAI

FOURIER TRANSFORMS

177

SPECIAl FOURIiR 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

/(x)

1 0 < a: < 6 0 a; > 6

»-»

a:2 + 62

,-bi

a;"—) g—bx

,-bI*

--1/2

a;-"

Bin bx

Bin bx

cos bx

tan-i (rc/b)

csc&z

e2x_i

fcM

1 — cos 6a

He-ba 2

a2+ 62

r(n) sin (n tan-W6)

(a2 + 62)n/2

463/2 "«

/4b

v^

ya"~l CSC (ntr/2) 2r(7i)

0 < m < 2

1 , fa + b

2^n;r^

B^a/2 a < 6 3r6/2 a> b

' 0 a < 6 7r/4 a=b r-/2 a > 6

2^*"''"

26**"*^ 26

f-^K?)~^

FOURIER TRANSFORMS

SPECIAL FOURIER COSINE TRANSFORMS

ELLIPTIC FUNCTIONS

INCOMPLETE ELirPTIC rNTEGRAL OF THE FIRST KIND

34.1

u = F{k

'*' = r

d9

s:

dv

where ^ = am u is called the amplitude of u and x = sin 0, and where here and below 0 < fe < 1.

COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND

34.2 K = F(ft,W2) = (

de

0 Vl - *^ sjn2 9

-i:

dv

= fii + (lT*= + (i^T'^ + (i^"^ +

a

INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND

34.3

Vl - A:2 sin2 tf d9 = I ^

n ''o Vl -

Vl-u2

ffPW

THE^SECOND KT

Vl - fc2 sin2 ^ d# =: I

0 ^0

■1 Vl - fc2^2 Vl-tj2

dv

= ii-(iT--feyf-(i^TT

INCOMPLETE ELLIPTIC INTEGRAL OF THE THIRD KIND

34.5

de

= /:

dv

0 (1 + n sin2 «) Vl - ^^ sin2 « ^^ (1 +mv2) V(l-■u2)(l-ft2i;2)

ELLIPTIC FUNCTIONS

34.6

COMPUTE ELLIPTIC INTEGRAL OF THE THIRD KIND

da _ r^ dv

•J n

Xir/2

(1 + m 8in2 «) vT-fc^sin^ ^o (1 + nv^) y/(\ - v^){l - k^v^)

LANDEN'S TRANSFORMATION

34.7

This yields 34.8

sin 201

*^"^ = fc + co8 20, *''■ ''^'"'^ " sm[2^,-4>)

F(fc.0) = f

d9

0 Vl - fc^ 8in2 tf

2 f*' 1 + feJ.

dffi

0 \/l-feJsin2(?i

where k^ — 2\Ic/{l + k). By successive applications, sequences kj, feg.^3. • - - and <f>i, ^g, 03, ... are obtained such that ft < fci < fcg < feg < ■ ■ - < 1 where lim fe„ = 1. It follows that

34.9

where 34.10

. 2\/fe , 2V^ ^ ^

The result is used in the approximate evaluation of F(ft,0).

JACOBI'S ELLIPTIC FUNCTIONS

From 34.1 we deiine the following elliptic functions.

34.11 X = sin (am w) = sn m

34.12 v1^^k2 — cos (am m) = cn«

34. 13 Vl - *=^*^ = Vl -ft2sn2M = dn w

"We can also define the inverse functions sn-' x, cn~^ x, dn-i x and the following

34.14 nsM ~

34.15 nc« —

34.16 ndw =

sn u

1 cnw

1 dnw

34.17

sc w =

34.18 sd« - ^

dnu

34.19 cd« = ^

dnu

snM

34.23 34.24

34.25

sn (m + v) en (m + v) dn (m + V)

SUM cni* dntJ + cnM sni) dna 1 — A;2 sn2 u 8n2 v

COM enV — snasnudnwdnu 1 - A;2 sn2 M sn211

dntt dnt? — ft^ snu sni; cna cn-u 1 — fc2 an2 y gn2 ^

ELLIPTIC FUNCTIONS

181

34.26 ^sna - cnudnM

34.27 ^ en u - - sn u dn M

34.28 -2-dn«

du

34,29 ^scu

— — fc2 sn M en u

— dc u nc M

34.30 34.31 34.32

SERIES EXPANSIONS

cnu = i-|^+(i + 4fc2) j^ - {1 + 4Ak2 + 16k*) fy + • ■ •

2 A

dnw = 1 - A:2|y+ fe2(4 + fc2)^_ fc2(i6 + 44fc2 + A^)^+ ...

34.33

lf'K,k = If f" "'"^

12 32 ^ 52

= .915965594.

Let 34.34 Then 34.35 34.36 34.37

PERIODS OF ELLli>TIC FUNCTIONS

K =

^0 Vl - fc2 gin2 0 ' J,

ff/2

0 \/l-fe'2sin2ff

snM has periods 4^ and 2iii:'

en u has periods 4K and Sif + 2iK'

dnu has periods 2K and 4iK'

where A:' = \1~!^

IDENTtTIES INVOLVING EUIPTIC FUNCTIONS

34.38 sn2tt + cn^w = 1

34.40 dn2u-A:2cn2« - fe" where )fc'- a/T^^

34.42

34.44

CH'^M =

dn 2m + en 2m

VI

1 + dn 2u — en 2m sn m dn m

+ cn2u

cnu

34.39 dn2u + fc2sn2u = 1 1 - en 2m

34.41

sn-'u —

1 + dn 2m

34.43 dn2„ = 1 - fcg + dn 2m + fe' en w

1 + dn 2m

34.45

Vf

— dn 2u ft sn u en u

+ dn2M

dnu

ELLIPTIC FUNCTIONS

mmmm

34.46 snO = 0 34.47 cnO = 1 34.48 dn 0 = 1 34.49 scO = 0 34.50 am 0 = 0

INTEGRALS

34.51 I snuiiu = -r In (dn m —/c en u)

34.52 j en u du = -j- cos~i (dn u)

34.53 I dnwdw = sin~^snM)

34.54 I sc u du

■ , In (dc u + y/T—l^ nc «)

34.55 I cs u du = In (ns m — ds m)

34.56 \ cdudu

-r In {nd M + fc ad u)

34.57 I dc u du

34.58 fsdudu

In (nc M + ac m) -1

In (na m — ca «)

sin~^ (fe edit)

34.59 JdaudM

34.60 I ns M dw = In (ds u — cs «)

34.61 I ncu

34.62 I nd u du

du =

1

1 , /, , sew In dc M +

VH^

coB~i (cdu)

LEGENDRE'S RELATION

34.63

where 34.64

34.65

EK' + E'K - KK' = v/2

E

- r^

fc2 8in2 e d$

•^ n

■n/2

dg

J-'tt/Z , /•jr/2 fla

0 -^0 Vir^fc^i

0 Vr^^fc2sin2ff sin^ff

ERROR FUNCTION erf (a;) = —r f e""

du

35.1 erf(x) = ~(x-^+^2, ^.3,

+

35.2 erf («)

1 -

1 -7^4 +

1-3 1*3'5

+

^^\- 2a:2 ' (2a;2)2 (2x2)3 35.3 erf(-x) = -erf(x), erf (0) ^ 0, erf('») = 1

COMPLEMENTARY ERROR FUNCTION erfc(a:) = 1 - erf (x) = —= f e "'d«

35.4 erfc («)

35.5 erfc (a;)

35.6 erfc (0)

2_ / _ a:3 gs _ z?

