Handbook of Elliptic Integrals for Engineers and

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berlicksichtigung der Anwendungsgebiete Band 67

Herausgegeben von

J. L. Doob

. A. Grothendieck . E. Heinz . F. Hirzebruch E. Hopf . H. Hopf . W. Maak . S. Mac Lane . W. Magnus J. K. Moser' M. M. Postnikov' F. K. Schmidt· D. S. Scott K. Stein

GeschaJtsjiihrende Herausgeber B. Eckmann und B. L. van der Waerden

Paul F. Byrd' Morris D. Friedman

Handbook of Elliptic Integrals for Engineers and Scientists Second Edition, Revised

With 22 Figures

Springer-Verlag Berlin Heidelberg New York 1971

Paul F. Byrd· Morris D. Friedman Aeronautical Research Scientist National Advisory Committee For Aeronautics (U. S. A.)

Geschaftsfiihrende Herausgeber:

Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zurich

AMS Subject Classifications (1970): 33 A 25

ISBN -13: 978-3-642-65140-3

e-ISBN -13: 978-3-642-65138-0

DOl: 10.1007/978-3-642-65138-0

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro-

duction by photocopying machine or similar means, and storage in data banks. Under § 54 01 the German Copyright Law where copies are made lor other than private use, a lee is payable to the publisher, the amount 01 the lee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1954 and 1971· Library 01 Congress Catalog Card Number 72-146515 . Softcover reprint of the hardcover 2nd edition 1971

Paul F. Byrd' Morris D. Friedman

Handbook of Elliptic Integrals for Engineers and Scientists Second Edition, Revised

With 22 Figures

Springer-Verlag New York Heidelberg Berlin 1971

Paul F. Byrd· Morris D. Friedman Aeronautical Research Scientist National Advisory Committee For Aeronautics (U. S. A.)

Geschaftsfiihrende Herausgeber:

Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zurich

AMS Subject Classifications (1970): 33 A 25

ISBN -13: 978-3-642-65140-3

e-ISBN -13: 978-3-642-65138-0

DOl: 10.1007/978-3-642-65138-0

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro-

duction by photocopying machine or similar means, and storage in data banks. Under § 54 01 the German Copyright Law where copies are made lor other than private use, a lee is payable to the publisher, the amount 01 the lee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1954 and 1971· Library 01 Congress Catalog Card Number 72-146515 . Softcover reprint of the hardcover 2nd edition 1971

To Bonita, Paul Jr., Bruce, Birgit, Richard and Isolde

Preface to the First Edition. Engineers and physicists are more and more encountering integrations involving nonelementary integrals and higher transcendental functions. Such integrations frequently involve (not always in immediately recognizable form) elliptic functions and elliptic integrals. The numerous books written on elliptic integrals, while of great value to the student or mathematician, are not especially suitable for the scientist whose primary objective is the ready evaluation of the integrals that occur in his practical problems. As a result, he may entirely avoid problems which lead to elliptic integrals, or is likely to resort to graphical methods or other means of approximation in dealing with all but the simplest of these integrals.

It became apparent in the course of my work in theoretical aerodynamics that there was a need for a handbook embodying in convenient form a comprehensive table of elliptic integrals together with auxiliary formulas and numerical tables of values. Feeling that such a book would save the engineer and physicist much valuable time, I prepared the present volume. Although the book is not a text, an attempt has been made to write it in elementary terms so that no previous knowledge of elliptic integrals, theta functions or elliptic functions is needed. A collection of over 3000 integrals and formulas, designed to meet most practical needs, is presented using Legendre's and Jacobi's notations, rather than the less familiar Weierstrassian forms. Many of these formulas are substitutions and recurrence relations for evaluating additional integrals which are not explicitly written. Sufficient explanatory material and cross-references are given to permit the reader to obtain the answers he requires with a minimum of effort. Short tables of numerical values are given for the elliptic integrals of the first and second kind, for Jacobi's "nome" q, for the function denoted by Heuman as A o , and for K times the Jacobian Zeta function. Tables of the last three functions are useful in the numerical evaluation of elliptic integrals of the third kind. Particular precautions, of course, have to be taken in a work of this kind to insure accuracy of the formulas. My co-author, Mr. MORRIS

VIII

Preface.

D. FRIEDMAl:', undertook the job of verifying each formula. Where ever possible, they were either derived independently in different ways or checked against more than one source. Criticisms of the material contained in the handbook and notice of any errors which may yet appear in it will be sincerely welcomed. It is impossible to acknowledge properly all the sources to which debt is owed. The bibliography, however, lists many books in which the derivation of some of the formulas can be found or where related material may be obtained. For friendly advice and valuable suggestions, I am under obligation to Professors A. ERDELYI, W. MAGNUS and R. C. ARCHIBALD, and to colleagues in the Theoretical Aerodynamics Section, Ames Aeronautical Laboratory, NACA. To my colleague, DORIS COHEN, I am grateful for a critical reading of the manuscript and for many suggestions leading to improvement of exposition and organization. Hearty thanks are also extended to Mrs. ROSE CHIN BYRD and Mrs. MARY T. HUGGINS for assistance in the task of preparing the tables of numerical values and to Mr. DUANE W. DUGAN for help in reading the proofs.

On behalf of both authors I wish finally to express gratitude to Springer-Verlag and to Professor K. KLOTTER of Stanford University who kindly called their attention to our work. The appearance of this book is in no small measure due to their cooperative attitude, their encouragement, and their genuine interest in the promotion of technical publications. Palo Alto, California. July 1953·

PAUL F. BYRD.

Preface to the Second Edition. I wish to acknowledge my debt to many correspondents for lists of errata and suggestions for improvement. All errors that have come to my attention have been corrected, but I shall be happy to receive any further corrections or suggestions. Since the appearance of the first edition in 1954, numerous computational methods for efficient calculation of the standard elliptic integrals and the Jacobian elliptic functions have been published. However, to improve this handbook by augmenting its contents with such work would actually require another volume and must be deferred. The need remain~ for an extensive table of integrals that can be expressed as combinations of standard elliptic integrals before one goes to the computer for numerical evaluation. Consequently, the first edition is reproduced essentially without change except for corrections. A new Supplementary Bibliography, however, gives many references to numerical approximations and computational algorithms. Saratoga, California. January 1971.