V^r 3-1! ^5-2! 7-3!

+

yfi^xV 23:2 ' (2a:2)2 (2x2)S 1, erfc{<") = 0

J"" fi-" —-

du

35.7 Ei{x) =

35.8 £i(a^) =

35.9 Eiix) -

^i 1 —7 — In X + I -

—y — \nx +

_ »—u

dw

+

I'l! 2-2! ' 3*3!

Ill 1-11+2|_|| +

35.10 EH-^) = 0

35.11 Si(x) =

SINE INTEGRAL

x3 _^ rS

Si{x) = f

({U

1-1! 3-3! 5*5! 7'7!

£ _ sing /l _ 3! 5! _ 2 a; la; arS a:5

+

35.12 Si{x) ~

35.13 Si(-x) = -Si(a:), Si(0) = 0, Si(«) = W2

cos a; / _ 2! 4! _

X \^ ^"^^

COSINE INTEGRAL Ci{x) = C'^^^du

35.14 Ci{x) = -y-lnx+ Ci^l^^^du

J,

35.15 Ci(x) = -y - \nx + ^ - ^^ + ^ ^ + ■-■

2-2! 4-4! 6-6! 8-8!

35.16 Ci(x) ~ ^^^(l-^+^ A-Sin^/ 2!,4! \

X \x x^^ x^ J X y- x'^^x* J

35.17 Ci('*) = 0

FRESNEL SINE INTEGRAL S{x) = J^ j'Jsinu^ du

\^V3-1! 7-3! 11-5! 15-7! ^

35.19 Six) ~ l-~^Jico.x2)(l~llA+ll3^_ __\ /j^_l^S^

2 ^/2^\ V 22a;5 2^x^ J ^ ^"^"' ' \2x^ 23ar'

35.20 S(-x) = -Six), 5(0) = 0, S{«) = -^

FRESNEL COSINE INTEGRAL C{x) = .pfcostt^dtt

35.2, C(.) =^(^-_^,_4_^_-.,...)

35

35.23 a-x) - -C(a;), C(0) - 0, C(~) - 1

RIEMANN ZETA FUNCTION ^a:) = TF + 4 + 4 +

35.24 «., = ^r"-Hilf_,,, ,>!

35.25 ni-a;) = 2i-^;r-^r(a:) co3(o-a:/2)j:(a:) [extension to other values]

-Iy2fc

22k-V2fc5 35.26 m) = ~7^J~ k = 1,2.3....

TRIANGLE INEQUALITY

36.1 36.2

|ai] - Ittal ^ lai + ojl ^ [aj + [oal |ai + fla + ■. ■ + a„| g ]a,I + [ojl + . -. + |a„|

CAUCHY-SCHWAR2 INEQUALITY

36.3 \a,b, + 0362 + ■.. + a„feJ2 ^ (|a,|2 + la^I^ + • - ■ + \aM\h\^ + I62P + ' •" + i6„|2)

The equality holds if and only if ai/6, - ag/fca = ■ • • = a„/6„.

INEQUALITIES INVOLVING ARITHMETIC, GEOMETRIC AND HARMONIC MEANS

If ^. G and H are the arithmetic, geometric and harmonic means of the positive numbers a,, a,,...,a,

H ^ G ^ A

then 36.4

where

36.5 A = "i + °2 + • • • + '^n

36.6 G = ^a,a^...a, 36.7 ^ .= 1 f 1 + 1 + ... + A

The equality holds if and only if aj = Og = • ■ • = a„.

H - nVo +a + •■• +

36.8

where 36.9

HOLDER'S INEQUALITY

kifri + ^262 + • ■ ■ + a„6„| ^ (|a,|p + la^lP + ... + Kji>)1/p(|6,|<. + [ft^i^ + ... + |(,j,ji/.

- + ^ = 1 P>1. 9>1

duces

The equality holds if and only if W-Vil-.l = Wp-./|6,| = • • • = |a,|.-V|M, For p = , = 2 it

re-

INEQUALITIES

CHEBYSHEV'S INEQUALITY

If tti ^ a2 ^ ' ■ ■ ^ a„ and 61 S 63 = ■ ■ • = &„> then

n

n

n

36.10

or

36.11 {ai + a2+ ■•■ +«„)(&! + &£+ ••■+&„) = n(ai&i + a262+ •■ ■ + a„6„)

MINKOWSKI'S INEQUALITY

If Qi, a^, ..., a„, bi, 6,, ... b„ are all positive and p > 1, then

36.12 {(ai+fe,)P+ (02+62)''+ ■■■ + (On+fiJ'*}^^'' ^ (ai + 02+ ■■■+0^^" + ib'l + bl+ ■■■ + 6^)i/p The equality holds if and only if O1/61 = 02/62 = ■ ■ - = o.n^f>n-

CAUCHY-SCHWARZ INEQUALITY FOR INTEGRALS

I/:

36.13 II f{x)g{x)dx

The equality holds if and only if f{x)/g(x) is a constant.

j''\f{x)\-^dx\Ij^\g{x)\^dx

HOLDER'S INEQUALITY FOR INTEGRALS

36.14 J \fix)g{x)\dx ^ if \f{x)\''dxi \j \9{x)\''dx

where l/p +1/g = 1, p > 1, g > 1. If p = g - 2, this reduces to 36.13. The equality holds if and only if \f{x)\f-^/\g{x)\ is a constant.

i/n

MINKOWSKI'S INEQUALITY FOR INTEGRALS

If P > 1, 36.15

<j \nx) + g(x)\^dx^ ^ ij \nx)\^dx^ + U \9(x)\'>dx

i/p

The equality holds if and only if f{x)/g{x) is a constant.