P. F. B.

Table of Contents. Page

VII

Preface to the First Edition. . . Preface to the Second Edition. .

IX

List of Symbols and Abbreviations

.XIV

Introduction. . . . . . . . . . Definitions and Fundamental Relations

8

110. Elliptic Integrals . . . . . . Definitions, p. 8. - Legendre's relation, p. 10. - Special values, p. 10. Limiting values, p. 11. - Extension of the range of tp and k, p. 12. Addition formulas p. 13. - Special addition formulas, p. 13. - Differential equations, p. 15. - Sketches of E(rp, k), F(rp, k), E(k) and K(k), p. 16. - Conformal Mappings, p. 17.

8

120. Jacobian Elliptic Functions. . . . . . . . . . . . . . . . . . . Definitions, p. 18. - Fundamental relations, p. 20. - Special values, p. 20. - Addition formulas, p. 23. - Double and half arguments, p. 24. - Complex and imaginary arguments, p. 24. - Relation to Theta functions, p. 24. - Approximation formulas, p. 24. - Differential equations, p. 25. - Identities, p. 25. - Sketches, p. 26. Conformal Mappings, p. 28. - Applications, p. 28.

18

130. Jacobi's Inverse Elliptic Functions. . . . . .' . . . . . . . . . . Definitions, P.29. - Identities, p. 31. - Special values, p. 31. Addition formulas, p. 32. - Special addition formulas, p. 32.

29

140. Jacobian Zeta Function . . . . . . . . . . . . . . . . . Definitions, p. 33. - Special values, p. 33. ----'Maximum value, p. 34.Limiting value, p. 34. - Approximation formula, p. 34. - Addition formulas, p. 34. - Special addition formula, p. 34. - Complex and imaginary arguments, p. 34. - Relation to Theta functions, p. 34. Sketches, p. 35. 150. Heuman's Lambda Function . . . . . . . . . . . . . . . . . . Definitions, p. 35. - Special values, p. 36. - Limiting value, p. 36.Addition formula, p. 36. Special addition formulas, p. 36. Relation to Theta functions, p. 37. - Sketches, p. 37. 160. Transformation Formulas for Elliptic Functions and Elliptic Integrals Imaginary modulus transformation, p. 38. - Imaginary argument transformation, p. 38. - Reciprocal modulus transformation, p. 38. Landen's transformation, p. 39. - Gauss' transformation, p. 39. Other transformations, p. 40.

33

Reduction of Algebraic Integrands to Jacobian Elliptic Functions

200. Introduction

. . . . . . . . . . . . . . . . . . .

210. Integrands Involving Square Roots of Sums and Differences of Squares Introduction, p.43. - Table of Integrals, p. 45.

35

38

42 42 43

Table of Contents.

XII

Page

230. Integrands Involving the Square root of a Cubic. Introduction p. 65. - Table of Integrals p. 68.

65

250. Integrands Involving the Square root of a Quartic. Introduction p. 95. - Table of Integrals p. 98.

95

270. Integrands Involving Miscellaneous Fractional Powers of Polynomials 148 Reduction of Trigonometric Integrands to Jacobian Elliptic Functions

162

Reduction of Hyperbolic Integrands to Jacobian Elliptic Functions

182

Tables of Integrals of Jacobian Elliptic Functions. . . . . . . .

191

310. Recurrence Formulas for the Integrals of the Twelve Jacobian Elliptic Functions. . . . . . . . . . . 191 330. Additional Recurrence Formulas. . . . . . .

198

360. Integrands Involving Various Combinations of Jacobian Elliptic Functions . . . . . . . . . . . . . . . . 211 390. Integrals of Jacobian Inverse Elliptic Functions.

221

Elliptic Integrals of the Third Kind

400. Introduction

223

. . . . . .

223

410. Table of Integrals . . . . Complete integrals, p. 225. -- Incomplete integrals, p. 232. Table of Miscellaneous Elliptic Integrals Involving Trigonometric or Hyperbolic Integrands

224

240

510. Single Integrals .

240

530. Multiple Integrals

245

Elliptic Integrals Resulting from Laplace Transformations

249

Hyperelliptic Integrals.

252

575. Introduction

252

576. Table of Integrals

256

Integrals of the Elliptic Integrals

272

610. With Respect to the Modulus

272

630. With Respect to the Argument

276

Derivatives

710. With Respect to the Modulus . Differentiation of the elliptic integrals, p. 282. the Jacobian elliptic functions, p. 283.

282 282 Differentiation of

730. With Respect to the Argument . . . . . . . . . . . . . . . . . 284 Differentiation of the elliptic integrals, p. 284. - Differentiation of the Jacobian elliptic functions, p.284. Differentiation of the Jacobian inverse functions, p.285. 733. With Respect to the Parameter . . . . . . . . . . . . . . . 286 Differentiation of the normal elliptic integral of the third kind, p. 286.Differentiation of other elliptic integrals, p. 287.

Table of Contents.

XIII Page

Miscellaneous Integrals and Formulas

288

Expansions in Series . . . . . • .

298

298 900. Developments of the Elliptic Integrals Complete elliptic integrals of the first and second kind, p. 298. - The nome, p. 300. - Incomplete elliptic integrals of the first and second kind, p. 300. - Heuman's function, p. 301. - Jacobian Zeta function, p. 301. - The elliptic integral of the third kind, p. 302.

907. Developments of Jacobian Elliptic Functions. . . . . . . . . . . 303 Maclaurin's series, p. 303. - Fourier series, p. 304. - Infinite products, P.306. - Other developments, p. 307. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . '0 • • • 308 1030. Weierstrassian Elliptic Functions and Elliptic Integrals . . . . . . 308 Definition, p. 308. - Relation to Jacobian elliptic functions, p. 309. Fundamental relations, p. 309. - Derivatives, p. 309. - Special values, p. 310. - Addition formulas, p. 310. - Relation to Theta functions, p. 310. - Weierstrassian normal elliptic integrals, p. 311.Other integrals, p. 312. - Illustrative example, p. 313.