37.1 cot a; =

X "^ 2a: j^2_^+^2_4^2 + x2_9^ +

37.2

CSCX - ^ ^^\^z_^ -2^2 + ^^—^

37.3 sec X = Av

' .+ =

.2 _ 4a;2 9^.2 _ 4a;2 ^ 2B;r2 - 4a:2

37.4 tan x = 8a; -I ^ h ^ I- . ^ u

lan ic »* \2 _ 4a;2 ^ 9^ - 43^2 + 26772 - 4a:2 ^

37.5

sec^a; =

4 J 1 + 1 + 1 , 1 ,

37.6 csc^ a; =

-1+ 1 + 1 + 1 I 1 +

a;2 (ar-;r)2 ^ (a: + ,r)2 ^ (a: - 2,r)2 ^ (a; + 2^)2 ^

37.7 coth X =

1 + 0-.}^L—^ 1 , 1 .

X U2 + ,r2 ^ ^ + 4^ "^ a:2 + 9^ +

37.8 csch X =

l_2x-^-^^ ^

X |a;2 + B^ a;2 + 4^ "f' ^2 + 9^

1 + ^

37.9 sechx = 4^

^+ 5

572 + 4x2 9^2 + 43.2 ^ 25^ + 43.2

37.10 tanhx =

^* "^ s-2 -f 4x2 "•■ 9^ + 4a;2 + 25:r2 + 4x2 "^

sina: =

.u_-Vi_- ,_^

38.2

C08X =

-^Vi-|S)(i-^.

38.3 sinh x

^(i + 5)(i + 5)(i+i.

38.4 cosh X =

-^)(-is)(^-^;

See also 16.12, page 102.

38.6 MX) = 1-^ 1-- 1-

where Xj, X2, X3, ... are the positive roots of Jq{x) = 0.

38.7 JM = .fi-gyi-E!yi_E^

where \i, X2, X3> ... are the positive roots of Ji(x) = 0.

38.8

sin X _ X

X X X X

COS - cos — COS -- cos —-2 4 8 16

38.9

2 2 4 4 6 6 1 '3* 3' 5'5'7

This is called Wallis' product.

188

BINOMIAL DISTRIBUTION

39.1

p>0, g>0, p + q = 1

POISSON DISTRIBUTION

39.2

*(«:)= 2 ^ X > 0

HYPERGEOMETRIC DISTRIBUTION

39.3

*(«) = 2

t/\n — t

t<x /r + s n

NORMAL DISTRIBUTION

39.4

X) = ^f e-'Ut

<i.(x) =

STUDENT'S t DISTRIBUTION

39.5

y/^ ~r(n/2)

1 V 2 / /•■ *^^* = JZZ T(n/2) J

1+^ n

-(n + I)/2

d(

CHI SQUARE DISTRIBUTION

39.6

1 r^

F DISTRIBUTION

39.7

*(«) =

r(n,/2) r(n2/2)

r t"./2(n2 + nit)-<"i+»*5/2d(

SPECIAL MOMENTS OF INERTIA

The table below shows the moments of inertia of various rigfid bodies of mass M. In all cases it is assumed the body has uniform [i.e. constant] density.

190

SPECIAL MOMENTS OF INERTIA

191

Length

Area

Volume

Speed

Density

Force

Energy

Power

Pressure

1 kilometer (km)

1 meter (m)

1 centimeter (cm)

1 millimeter (mm)

1 micron (ju)

1 millimicron (m^u)

1 angstrom (A)

— 1000 meters (m)

— 100 centimeters (cm) = 10-2m

^ 10-3 m

— 10-8m = 10-!'m = 10-10 m

1 inch (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 in. 0.3937 in. 39.37 in. 0.6214 mile

1 square meter (m2) ^ 10.76 ft^ 1 square foot (ft2) - 929 cm2

1 square mile (mi2) = 640 acres 1 acre = 43,560 ft2

1 liter (/) = 1000 cm^ = 1.057 quart (qt) - 61.02 in3 = 0.03532 ft^

1 cubic meter (m^) = 1000 I = 35.32 ft^

1 cubic foot (ft3) - 7.481 U.S. gal = 0.02832 m^ - 28.32 I

1 U.S. gallon (gal) ^ 231 in^ - 3.785 I; 1 British gallon = 1.201 U.S. gallon = 277.4 in*

1 kilogram (kg) =^ 2.2046 pounds (lb) = 0.06852 slug; lib = 453.6 gm = 0.03108 slug 1 slug = 32.174 lb = 14.59 kg

1 km/hr ^ 0.2778 m/sec ^ 0.6214 rai/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 Ib/ft^ = 1.940 slug/ft3 1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft^ - 0.5154 gm/cm^

1 newton (nt) - 10^ dynes = 0.1020 kgwt = 0.2248 Ibwt

1 pound weight (Ibwt) ~ 4.448 nt = 0.4536 kgwt = 32.17 poundais

1 kilogram weight (kgwt) ^ 2.205 Ibwt = 9.807 nt

1 U.S. short ton = 2000 Ibwt; 1 long ton = 2240 Ibwt; 1 metric ton = 2205 Ibwt

1 joule = Intm = 10^ ergs = 0.7376 ft Ibwt ^ 0.2389 cal = 9.481 X lO^^ Btu

1 ft Ibwt ^ 1.356 joules ^ 0.3239 cal = 1.285 X 10-3 Btu

1 calorie (cal) = 4.186 joules = 3.087 ft Ibwt = 3.968 X 10-3 Btu

1 Btu (British thermal unit) - 778 ft Ibwt = 1055 joules = 0.293 watt hr

1 kilowatt hour (kw hr) = 3.60 X 10« joules - 860.0 kcal = 3413 Btu

1 electron volt (ev) = 1.602 X IQ-^^ joule

1 watt = 1 joule/sec = 10' ergs/sec = 0.2389 cal/sec

1 horsepower (hp) ^ 550 ft Ibwt/sec = 33,000 ft Ibwt/min = 745.7 watts

1 kilowatt (kw) = 1.341 hp = 737.6 ft Ibwt/sec = 0.9483 Btu/sec

1 nt/m^ - 10 dynes/cm2 = 9.869 X 10-8 atmosphere = 2.089 X 10-2 lbwt/ft2 1 lbwt/in2 - 6895 nt/m^ = 5.171 cm mercury = 27.68 in. water

1 atmosphere (atm) = 1.013 X 10^ nt/m2 = 1.013 X 10^ dyne3/cm2 = 14.70 lbwt/in2 = 76 cm mercury = 406.8 in. water

Part n

TABLES

SAMPLE PROBLEMS

ILLUSTRATING USE OF THE TABLES

COMMON LOGARITHMS

1. Find log 2.36.

We must find the number p such that lO" - 2.36 = A'. Since 10" = 1 and 10^ = 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, ie. 2.36 = 100-3'29.