1050. Theta Functions . . . . . . . . . . . . . . . . . . . . . . . 31 5 Definitions, p. 315. - Special values, p. 316. - Quasi-Addition Formulas, p. 317. - Differential equation, p. 317. - Relation to Jacobian elliptic functions, p. 318. - Relation to elliptic integrals, p. 318. 1060. Pseudo-elliptic Integrals. . . . Definition, p. 320. - Examples, p.321.

320

Table of Numerical Values 322 Values of the complete elliptic integrals K and E, and of the nome q with respect to the modular angle, p. 323. - Values of the complete elliptic integrals Ir, K', E, E', and of the nomes q and q' with respect to k Z, p. 324. - Values of the incomplete elliptic integral of the first kind, F(rp, k), p. 325. - Values of the incomplete elliptic integral of the second kind, E(rp, k), p. 331. - Values of the Function KZ(P, k), p. 337. - Values of Heuman's function Ao(/J, k), p. 345. Bibliography. . . . . . . .

351

Supplementary Bibliography

353

Index

355

List of Symbols and Abbreviations. The following table comprises a list of the principal symbols and abbreviations used in the handbook. Notations not listed are so well understood that explanation is unnecessary. Symbol or Abbreviation

== amu

am (u, k)

am- l (y,k)

Meaning Parameter of elliptic integral of the third kind

110

Amplitude u

120

Inv.erse amplitude y

130

cnu dnu

cdu cd- l (y, k) cn (u, k) "'" cn u

Cosine amplitude u; Jacobian elliptic function

cos-l,p

cs- l (y, k) dcu dc- l (y, k)

cnu sn u

120

dnu cnu

120

Delta amplitude u; Jacobian elliptic function

dn u (y, k)

dnu snu

ds u (y, k)

Roots of polynomial written in Weierstrassian form

E(91,k) co::E(u)

E' (91, k)

~

E (91, k')

~

E (k) =: E

E'

== E (n/2, k)

E(k')

F (91, k)

120

Inverse trigonometric function, often written arc cos 91

cs u

ds- l

120 130

130

cn-l (y, k)

dn-l

Section

"=

u

Legendre's incomplete elliptic integral of the second kind; (91 = amu) Associated incomplete elliptic integral of the second kind Complete elliptic integral of the second kind Associated complete elliptic integral of the second kind Incomplete elliptic integral of the first kind

130 130 120 130 120 130 1030

110

xv

List of Symbols and Abbreviations. Meaning

Symbol or Abbreviation

Section

00

== " (a)m (b)m L..J (c)mm!

F(a,b;c;z)

zm, hypergeometric series

900

m=O 00

= "

G

L..J

m=O

( (- 1)m)2 "'" 0.91596559, Catalan's 2m 1

+

615

constant

r(z)

== :

tr [(1 + ~ )'(1 + ~-rl m=l

=F

Z

1030

0, -

1, - 2, ... , Gamma function

1050

Eta functions of Jacobi 00

~L m!r(y~m+1) (~r+2m 1n=O

== e- Y 1Y >Yl >a. Here I may also be split into two integrals, e.g.,

)/

f Y (t )/.

f

00

dt l -

al) (t 2

-

bl )

=

)I.

If(t2-

f

00

dt al) (t l

Hence we may employ 215.00 twice.

-

bl) -

Y

Y (t 2

-

dt a2 ) (t 2

-

bl) •

Introduction.

3

The above also applies to the table of integrals involving trigonometric or hyperbolic integrands. We give now the following five examples to illustrate in detail how the handbook may be used for rapid evaluation of elliptic integrals encountered in geometrical and physical problems.

ExampleI. The length of arc of a hyperbola (1 ) measured from the vertex to any point (x, y) is determined by the integral

(2)

s=

V ;P f

where a2 = fJ2,

(3)

OC2

y

2

o

b2 -

OC2

P' 2 +p

'

(b < a).

From 221.04: and 313.02, it is seen that

Example II. An integral which arises in finding the decrease in lifting pressure near the tip of a sweptback wing flying at supersonic speed has the form

Introduction.

4

From 257.39, with a=t1 , b=to, c=-~, d=-1, m=1 andp={JY/X, we obtain (noting that the integral is complete)

J

1 - (b-a) sn 2 u

K

I

(8)

=

2

(p - a)

where k2

(9)

Since 1 > I

(10)

1

V(a -

c) (b - d)

= Ja -

=

b) (c - d) (a - c) (b - d)

ra =: :~ ~~ =: ~ =

b-

°

du

d

(p - d) (a - b) su2 U (a-p)(b-d)

1 _

'

(t1 - to) (1 - t1) . 2t1 (to 1)

+

> k2 , we use 413.06. Equation (8) thus becomes

____ ~ .. ~~_

c)0-

(P - d) V(a -

d)

2XK

:rtAo(~.k) V(a - p) (p - d) (b - P) (P -

V(X' t~ -

fl2 Y2) (X

c)

+ fl Y) (X to -

fl Y) ,

where (11)

~ = sin-1VJP -

c) (b - d)_ (P - d) (b - c)

= sin-1

(fl Y (fl Y

+ 11 X) (to + 1) + X) (to + t l)

and Ao (~, k) is defined in ISO.

Example III. To obtain the gravitational potential V of a homogeneous solid ellipsoid at the exterior point (X, Y, Z), one is led to the integral

(12)

I

v=

n f2 rx {J y

J(

1-

y,

(/.;~ t -

(J2: t -

I/~ t) X

dt

X

V((/.2 + t) (fl2 + I) (1'2 + I)

where f2 is the density and Yl is the largest root of (

13

+ Y1 +

X2

) (/.2

fl2

Z.

y2

+ Y1 + y2 + Y1

=

1,

(2 (J2 2) rx > >y .

Equation (12) thus involves the evaluation of four integrals of the form 00

(14)

I=f y,

R(t)dl V(t - a) (I - b) (I - c) ,

where a = _y2, b = - {J2, C= -rx2, (Yl > a> b > c) and R(t) is a rational function of t. From 238.00 we have immediately

Introduction.

5

where (16)

With m= 1, one obtains next, from 238.03 and 310.02,

!

!sn udu

~

~

dt = 2 (t-c)V(t-a)(t- b) (t-c) (a-c)Va-c

)/,

=

(17)

2

kB(a-c)Va- c 2 [u1

-

2

0

[u _ E(u)]~' 2 [F(q>, k) - E(q>. k)]

E(am u 1 , k)]

k 2 (a - c) Va - c

kl (a - c) Va - c

From 238.04 and 318.02 it is readily seen that

!