2. Find (a) log 23.6, (6) log 236, (c) log 2360.

From Problem 1, 2.36 = 10''-3729, jhen multiplying successively by 10 we have 23.6 = 10'-3'29, 236 = 102-3729^ 2360 = 103-3729

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 (6) 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, (6) log .0236, (c) log .00236.

From Problem 1, 2.36 = 10''-3'='29. Then dividing successively by 10 we have

.236 = 10*>3ra9-J = 1Q9.3729-10 = 10~-8271

.0236 = 100-3729-2 = X08-3729-10 == 10-16271

.00236 = 1003729-3 - IO'3729-IO = 10-2-6271

Then

(a) log .236 ^ 9.3729 - 10 -.6271

(6) 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.

4. Verify each of the following logarithms.

(a) log87.2. Mantissa - .9405. characteristic - 1; then log87.2 = 1.9405.

(b) log 395,000 = 5.5966.

(c) log .0482. Mantissa - .6830, characteristic - 8 - 10; then log.o .0482 - 8.6830 - 10. (rf) log .000827 - 6.9175 - 10.

5. Find log 4.638.

Mantissa of log 4640 .. .6665 Mantissa of log 4.638 = .6656 + (.8)(.0009)

Mantissa of log 4630 = .6656 _ (.RRt * * j- -^ — .d663 to four digits

Tabular difference - .0009 Then log 4.638 = 0.6663

(6656+')!"'' '"' P^°P-*--^ P-ts table on page 202 can be used to give the mantissa directly

6. Verify each of the following logarithms. (a) log 183.2 ^ 2.2630 (2625 + 5) (6) log 87,640 - 4.9427 (9425 + 2)

(c) log .2548 =9.4062-10 (4048 + 14)

(d) log .009848 = 7.933 - 10 (9930 + 3)

COMMON ANTILOGARITHMS

7. Find (a) antilog 1.7530, (b) antilog (7.7530 - 10)

digits before the decimal point. T.^TeZ^f, .ll^Z'i^ '' '' ^'^^^ ^'^ '^^

^'* ac'tlris'ril'V-'lo^th?' '"h' 5662 corresponding to the mantissa .7530. Then since the char-

Thnrfl I' ""^^"^ """'* ^"^^^ **° ^«^°s immediately following the decimal noint

Ihus the required number is .005662. ^"'luwing me aecimal point.

8. Find antilog (9.3842 - 10).

on page%rwlTa;f'' "" '^'"^^" ''''' ^"' •'''' »"^ ^ ™-* "^ interpolation. From the table

Number corresponding to.3850 = 2427 Given mantissa = 3842

Number corresponding to .3840 = ^42]_ Next smaller mantissa - .3840

Tabular difference = 6 Difference = "i^^

Then 2421 + ^(2427-2421) ^ 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.

(a) antilog 2.6715 = 469.3

(b) antilog 9.6089-10 = .4063

(c) antilog 4.2023 = 15.930

COMPUTATIONS USING LOGARITHMS

j^ ^ ^ (784.6)(.0431) ^ j^gp ^ log 784.6+ log .0431 - log 28.23. 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.

(784.6)(.0431) ^ (102.B347)(10a.e345-10) ^ ^q2.,M7 + S.^Z,B-10-IA,07 = 10.0785 ^ 1198

28.23 101-4507

11. P = (5.395;8. logP = 8 log 5.395 = 8(0.7320) - 5.8560, and P ^ 717,800.

12. P = V38^2 = (387.2)>/2. logP = | log 387.2 = ^(2.5879) = 1.2940 and P = 19.68.

13. P = ^.08317 = (.08317)1/5. log P - ^ log .08317 = ^(8.9200-10) = f, (48.9200 - 50) - 9.7840-10 and P = .6081.

14. P = V-003654 ( 18-3 7P j^^p ^ 4 log .003654 + 3 log 18.37 - (4 log 8.724 + ^ log743.8)

(8.724)4 ^743.8

Numerator N Denominator D

I log .003654 - ^(7.5628-10) 4 log 8.724 = 4(0.9407) = 3.7628

= ^(17.5628-20) = 8.7814-10 ^ log 743.6 = 1(2.8714) = 0.7178

3 log 18.37 ^ 3(1.2641) = 3.7923 Add: log D = 4.4806

Add: logiV = 12.5737-10

logN = 12.5737-10 (-) logZ) - 4.4806

logP =^ 8.0931-10. Then P = .01239

NATURAL OR NAPIERIAN LOGARITHMS

15. Find (a) In 7.236, (6) In 836.2, (c) In .002548. (a) Use the table on page 225.

In 7.240 = 1.97962 In 7.230 = 1.97824

Tabular diflference = .00138 Then In 7.236 ^ 1.97824 4- ^(.00138) = 1.97907

In terms of exponentials this means that e^-^'^o' - 7.236.

(6) As in part (a) we find

In 8.362 = 2.12346 + 3^^(2.12465 - 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 ee-'^ss? = 836.2.

(c) As in part (a) we find

In 2.548 - 0.93216 + 3%{0.93609 - 0.93216) = 0.93530

In .002548 = In (2.548 X lO"-*) = In 2.548 - 3 In 10 = 0.93530-6.90776 - -5.97246 In terms of exponentials this means that g-5-97246 - .002548.

TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)

16. Find (a) sin 74°23', (6) cos 35°42', (c) tan 82°56'.

(a) Refer to the table on page 206.

sin 74°30' = .9636

sin 74°20' = .9628

Tabular difference .0008

'^*'*" sin 74<'23' - .9628 + ^(.0008) = .9630

{&) Refer to the table on page 207.

cos35''40' .8124

cos 35^60' = .8107

Tabular difference — .0017

"T^^^" cos 35^42' - .8124 - ^(.0017) - .8121

" cos 35^42' ^ .8107 + ^(.0017) = .8121

(c) Refer to the table on page 208.

tan82°60' = tan 83°0' = 8.1443 tan82°50' ^ 7.9530

Tabular difference = .1913

■^^^^ tan82°56' = 7.9530 +-^^(.1913) = 8.0678

17. Find (a)cot45°16', (6) sec 73°48', (c) esc 28°33'.

(a) Refer to the table on page 209.

cot45°10' = .9942 cot45°20' := .9884

Tabular difference = .0058 ^*^^" cot45°16' = .9942 - ^(.0058) = .9907

°' cot45'^I6' - .9884 + ^(.0058) = .9907

(6) Refer to the table on page 210.

sec73°50' - 3.592

sec73°40' = 3.556

Tabular difference =: .036

'^''^" sec 73^48' = 3.556 + ^(.036) = 3.585

(c) Refer to the table on page 211.

csc28'=30' - 2.096

csc28''40' = 2.085

Tabular difference = ,011

■"•^ csc28='33' - 2.096 - ^(.011) = 2.093

°' csc28°33' = 2.085+ ^^(.011) - 2.093

INVERSE TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)

18. Find (a) sin-i (-2143), (6) cos-> (.5412), (c) tan-i (1.1536).

(a) Refer to the table on page 206.

sin 12^30' = .2164

sinl2°20' = .2136

Tabular difference = .0028

Since .2143 is -^l-^^ ~ -2136 _ j ^^ ^^^ ^ between .2136 and .2164, the required angle is

.0028 *

12'*20' + i(lO') = 12°22.5'.