~

~

(18)

dt (t-b)V(t-a)(t- b) (t-c)

= =

[E() u1

2 k l k'2(a- c)Va-c 2

k 2 k'l(a- c)Va-c

=

2

2

(a-c)Va-c

k'2 u1 - k2

-

...

!sd udu 0

snUICnul dnu1

[E(q;, k) _ k'2F(q;, k) _ k2

]

sinq>cosq> ]

V1-k 2 sin2 q>

,

and with 238.05 and 316.02 that

~

(19)

{! ~

dt (t - a) V (t - a) (t - b) (t - c) =

2

=

k'l(a- c)

2

(a - c) Va - c

!"~n2UdU 0

i [tanq;V1-k2sin2q;-E(q;,k)],

where k andq; are given by (16). We finally have, making use of (15)-(19),

(20)

Introduction.

6

ExampleIV. An integral which occurs in finding the spanwise distribution of circulation on a slender wing mounted on a cylindrical body is

(21)

J=

J y

ex

or

tl

(14 - (J') dt cr(1) (t l (J'fcr.l ) ,

V(tl -

(22) Using 216.06 and 216.09, we may at once write (22) as

J=

(23)

[j'dC

ex.

2U

du - k 2 !~d2 U duj,

where (y2 _ cr.!) cr.1 cr.2 yl_{J' .

cp = amu1 = sin-1

(24)

The two elliptic integrals in (23) are evaluated by employing 321.02 and 320.02. We then have

I

J = ex. [u1 -

(25)

E(Ur)

= ex. k' 2 tn ~e1 nd

+ dnu Ur =

tnu1

U1 cr.k'Btanm

1

-

+. E(u1) -

k 2 snUr cdUr]

-:;;==~~'r=

V1 -

k 2 sin2 qJ ,

or, applying (24), (26)

The final evaluation thus involves only elementary functions. Integral (21) is an example of an elementary integral which has all the appearance of being elliptic. That this would lead to elementary functions, however, could have been shown in another way. For, if we substitute t 2/ex. 2 = t1 , it may be written

(27) where

k 2 = OC4/f34 and R(tr) = cr.' t~ - 2 P!.2cr.fJ

I

tl

•

Equation (27) now has the form of 1060.05 and satisfies the relation R(t1)

hence it is pseudo-elliptic.

+ R( 1/k~ t1) =

0;

Introduction.

7

Example V. In considering the motion of a simple pendulum whose angular displacement from the position of stable equilibrum is cpo one is led to an integral of the type

t=

(28)

f

'P

o

d{}

1"a+ bcos{} •

where a and b are certain constants which depend on the acceleration of gravity. on the value of dcpJdt at cp=o. and on the length of the pendulum. There are three different types of motion according as a >b. a b > O. one employs 289.00 and obtains immediately (29)

where k 2 =2bJ(a+b). When b >a >0, we have from 290.00

t = V+F(e,k)

(3 0)

where k 2 =(a+b)J2b, and e=sin- 1 ~1a-+C~SIP)

.

Finally, if a = b, there results from either 289.00 or 290.00

t = _1_F(!L

(3 1)

1"2 a

2

1).

which, upon using 111.04, reduces to

t=

1 1/f2a

In (IP tan2

+ sec-IP) . 2

Definitions and Fundamental Relations. Elliptic Integrals. Definitions. The integraP 110.01

is called an elliptic integral if the equation

(a o and a1 not both zero), has no multiple roots and if R is a rational function of t and of the square root Va o t4 + a 1 t3 + a 2 t2 + a3 t + a4 •

The Three Canonical Forms of the Elliptic Integrals. It is always possible to express 110.01 linearly in terms of elementary functions 2 and of the following three fundamental integrals: The normal elliptic integral of the first kind:

110.02

The normal elliptic integral of the second kind:

110.03

1 There are special cases of integrals in this form that can be expressed in terms of a finite number of elementary functions. Such integrals are said to be pseudoelliptic (see 1060). 2 Elementary functions are algebraic, trigonometric, inverse trigonometric, logarithmic and exponential functions.

Elliptic Integrals.

9

The second form of each of the above integrals is written in Legendre's notation, while the first and third forms are in Jacobi's notation. In their algebraic forms, these three standard integrals possess the following important properties: The first is finite for all rea: or complex values of y, including infinity; the second has a simple pole of order 1 for y = 00; and the third is logarithmically infinite for y2 = 1/oc2 . The Modulus. The number k is called the modulus. This number may take any real or imaginary value in theoretical investigations. In applications to engineering and physics, however, transformations are generally employed to make 0 < k < 1 (d. 160 and 162). The Complementary Modulus. The number k' is referred to as the complementary modulus and is related to k by 110.05 k' = k2 •

V1-

The Argument. The variable limit y or q; in 110.02-110.04 is the argument of the normal elliptic integrals. The argument may of course be either real or complex, but it is usually understood that O 1 .

K(O) = K' (1) = n/2, E(O) =E'(1) =n/2.

I {

tanh-1 (Vex· _- 1 tanlP) , VexS - 1

if a.2 < 1

1 +k'

2

+ (1- k')].

= sincp,

= In (tancp + seccp),

II( cp, a.2 ,1) = -1- [In (tan cp + sec cp) S 1-ex

IX

In

Vi +

exsin lP ] ----,---

1 - exSInIP'

[ex· > 0, =_1_. i-ex

[In(tan m T

ex2

* 1] ,

+ seccp) + lexlarctan(lexlsincp)J,

[ex2 < 0].

Elliptic Integrals.

111.05

11

E(1I:12,1) =E(1) =E'(O) = 1, { F (11:/2,1) = K(1) = K' (0) = 00, II (11:12, (1.2, 1) = II ((1.2, 1) = 00. II(q;, 0, k)

= F(q;, k),

111.06

1

111.07

F(iq;,O) =iF({}, 1) =iln(tan{} +sec{}) =iq;, [sinhq;= tan{}] { E(iq;,O) = iln (tan{) + sec{}) = iq;, II(iq;, oc2 , 0) = i [In (tan {} +sec{}) - oc2 II({}, 1 - oc2 , 1)J/(1 -oc2).