(6) Refer to the table on page 207.

cos57°10' ^ .5422

cos 57=^20' = .5398

Tabular difference - .0024 Then cos-M-5412) = 57°20'- i^^^^^^^^dO') - 57^4.2'

or eos->(.5412) - 57n0' + i^l?^=2^(10'} - 57^4.2'

(e) Refer to the table on page 208.

tan49°10' = 1.1571

tan49°0' ^ 1.1504

Tabular difference — .0067 Then tan-M1.1536) = 4900-+ 1-1536^-1.1504 ^^p,^ ^ ^904.8'

Other inverse trigonometric functions can be obtained similarly.

TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS (RADIANS)

19. Find (a) sin (.627), (6) cos (1.056), (c) tan (.153).

(a) Refer to the table on page 213.

sin (.630) - .58914

sin (.620) = .58104

Tabular difference — .00810

Then sin (.627) - .58104 + -^^(.00810) = .58671

(6) Refer to the table on page 214.

cos (1.050) = .49757

cos (1.060) = .48887

Tabular difference = .00870 Then cos (1.056) = .49757 - ^(.00870) = .49235

or cos (1.056) = .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.

20. Find sin"' (.512j in radians.

Refer to the table on page 213.

sin (.540) ~ .51414

sin (.530) = .50553

Tabular difference = .00861

.512 - .50553 .00861

Similarly the other inverse trigonometric functions are obtained.

Then sin-M.512) - .530 + iii^--J?^ f.Ol) - .5375 radians

COMMON LOGARITHMS OF TRIGONOMETRIC FUNCTIONS

21. Find (a) log sin 63 = 17', (6) log cos 48="44'.

(a) Refer to the table on page 217.

Iogsin63°20' = 9.9512 - XO

logsin63°10' = 9.9505-10

Tabular difference = .0007

Then log sin 63°17' = 9.9505 - 10 -f -j:^(.0007) = 9.9510 - 10

(6) Refer to the table on page 219.

log cos 48°40' = 9.8198 - 10

logcos48°50' = 9.8184-10

Tabular difference = .0014

Then logcos48°44' = 9.8198 - 10 - 3^(.0014) = 9.8192-10

o^ logcos48°44' ^ 9.8184 - 10 +-j^(.0014) = 9.8192-10

Similarly we can find logarithms of other trigonometric functions. Note that log sec x = -log cos x, log cot ic — -log tan a;, log esc ac — -log sin x.

22. If log tan a: - 9.6845-10, finds:.

Refer to the table on page 220.

Iogtan25°50' = 9.6850-10

]ogtan25°40' = 9.6817-10

Tabular difference = .0033

Then :. = 25°40- + ^-^«^5-9.6817 ^^Q,^ ^ ^^.^^ g,

.UUoo

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

CONVERSION OF RADIANS TO DEGREES, MINUTES AND SECONDS

24. Find 2.5-17 radians in degrees, minutes and seconds.

Refer to the table on page 222.

2 radians - 114° 35' 29.6"

.6 = 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

2G. Find (a) e->-^'*, (fc) e--i5B.

(a) Refer to the table on page 226.

e5.30 ^ 200.34 e5.2o ^ 181.27

Tabular difference = 19.07 Then «5.24 = I81.27 + -^(19.01) = 188.90

(6) Refer to the table on page 227.

e--i50 = ,86071

e-.i6o ^ .85214

Tabular difference = .00857 Then e" >S8 = .86071 - ^^(.00857) = .86385

or e->58 = .85214 + ^(.00857) = .85385

27. Find (ct) sinh (4.846), (b) sech (.163). (a) Refer to the table on page 229.

sinh (4.850) = 63.866 sinh (4.840) = 63.231

Tabular difference ~ .635 Then sinh (4.846) = 4.840 + ^(.635) = 5.221

(6) Refer to the table on page 230.

cosh (.170) = 1.0145

cosh (.160) = 1.0128

Tabular diflference = .0017

Then cosh (.163) = 1.0128 + ^(.0017) = 1.0133

and so sech (.163) = = — -— = .98687

^ cosh (.163) 1.0133

28. Find tanh-'(.71423).

Refer to the table on page 232.

tanh (.900) = .71630

tanh(.890) - .71139 Tabular difference = .00491

Then tanh-'(.71423) = .890 + ^HM^^L^Zll^dO) = .8958

.00491 '

INTEREST AND ANNUITIES

29. A man deposits $2800 in a bank which pays 5% 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 = $2800(1 +.0125)32 = $2800(1.4881) = $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 k - 10 • 2 ^ 20 payment periods at interest rate r = .06/2 = .03 per period, the present value is

P = $12,000(1 +.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

" (1 + .04)20 _ I .04

$500(29.7781) = $14,889.05

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

.015 using the table on page 243.

L .0

- (1.015)

= $120(34.0426) = $4085.11

TABLE

1

FOUR PLACE COMMON LOGARITHMS

logioA^ or log A^

TABLE

2

FOUR PLACE COMMON ANTILOGARITHMS

10^ or antilog p

Table 2

(continued)