111.08

F (iq;, 1) = i{} = 2 tan -1 e'" - 11:/2, [sinhq; = tan {}] { E (iq;, 1) = i tan{} = i sinhq; = sin iq;, II(iq;, oc 2 , 1) = i[{} - (1.2 II({), 1 - oc2 , O)J/(1 - oc2).

111.09

{

111.10

{

fl(q;, 1, k) = [k'2 F(q;, k) - E (q;, k) + tan q; Vi - k2sin 2 q; ]/k'2, II(q;,k 2, k) = [E (q;, k) - k' sin q; cos q;/V1 - k 2sin 2 q; ]/k'2.

F [sin-1 (11k), kJ = F [11:12 + i cosh-1 (11k), kJ =K + i K', E [sin-1 (11k), kJ = E + i (K' - E'). K' = K,

K: K

when

Vi -

= ViK = 2K

k= 1 'k = 3 - 2V2,

V32 k ,

K'=V3K.

111.11

k = V2/2,

1

- n13 E -12K + E' = n

13

4K'

)

-

when k = (V3 -1)/2V2.

K,

-+-1!2 13 k K' .

Limiting Values. 112.0 I

Lim(K -ln~) k ...... l k

112.02

K -- E = L'Im----=1I:4. E - k'2 K I · m Ll

112.03

Lim (E - K) K' = O.

112.04 *

Lim

112.05

Lim

k ...... O

=

O.

k_O

~

~

k ...... O

k ...... O

"' ...... 0

e-(nK'IK)

k2 E ~tp, k) smtp

-(nK,K')

= Lim_e-._,k ...... l

= Lim "' ...... 0

k

F ~tp, k) smtp

2

= 1/16.

= Lim '1' ...... 0

II (lfi.' 'X 2 , k) smtp

= 1.

* The term e- 1 :

= K + iF(A, k'), E ('If, k) = E + i [F (4, k') - E (A, k') +

F('If, k)

A

±

2 A""""'"] (1 - k 2 sin 2 (J) sin A cos A V-l-_-kccc' ' 2--'si'-nOOcos2 A k 2 sin 2 {J sin2 A '

In the special case when sin 'If (

+k

. -1 = SIn

Vsin 2 'IjJ-l' k'sin'IjJ

k'2 sin A cosA Vl-k'2 sin 2A

J,

.

1 For a 10-place tabulation, see K. Pearson's Tables at the Complete and Incomplete Elliptic Integrals (reissued from Tome II of Legendre's "Traite des Fonctions Elliptiques") London 1934; and for a short 6-place table, see Appendix here.

Elliptic Integrals.

115.03

F(1p, k) E(1p, k)

{

= =

F({J, k) E({J,k)

+ i K', + cot{J V1 -

13

k2 sin 2 {J

+i

(K' - E'l,

where 11k

~

sin1p
:'

~1.0~------+_---n~4-----~

~ 0.5

o

rp=JO· O'sr-----~~------+_----_1

11'-0' JO

•

sin- 'k Fig. 3.

60'

JO'

GO' Fig.4.

90'

17

Elliptic Integrals. UQ

I'--- ~

""'-

'" ""'- "'-

~.

""

/

V ~

1.()()

o

0.1

0.2

aJ

o.~

0.5

IcZ_

'"

b.6 0.7 0.8

10

1"\

Qg

/"

/

1.8 16 ()

10

0.1

02

OJ

~~

o.~

as 06 Jc2 -

0.7

0.8

0.9

~~

Conformal Mappings. Some conformal mapping properties of the elliptic integrals are illustrated in the following sketches. Mapping of the half-plane 1m (z) > 0 onto a rectangle in the C-plane: 119.01

z = snC.

J_II-K+iK' I_~I~I K I K+iK' I~ z

0

:

-1jk

A :

- 11k

ZT

- 1

1 I

0

I

11k

00

t;-plane

LK=i-

0

c¢----

8

-1

1/k

fl

A

Fig. 7.

Mapping of the function

•

c=I

119.02

1 - k 2 z~ --_~2dZl = 1 - "1:

o

E (z, k): s-plane

Z-plane

o

o

A

-11k

-1

I

8

c

C 11k Fig. 8.

Byrd and Friedman, Elliptic Integrals.

2

1.0

Definitions and Fundamental Relations.

18

Mapping of two horizontal lines onto two vertical lines:

119.03 t=

!

! [E (k{, k') -

b=

[E (k) - (1 - k 2 a2 ) K (k)J,

k 2 a2 F (k{, k') J,

Z-p/ane A -11k

BCD E -a

-T

I

F

a

I/k Fig. 9.

where

k{ =

V1 -kr

and a is determined for given k by the relation a2

=

E(k') k2K(k') ,

k'

=

V1 -

k 2•

Jacobian Elliptic Functions. Definitions. Instead of considering the elliptic integral

120.01 as the primary object of study, ABEL and JACOBI reversed the problem and investigated the inversion of this integral. Inverse functions l may thus be defined by Yl =sin1p=sn(u, k) and 1p=am(u, k), or briefly Yl = sn u, 1p = am u when it is not necessary to emphasize the modulus; these may be read sine amplitude u and amplitude u. The function sn u is an odd elliptic function of order two. It possesses a simple pole of residue 11k at every point congruent to iK' (mod 4K, 1 This has an analogy with trigonometric functions. The singly-periodic function y = sin u, for example, defines the in version of the integral

y

U=J1I o

dt • t 1 - /2

[Y;:;;1].

Jacobian Elliptic Functions.

19

ZiK') and a simple pole of residue -11k at points congruent to zK + iK' (mod 4K, ZiK'). Two otherfundions can then be defined by cn (u, k) = V1 - y~ = cosip, dn(u,k)=V1-k2y~=L1ip=V1-k2sin2ip, requiring that sn(O,k)=O, cn (0, k) = 1 and dn (0, k) = 1. The function cn u is an even function of on;l.er two. It has simple poles at every point congruent to iK' and 2K +iK' (mod 4K, zK + ZiK') with residues - ilk, ilk respectively. We note also that dn u is an even elliptic function of order two, having singular points congruent to iK' or 3iK' (mod ZK, 4iK'). These points are simple poles of residue - i and i respectively. The functions sn u, cn u, and .dn u are called Jacobian elliptic functions 1 and· are one-valued functions of the argument u. These functions, like the trigonometric functions, have a real period, and, like the hyperbolic functions, have an imaginary period. They are thus doubly periodic 2, having the periods (4K, 2i K'), (4K, ZK +Zi K') and (2K, 4i K') respectively. The modulus is denoted by the number k and the complementary modulus by k' = V1- k 2. The quotients and reciprocals of sn u, cn u and dn u are designated m Glaisher's notation by

120.02

nsu

1 == --' sn u '

tnu=scu=

ncu

== --' cnu '

-== tnu

1

1

ndu == ~--'

snu. cnu '

sdu= snu .

cnu csu=--' snu'

cdu=cnu.

dsU=dnu. sn u '

dnu'

dnu'

dnu'

dcu

==

dnu .

cnu '

We thus have, in ail, twelve Jacobian elliptic functions 3. These functions have many direct applications in physical problems but are of interest to us principally for the purpose of evaluating elliptic integrals. The most important properties of the functions are summarized by the formulas given in the remainder of this section. 1 The best tabulation of sn u, cn u and dn u appears in G. W. Spenceley and R. M. Spenceley's Smithsonian Elliptic Functions Tables (Smithson. Misc. ColI., v. 109) Washington, 1947. 2 The Jacobian elliptic functions sn u, en u, dn u are of course only special cases of a more general class of functions which possess similar properties. If 2W and 2w' are any two numbers whose ratio w'jw is not purely real, then any function f(u), which is analytic except at poles and which has no singularities other than poles for any finite value of u, is called an elliptic function if the doublyperiodic relation t(u 2w) = t(u 2w') = t(u)

+

+

is satisfied for all values of u for which f (u) is defined. 3 For an attractive and unique treatment of these functions, exhibiting them as functions constructed on a canonical lattice, see Jacobian Elliptic Functions by E. H. NEVILLE,Clarendon Press, Oxford 1951. 2*

Definitions and Fundamental Relations.

20

Fundamental Relations.

121.00

sn2 u + enS u = 1, k 2 snll u + dn2u = 1 , dn2 u- kll enIIU -- k'll ,

1

k'Bsnll u

121.02

dnu =Atp = V1-kllsin2tp,

u = am-l (II', k)

121.01

J~: =F(tp,k), tp

=

+ en2 u = dn2 u.

o

amu

= am(u, k) =11'.

-1:;;;;snu:;;;;1, -1:;;;;enu:;;;;1, k' :;;;; dnu :;;;; 1,

[u real]

1 -

00

< tnu
0 onto a rectangle in the z-plane:

129.50

[d. 119.01]

C=snz. (-plane

D

£1 A

o

A

-11k

-1

8

C

'1k

Fig. 17.

Mapping of a circular ring onto the unit circle with a symmetrical slit:

129.51 where

k (q) = LZ

= 4 llq V~

"" [- -++-lm_ - ; IT 1

]'

q2m

1

1

m~l

Z-plane

-/

Fig. 18.

Other Mappings. The function C= cn z maps the rectangle - K < R P . (z) < K. 0< 1m (z) < K' onto the right half-plane RP. (C) > 0 with a cut along the linear segment 0 < , ~ 1. The function, = sn 2 z maps the rectangle 0 < RP. (z) < K, - K' < 1m (z) < K' onto the entire '-plane with the slits - oo:s:C;;;: 0, 1 ;;;:C ~ 00. The transformation C= (sn z) /( 1 cn z) maps the rectangle - K < RP.(z)1J.

130.07

y

130.08

sd-1(y, k) =f

dt V(1-k'2t 2)(1 +k 2t 2)

o

=F[sin-1 V y2/(1 +k2y2),kJ, [1/k';;;;:y>O].

y

130.09

nd -1 (y, k) =f~_====d=t=~~ 1 V(t 2 - 1) (1- k'2t 2)

F[sin -1 V(y2 - 1)/k2 y2, kJ , [1/k':;;;:; y>1].

00

130.10

ns-1(y, k) =f--~-----=- = F[sin- 1 (1/y), kJ, V(t 2 - 1) (t 2 - k 2)

y

[y

~

1 >k].

00

130.11

ds-1(y,k)=f y

dt V(t 2 - k'2) (t 2

+ k 2)

=F[sin-1 V1/(y2+k 2 ),kJ, [y~k'>OJ.

Jacobi's Inverse Elliptic Functions.

=jv

31

00

130.12

cs- (y, k) 1

(1

dt

+ t 2) (t 2 + k'2)

= F[sin -1 V1/(1

+ y2),llJ, [00 > y ~

Y

OJ.

130.13

[O:S;;yy ~ b >0) k 2 = _a_2 __

. -lV

00

212.00

j

+

2

sn Ui = sing;.

+y

a

1

Va + b2 '

g=

a2 b2 -2--2 '

g; = am U 1 = SIn

1v

a2 + b2 '

'"

dt

(a 2 + t 2) (t2 _ b2)

=gjdu=gu1=gsn-1(Sing;,k)

Y

0

=

gF(g;,k),

[cf.211.00].

'"

= gjdn 2 Udu=gE(u1 )

212.01

o

= g E(g;, a2

212.02

t2

+bt = g j

_

2 2

'"

o

where

ex: 2

=

(a 2

du

1 - oc2 sn 2 u

=

k).

g lIeg;, ex:2 , k) ,

[See appropriate case in 400.]

+ p)/(a + b 2

2 ).

212.03

212.04

[See 361.29.]

212.06

212.07

J

dt t2 _ b 2

OO

Y

j'"

1fa 2+t2 Vt2 _ b2 = g nc 2 U d U ,

(y,* b).

[See 313.02.]

0

j

00

212.08

Y

dt a 2 +t2

[See 312.02.J

48

Reduction of Algebraic Integrands to Jacobian Elliptic Functions.

f tB

+

OOdt

212.09

as t S tZ _bS

f'"

[See 315.02.]

=gnd2 udu.

:Y

0

212.10

(y =1= b) .

f

tB_b Z

212.11

2 +t = g cd udu.