FOUR PLACE COMMON ANTILOGARITHMS

10^ or antilog p

p

.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

0

3162 3170 3177 3184 3192

3236 3243 3251 3258 3266

3311 3319 3327 3334 3342

3388 3396 3404 3412 3420

3467 3475 3483 3491 3499

3548 3556 3565 3573 3581

3631 3639 3648 3656 3664

3715 3724 3733 3741 3750

3802 3811 3819 3828 3837

8890 3899 3908 3917 3926

3981 3990 3999

4074 4083 4093

4169 4178 4188

4266 4276 4285

4365 4375 4385

4467 4477 4487

4571 4581 4592

4677 4688 4699

4786 4797 4808

4898 4909 4920

5012 5023 5035

5129 5140 5152

5248 5260 5272

5370 5383 5395

5495 5508 5521

5623 5636 5649

5754 5768 5781

5888 5902 5916

6026 6039 6053

6166 6180 6194

6310 6324 6339

6457 6471 6486

6607 6622 6037

6761 6776 6792

6918 6934 6950

7079 7096 7112

7244 7261 7278

7413 7430 7447

7586 7603 7621

7762 7780 7798

7943 7962 7980

8128 8147 8166

8318 8337 8356

8511 8531 8551

8710 8730 8750

8913 8933 8954

9120 9141 9162

9333 9354 9376

9550 9572 9594

9772 9795 9817

4009 4018

4102 4111

4198 4207

4295 4305

4395 4406

4498 4508

4603 4613

4710 4721

4819 4831

4932 4943

5047 5058

5164 5176

5284 5297

5408 5420

5534 5546

5662 5675

5794 5808

5929 5943

6067 6081

6209 6223

6353 6368

6501 6516

6653 6668

6808 6823

6966 6982

7129 7145 7295 7311 7464 7482 7638 7656 7816 7834

7998 8017

8185 8204

8375 8395

8570 8590

8770 8790

8974 8995

9183 9204

9397 9419

9616 9638

9840 9863

0

3199 3206 3214 3221 3228

3273 3281 3289 3296 3304

3350 3357 3365 3373 3381

3428 3436 3443 3451 3459

3508 3516 3524 3532 3540

3589 3597 3606 3614 3622

3673 3681 3690 3698 3707

3758 3767 3776 3784 3793

3846 3855 3864 3873 3882

3936 3945 3954 3963 3972

4027 4036 4046 4055 4064

4121 4130 4140 4150 4159

4217 4227 4236 4246 4256

4315 4325 4335 4345 4355

4416 4426 4436 4446 4457

4519 4529 4539 4550 4560

4624 4634 4645 4656 4667

4732 4742 4753 4764 4775

4842 4853 4864 4875 4887

4955 4966 4977 4989 5000

5070 5082

5188 5200

5309 5321

5433 5445

5559 5572

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

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

5093 5105 5117

5212 5224 5236

5333 5346 5358

5458 5470 5483

5585 5598 5610

5715 5728 5741

5848 5861 5875

5984 5998 6012

6124 6138 6152

6266 6281 6295

6412 6561 6714 6871 7031

7194 7362 7534 7709 7889

8072 8260 8453 8650 8851

9057 9268 9484 9705 9931

6427

6577 6730

6887 7047

7211

7379 7551

7727 7907

8091 8279 8472 8670

8872

9078 9290 9506

9727 9954

6442 6592 6745 6902 7063

7228 7396 7568 7745 7925

8110 8299 8492 8690 8892

9099 9311 9528 9750 9977

9

Proportional Parts 123456789

2 3 4 4 5 6

3 3 3 3 3

3 3 3 4 4

4 4 4 4 4

2 4

2 4

2 4

2 4

12 4 1 2 4 1 2 4 1 3 4 1 3 4

13 4 1 3 4 1 3 4 1 3 4 1 3 4

4 5 5 5 5

5 5 5 5 5

6 6 6 6 6

6 6 7 7

2 3 4 6 5 6 7

4 5 6 6

4 5 6 7

4 5 6 7

4 5 6 7

4 5 6 7

4 5

4 5

4 5

4 6

5 6

5 6

5 6

5 6

5 6

5 6

5 7

5 7

5 7

6 7 6 7

6 7 9 10 12 13

6 8 9 11 12 14

6 8 9 11 12 14

6 8 9 11 13 14

6 8 10 11 13 15

7 8 10 12 13 15 7 8 10 12 13 15 7 9 10 12 14 16 7 9 11 12 14 16 7 9 11 13 14 16

2 5 7

7 9 11

8 9 11 8 10 12 8 10 12 8 10 12

8 10 12

8 11 13

9 11 13 9 11 13 9 11 14

13 15 17

13 15 17

14 15 17 14 16 18

14 16 18

15 17 19 15 17 19

15 17 20

16 18 20 16 18 20

12 3 4 5

1

2 3

4

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

0'

10'

20'

30'

1.0000 1.0000 1.0000 1.0000

.9998 .9998 .9997 .9997

.9994 .9993 .9992 .9990

.9986 .9985 .9983 .9981

.9976 .9974 .9971 .9969

.9962 .9945 .9926 .9903 .9877

.9848 .9816 .9781 .9744 .9703

.9659 .9613 .9563 .9611 .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

.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

.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

.9954 .9936 .9914 .9890 .9863

.9833 .9799 .9763

.9724 .9681

.9630 .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

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

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

45^

.7071 .7050 .7030 .7009 .6988 .6967

45"

46

47

48

49

50° 51 52 53

54

55°

56

57

58

59

60°

61

62

63

64

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

10'

.7050 .6926 .6799 .6670 .6539

.6406 .6271 .6134 .5995 .5854

.5712 .5568 .5422 .5275 .5125

.4975 .4823 .4669 .4514 .4358

20'

.7030 .6905 .6777 .6648 .6517

.6383 .6248 .6111 .5972 .5831

.5688 .5544 .5398 .5250 .5100

.4950 .4797 .4643 .4488 .4331

30'

.7009 .6884 .6756 .6626 .6494

.6361 .6225 .6088 .5948 .5807

.5664 .5519 .5373 .5225 .5075

.4924 .4772 .4617 .4462 .4305

40'

.6988 .6862 .6734 .6604 .6472

.6338 .6202 .6065 .5925 .5783

.5640 .5495 .5348 .5200 .5050

.4899 .4746 .4592 .4436 .4279

.2588 .2419 .2250 .2079 .1908

.1736 .1564 .1392 .1219 .1045

.0872 .0698 .0523 .0349 .0175

.2560 .2391 .2221 .2051 .1880

.1708 .1536 .1363 .1190 .1016

.0843 .0669 .0494 .0320 .0145

.2532

.2363 .2193 .2022 .1851

.1679 .1507 .1334 .1161 .0987

.0814 .0640 .0465 .0291 .0116

.2504

.2334 .2164 .1994 .1822

.1650 .1478 .1305 .1132 .0958

.0785 .0610 .0436 .0262 .0087

.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

.2447 .2278 .2108 .1937 .1765

.1593 .1421 .1248 .1074 .0901

.0727 .0552 .0378 .0204 .0029

.0000

207

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

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

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

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 .5668

.5890

.6128 .6371 .6619 .6873

.7133 .7400 .7673 .7954 .8243

.8541 .8847 .9163 .9490 .9827

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

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

1.0000 1.0058 1.0117 1.0176 1.0235 1.0295

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'

l.OOOO 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

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

11.430 11.826

14.301 14.924

19.081 20.206

28.636 31.242

57.290 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

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

13.197 13.727

17.169 18.075

24.542 26.432

42.964 49.104

171.89 343.77

208

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

0'

57.290 28.636 19.081 14.301

10'