-2--2

a

[See 321.02.]

".

[See 320.02.]

o

212.12 where 212.13 where

fV 00

212.14

f

R(t2) dt

".

+b

~=.:==;=;::===:= = g R [(a 2 (a 2 t 2 ) (t 2 - b2 )

:Y

+

2)

ds 2 u] du,

0

where R (t 2 ) is any rational function of t2 • Integrands involving Va2+t2 and Vb 2 _t 2, (b cn2 u

= .!: b2

fV

213.01

213.02 where

2

COS-1 (yJb)

b-:;r::=o.===d===t=====-====(a 2 t 2 ) (b 2 - til

:Y

a+b

-

cp = am U 1 = 213.00

b2

k2 _ '

+

=

j 1

(p - t 2)

1

= Va 2 + b2

en U 1

'

= COScp.

U1

t2) (b B - t2)

= P

~ j: -~:sn2 bZ

=

b2 /(b 2

-

14

0

= P ~ bB II(cp, (1..2, k), (1.2

~ 0)

=gcn-1(eoscp,k)=gF(cp,k).

0

1f(a2d~

,

g

gf"~ u = g

{j :Y

2 '

>y

P), P =1= b2 •

[See 400.]

Integrands involving Va 2 +12 and Vb 2 - t 2 , (b> y ;;;:0).

213.03

a dt = b gJ'"sn udu. J V7ii=t2 b

2

2 +tS

Y

213.04

J

J

__

b

[See 310.02.]

2

0

".

fbi -

dt 1 t2 a 2 +tl Vaz+t l =k2g

Y

49

[See 318.02.]

sd2 udu.

0

213.05

[See 321.02.]

213.06

[See 312.05.]

213.07

[See 320.02.]

".

(as:' b2)m Jnd!'" u du.

213.08

o

J b

213.09

V +dt12) (b2 _

t2 ". (a 2

Y

b

213.10

J~ t 2

Y

b2 a

_t2

+t

-2--2

=g

;m Jnc "'udu,

[See 315.05.]

~

t2)

=

2

b

J".tn udu, 2

y=l=O.

[See 313.05.]

0

[See 316.02.]

y=l=o,

o

213.11 [See 336.03. J where

J b

213.12

Y

(PI - t2 ) dt (P-t2)V(a8+t2) (b 2 _t2)

=

J ~

PI - b2

p-b2 g

0

1 - ct~ snB U 1-a.2 sn2 u

'

[See 340.01.]

where 213.13 where R(t2) is any rational function of t2. Byrd and Friedman, Elliptic Integrals.

du

4

50


Integrands involving Va 2 - t 2 and Vb 2

+t g

1p =

214.00

214.01

1

y

J o

V(a

t 2 ) (b 2

2 -

1 o

where

214.03

+ t2)

V+

(1.2

b2 a2

1

= ---:=:== 2 2

Va + b

sn U1 = sin 1p

= gJdu=gu1 0 = g sn-1(sin 1p, k) = g F(1p, k).

__

J _d_tP - t2

+ b2) + b2) ,

-

t2 t2

'"

-

~J-_d_U__ - ~Il( 1X2 k) P 1 - (X2 sn2 U P 1p" , 0 (See 400.]

= a2 (P +b2 )/p (a 2 + b2 ), P =1= O.

--dt=a gJ'"cd udu. JYV~ +t b2

o

> 0)

(a :;;;: y

'"

dt

Y

214.02

y2(a2 a 2(y2

am U 1 = sin- 1

2,

2

[See 320.02.]

2

2

0

214.04

[See 310.02.]

214.05

(See 318.05.J

214.06

[See 319.05.J

214.07 214.08

y=l=a.

J

y

b2

dt

+ t2

V +t2 = lii-=-!2 -b2

o

J'"

__

'"

2 +t2 =-~-fJnc2udu, y=l=a, J _d_t_Vb a2 _ t2 a2 _ t2 a2

o

[See 312.02.J

0

Y

214.09

a2 g

~ cn 2 U duo

[See 316.02.J

0

[See 313.02.]

Integrands involving

Vt -a 2

2

Vi2-=- b2,

and

(00 >y :;;;: a> b >0).

51

214.10 [See 315.02.]

214.11 214.12

JV(a -t y

2

o

j

y

214.13

o

2)

(b 2 + t2 ) dt

dt (P _ la)m V(a 2 _ 12) (b 2

= a2 b2 g Jcd2 u nd2 udu. [See 361.16.] ~

0

g

+ t2) = pm

j 0

'"

dn 2m udu (1 - a.2 sn2 u)m ,

[See 339.04.]

where 214.14 where

where R (t2) is any rational function of Integrands involving

Vt

Vt

a 2 and

2-

[2.

2-

b2 , (00 > y ~ a

> b > 0)

a sn2 u=12 ' 2

q;=amut=sin-1(a!y), 00

215.00

j Y

215.01

'"

= gjdU=gUt=gsn-1 (sinq;, k)=gF(q;,k).

de

a2) (/2 - b2)

0

jOOdt2t YG2_b2 =g j'" dn t - a -2--2

Y

215.02

V(/2 -

snu1 = sinq;.

2

udu=gE(u1 )=gE(q;,k).

0

'"

=gj o

d2u 2 l-a.snu

=gll(cp,rx2 ,k), [See 400.]

where 4*

52


JOO

215.03

dt

II_b'

j'"

1~

[See 320.02.J

V~=g ed2 udu. 0

y

215.04

JOO

215.05

dt

-2--. I -a

vg2 _ bl -.--, = lo-a

y

f'"

g de2 u du,

y =1= a.

[See 321.02.]

"

215.06

j t2m Y(tl - dtal) (t2 00

215.07

j sn2m ~

bl )

Y

=

1

a2m +1

U

[See 310.05.]

0

_ j'" jOOdttB 1~ V~ -g en udu.

215.08

d u.

2

.

[See 312.02.]

0

y

215.09

y=l=a.

[See 313.02.]

215.10

215.11

[See 315.02.]

215.12 [See 337.04.]

where

215.13 where 00

215.14

j y

".

R(t2) dt = jR(a2ns2u) y(11 - al) (12 - b2) g 0

where R(t2) is any rational function of t2 •

du,

Integrands involving VtZ-aS and Vt 2 -bZ, (y>a>b>O).