343.77 49.104 26.432 18.075 13.727

11.430 11.059

9.5144 9.2553

8.1443 7.9530

7.1154 6.9682

6.3138 6.1970

20'

171.89 42.964 24.542 17.169 13.197

10.712 9.0098 7.7704 6.8269 6.0844

30'

114.59 38.188 22.904 1G.350 12.706

10.385 8.7769 7.5958 6.6912 5.9758

40'

85.940 34.368 21.470 15.605 12.251

50'

68.750 31.242 20.206 14.924 11.826

10.078 9.7882

8.5555 8.3450

7.4287 7.2687

6.5606 6.4348

5.8708 5.7694

5.6713 5,5764 5.4845 5.3955 5.3093 5.2257

5.1446 5.0658 4.9894 4.9152 4.8430 4.7729

4.7046 4.6382 4.5736 4.5107 4.4494 4.3897

4.3315 4.2747 4.2193 4.1653 4.1126 4.0611

4.0108 3.9617 3:9136 3.8667 3.8208 3.7760

45°

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

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

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

3.6059 3.3759 3.171G 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

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

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

1.4106 1.4019 1.3934 1.3848

1.3597 1.3514 1.3432 1.3351

1.3111 1.3032 1.2954 1.2876

1.2647 1.2572 1.2497 1.2423

1.2203 1.2131 1.2059 1.1988

1.1778 1.1369 1.0977 1.0599 1.0235

1.1708 1.1303 1.0913 1.0538 1.0176

1.1640 1.1237 1.0850 1.0477

1.0117

1.1571 1.1171 1.0786 1.0416 1.0058

1.0000 .9942 .9884 .9827 .9770 .9713

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

0'

10'

20'

30'

40'

50'

1.000 .9942 .9884 .9827 .9770 .9713

.9657 .9601 .9545 .9490 .9435 .9380

.9325 .9271 .9217 .9163 .9110 .9057

.9004 .8952 .8899 .8847 .8796 .8744

.8693 .8642 .8591 .8541 .8491 .8441

.8391 .8342 .8292 .8243 .8195 .8146

.8098 .8050 .8002 .7954 .7907 .7860

.7813 .7766 .7720 .7673 .7627 .7581

.7536 .7490 .7445 .7400 .7355 .7310

.7265 .7221 .7177 .7133 .7089 .7046

.7002 .6959

.6745 .6703

.6494 .6453

.6249 .6208

.6009 .5969

.5774 .5735

.5543 .5505

.5317 .5280

.5095 .5059

.4877 .4841

.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

.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

.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

.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

.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

.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

90° .0000

209

1

2 3 4

5°

6 7 8 9

10°

11

12

13

14

16 17 18 19

20°

21

22

23

24

0'

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

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

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

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

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

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

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

1.103 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

1.155 1.153

1.143 1.142

1.133 1.131

1.122 1.121

1.113 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

1.097 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

1.147 1.145

1.136 1.134

1.126 1.124

1.116 1.114

1.106 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

TABLE

9

NATURAL TRIGONOMETRIC FUNCTIONS (in radians)

Table?

(continued)

NATURAL TRIGONOMETRIC FUNCTIONS (in radians)

Table 9

(continued)

NATURAL TRIGONOMETRIC FUNCTIONS (in radians!

TABLE

10

log sin X (a: in degrees and minutes)

[subtract 10 from each entry]

Table 10

(continued)

lOQ sin X {x in degrees and minutes)

[subtract 10 from each entry]

TABLE

n

log COS X {x in degrees and minutes)

[subtract 10 from each entry]

Table 11

(continued)

■Og cos X {x in degrees and minutes)

[subtract 10 from each entryl

TABLE

12

log ton X (a; in degrees and minutes)

[subtract 10 from each entry]

Table 12

(continued)

log ton X (a; in degrees and minutes)

[subtract 10 from each entry]

TABLE

13

CONVERSrON OF RADIANS TO DEGREES, MINUTES AND SECONDS OR FRACTIONS OF DEGREES

TABLE

14

CONVERSION OF DEGREES, MINUTES AND SECONDS TO RADIANS

TABLE

15

NATURAL OR NAPIERIAN LOGARITHMS

log^ X or In X

X

0

6

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

.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

.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

.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

.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

.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

.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

.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

.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

.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

.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

Table 15

(continued)

NATURAL OR NAPIERIAN LOGARITHMS

log, X or In X

Table 18a

(continued)

HYPERBOLIC FUNCTIONS

sinh X

Table 18b

(con tinned)

HYPERBOLIC FUNCTIONS

cosh X

Table 18c

(continued)

HYPERBOLIC FUNCTIONS

tanh X

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

nl

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

1,124,000,727,777,937,680,000

25,852,016,738,892,566,840,000

620,448,401,733,421,599,360,000

15,511,210,043,336,539,984,000,000

403,291,461,126,724,039,584,000,000

10,888,869,450,421,549,068,768,000,000

304,888,344,611,803,373,925,504,000,000

8,841,761,993,742,297,843,839,616,000,000

265,252,859,812,268,935,315,188,480,000,000

8.22284 X 1033 2.63131 X 1035 8.68332 X 1036 2.95233 X 1038

1.03331 X 10-to 3.71993 X 10^1 1.37638 X 10« 5.23023 X 10^" 2.03979 X 1046

234

TABLE

21

BINOMIAL COEFFICIENTS

w! _ n{n-!)■ • ■jn-k + l) _

klin-k)\ ~ kl

n n~k

0!=1

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 41.

Table 21

(continued)

BINOMIAL COEFFICIENTS

n! _ n(w - 1) - ■ • (n- A: + 1)

kl{n-k)\ " ft!

n n — k

0! = 1

For k > 16 use the fact that

71

n — k

Tabre 22 (continued)

SQUARES, CUBES, ROOTS AND RECIPROCALS

COMPOUND AMOUNT: (1 + r)"

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 = Pd+r)-*.

TABLE

24

PRESENT VALUE OF AN AMOUNT: (1 + r) "

The present value P which will amount to A in « years at an interest rate of r (in decimals) compounded annually is P =^ -A(l + r)".

1%

li%

i^-%

2%

H'

3%

4'c

6%

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

.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

.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

.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 .51939 .51171 .50415 .49670 .48936 .48213 .47500

.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

.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

.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

.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 .26342 .24367 .23430 .22529 .21662 .20829

.20028 .19257 .18517 .17805 .17120 .16461 .15828 .15219 .14634 .14071

.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

.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

AMOUNT OF AN ANNUITY:

(l + r)«-l

If a principal P is deposited at the 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.

TABLE

26

PRESENT VALUE OF AN ANNUITY:

1- (l + r)-"

An annuity in which the yearly payment at the end of each of n years is A at an

Tl — (1 -f- r)~" interest rate r (in decimals) compounded annually has present value A

BESSEL FUNCTIONS

lAx)

BESSEL FUNCTIONS

Ker (x)

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 hy v = 3.14159... .

TABLE

40

EXPONENTIAL, SINE AND COSINE INTEGRALS

Ei{x) = C ^du, St{x) = r ^^^du, Ci{x) =: f

cos«

du

TABLE

41

LEGENDRE POIYNOMIALS Pn{x) [Po{x) = l, Piix) = x]

TABLE

LEGENDRE POLYNOMIALS P„(cosd) [Po(cos5) = l]

TABLE

43

COMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS

K

, _ - , E = \ yj\ ~- &2 sin^ e d9, fc = sm t&

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

K

E

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

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

TABLE

44

INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND

J"** dd

0 y\ — k^ %m^B

TABLE

45

INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND

J,* \/l - fc2 sin^ e dd, k = sin^ 0

TABLE

46

ORDINATES OF THE STANDARD NORMAL CURVE

1

y =

^/%^

-x'/2

TABLE

47

AREAS UNDER THE STANDARD NORMAL CURVE

from —=o tox

erf(a:) =-^ C e-^^^ dt

Source: R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research (6th edition, 1963), Table III, Oliver and Boyd Ltd., Edinburgh, by permission of the authors and publishers.

Source: Catherine M. Thompson, Table of percentage points of the x^ distribution, Biometrika, Vol. 32 (1941), by permission of the author and publisher.

9Sth PERCENTILE VALUES FOR THE > DISTRIBUTION

tti = degrees of freedom for numerator na = degrees of freedom for denominator (shaded area = .95)

Source: G. W. Snedecor and W. G. Cochran, Statistical Methods (6th edition, 1967), Iowa State University Press, Ames, Iowa, by permission of the authors and publisher.

Source: G. W. Snedecor and W. G. Cochran, Statistical Methods {6th edition, 1967), Iowa State University Press, Ames, Iowa, by permission of the authors and publisher.

261

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 wiU be clear from

the context.

Symbols

Ber„(a:), Bei„(a!) 140

B{m,n) beta function, 103

B„ Bernoulli numbers, 114

C{x) Fresnel cosine integral, 184

Ci{x) cosine integral, 184

e natural base of logarithms, 1

61,62,63 unit vectors in curvilinear coordinates, 124

erf (x) error function, 183

erfc (x) complementary error function, 183

E = E{k, it/2) complete elliptic integral of second kind, 179

Eik, ,p) incomplete elliptic integral of second kind, 179

EHx) exponential integral, 183

E„ Euler numbers, 114

F(a, b; c; x) hypergeometric function, 160

F{k, 0) incomplete elliptic integral of first kind, 179

y, ^-1 Fourier transform and inverse Fourier transform, 175, 176

^1, fh, Aa scale factors in curvilinear coordinates, 124

Hnix) Hermite polynomials, 151

H^l\x), H'-1\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

Jf^{x) Bessel function of first kind, 136

K = F{k.Trl2) complete elliptic integral of first kind, 179

Ker„ (a:), Kei„ {x) 140

K^{x) modified Bessel function of second kind, 139

In X or loge X natural logarithm of x, 24

log X OT logio X common logarithm of x, 23

L^{x) Laguerre polynomials, 153

Ln {«) associated Laguerre polynomials, 155

^, ^-1 Laplace transform and inverse Laplace transform, 161

P^{x) Legendre polynomials, 146

^^(2:) associated Legendre functions of first kind, 149

Qnix) Legendre functions of second kind, 148

Q"(x) associated Legendre functions of second kind, 150

r cylindrical coordinate, 49

r polar coordinate, 22, 36

r spherical coordinate, 50

Six) Fresnel sine integral, 184

Si(x) sine integral, 183

r„(a:) Chebyshev polynomials of first kind, 157 U„{x) Chebyshev polynomials of second kind, 158 YAx) Bessel function of second kind, 136

INDEX OF SPECIAL SYMBOLS AND NOTATIONS

Greek Symbols

y Euler's constant, 1

r{x) gamma function, 1, 101

f(a;) Riemann zeta function, 184

6 cylindrical coordinate, 49

e polar coordinate, 22, 36

e spherical coordinate, 50

«■ 1

•p spherical coordinate, 60

*(p) the sum 1 + - + ^ + ■ ■

'^{x) probability distribution function, 189

+ -, *(0) = 0, 137

Notations

A = B A equals fi or A is equal to B

A > B A is greater than B [or B is less than A]

A < B A is less than B [or B is greater than A]

A ^ £ A is greater than or equal to B

A '^ B A is less than or equal to B

A "^ B A is approximately equal to B

A -^ B A is asymptotic to B or A/B approaches 1, 102

A if A ^ 0

\A\ absolute value of A = n \ factorial n, 3

'' = g=/».

y =

_ d^y_

dx2

= f"{x), etc.

Dp =

dxP dy

df sf d-^f

, etc.

dx' dy' dx dy

d{x. y, z)

a(ui,M2,«3)

/

X

f{x) dx \'dt

A>B

AXB

V

V2 = V • V

V* = V2{V2)

-A if A ^ 0

binomial coefficients, 3

derivatives of y or f{x) witti respect to x, 53, 55

pth derivative with respect to x, 55 differential of y, 55 partial derivatives, 56

Jacobian, 125 indefinite integral, 67

definite integral, 94

line integral of A along C, 121

dot product of A and B, 117 cross product of A and B, 118

del operator, 119 Laplacian operator, 120 biharmonic operator, 120

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, X49

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, 107,114,116

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, 169

265

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 [sec 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, 26 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, 64

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]

267

Differentiation (cont.)

general rules for, 63

of integrals, 95 Diodes, 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, 261 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

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, 161, 153,

155, 157, 158 Geometric formulas, 6-10 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, 162

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, 67 [see also Integrals]

constants of, 57

general rules of, 57-59 Integration by parts, 57

generalized, 69 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

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 £:raphs 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, 164

orthogonality of, 154

recurrence formulas for, 163

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

INDEX

Orthogonality and orthogonal aeries, 144, 145,

147, 150, 152, 154, 156, 168, 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, 62

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, 36

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

liable 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

271

Series, arithmetic (cont.) orthogonal [see Orthogonality and orthogonal series] power, HO, 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, 96 Triangle, plane [see Plane triangie]

spherical [see Spherical triangle]

Triangle inequality, 185 Triangular wave function, 172 Trigonometric functions, 11-20

addition formxilas 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

z intercept, 34

y axis, 11

y intercept, 34

Zero vector, 116

Zeros of Bessel functions, 250

Zeta function of Riemann, 184

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