Vi2- a

tS-a S t· - b-

2

and

Vt

b2 , (y >a >b > 0)

2 -

b2

sn 2 u = -.--.-,

k 2 = -Z a

53

g = 1/a,

'

sn Ut = sin 11' •

", =gJdu = gUt = g sn-1 (sin1p, k)

216.00

o

=

Jdn

gF(lp, k).

w.

=

216.01

k~S

o

2U

du

=

k~2 E(~)

= k~2

E(lp, k) .

216.02 [See 400.] where 216.03

[See 313.02.]

216.04

[See 316.02.]

216.05

[See 310.02.J

216.06

[See 321.05.]

216.07

[See 315.02.]

JIS V y

216.08

--

dt

a

J

".

12 - as '2 12_bs-k g

0

sd2 u d U.

[See 318.02.]

54


216.09

[See 320.05.]

,

216.10

~

JV(tli

-

all) (tll - bll ) dt

= (all - bll)lI g J tn ll u nell udu. 0

a

[See 361.07.]

j en ~

216.11

g (al _ bl)'"

o

(P - al)'"

where

j

216.13

..

where

j'"

g

216.12

a.1I

=

,

o

u d u. [See 312.05.]

2",

cnl"'udu

(1 - a.2 snl u)'" ,

[See 338.04.]

~

(Pl-tl)dt _ (P - tl) (t 8 - al) (tl - b2) -

V

(P - bll)/(P - all),

=

a.~

Pl-al jt-a.f sn1u du P - a l g 1 - a.1 sn2 u ' 0

[See 340.01.]

(PI - bll)/(PI - a2 ),

P=l= all.

216.14 where R(tll) is any rational function of tll.

Integrands involving Va 2 -

217.00

I' j

b

217.01

V(a

2 -

dt t·) (tl

-

bl )

t and Vt 2

2-

b2 , (a:;;:: y >b > 0)

~ =gjdu=gUt=gSn-l(sinq.>,k) =gF(q.>,k),

0

=

[cf.218.00].

;1 j '" dn udu = ;1 E(Ut) 2

o

= :.

E(q.>,k).

Integrands involving ya2 -t2 and

Yt

2

_b 2 , (a '2;y>b >0).

55

217.02

where

217.03

[See 320.02.J

217.04

[See 318.02.J

217.05

[See 315.05.J

217.06

f

y

f nc udu

t2dt k l2 ~------c======g(a 2 _ t2) (a 2 _ t2) (til _ b2) k2

Y

217.07

f ~dt_ t Vat2 2

f

b

2 _

'

y=l=a.

[See 313.02.J

k,aft d

[See 316.02.J

'"

bt2 -2

2 gnu u, y =1= a.

0

b

217.08

__

2

0

b

Y

~

Y

J.sn-u

dtYG2b2 = g k2 --t2 a2 _ t2

'"

d u.

[See 310.02.J

0

217.09

217.10

(a2.! b2)m

fo '"dc

y =1= a.

217.11

2m udu,

[See 321.05.J

[See 312.02.J

217.12 [See 339.04.]

where

56



Integrands involving sn

2

U

=

Va t and Vt 2 -

as - t2 a2 _ b2

2

k2

b2 , (a> y ~ b > 0)

2 -

= ~~ ' a2

. Vas'

."=am~=sln-l T

y2 as _ b2 '

g = 1/a,

sn~

= sin1p

... =gJdu =

218.00

gUl

= gsn-l (sin1p,k)

= g F(1p, k),

o

[ct. 217.00.]

218.01

where [See 312.02.]

218.03

218.04

218.05

~2~ -gJUItn -2 J-t2-dtb t-b a

Y

-2--2

_

0

2

udu,

y=l=b.

[See 316.02.]

Integrands involving V~2 - t2 and Vii _b 2 , (a> y ~ b > 0).

218.06

57

[See 314.05.]

218.07

y=l=b.

[See 321.02.] 218.08

- b =gk2 f"'cd2udu. fadtt vg2 a - t 2""

2 -2--2

Y

[See 320.02.]

0

218.09

[See 310.02.]

218.10

[See 318.02.]

218.11

a

fV(a 2- t2) (t2_b 2) dt = (a 2Y

~

b2)2g f sn2 u 0

cn2 u duo [See 361.01.]

218.12

[See 315.05.]

218.13 [See 336.03.] where a

218.14

f

(t2_PI}dt (t2_p}V(a2-t2}(t2_b2)

=

Y

where

/X~= (a 2 -b2 )/(a2 -Pl)'

~

a2 -PI f1-(l(~sn2udu a2 -p g 1-(l(2sn2 u ' 0

/X 2 =

[See 340.01.]

(a 2 -b2 )/(a2 -p), p=l=a 2 •

218.15 where R(t2 ) is any rational function of t2 •

58


Integrands involving Va 2 -

t2

g= 1/a,

rp = amu1 = sin-l (y/b) ,

snu1 = sinrp

1I v

=b2 '

",

y

=gIdu=gu1=gsn-1(sinrp,k)

dt

(a2 _ /2) (b2 _ /2)

o

219.01

(a> b;;;;; y > 0)

t 2,

-

/2

sn 2 u

219.00

and Vb 2

= gF

0

[d. 220.00.]

(rp, k),

~2::::t2 dt=a IU'dn udu=aE(u )=aE(rp,k). IYV~ 2

o

1

0

LI

u,

o

J..-IJ( (J,,2 k) p cp" , [See 400.J

IU' v~dt = bk cn udu. IY1~

[See 312.02.]

219.02

p

du

-- -

1-(X2sn 2 u -

where 219.03

2

o

0

219.04

[See 318.02.]

219.05

[See 310.05.]

219.06

219.07

219.08

I

y

219.09

o

a2 dt _ /2

V'

I

u,

2 ab

2 _

tt2

2

-

k2 g

0

d2 U d U •

C

[See 320.02.J

Integrands involving y

j

219.10

Va 2 -- t2

(a> b > y :2: 0).

59

~

b2

dt

__

t2

lja2=t2 g fd 2 Vb2=t2- = k2 c U du,

o

y

=F b.

[See 321.02.]

0 y

219.11

and Vb 2 - t2 ,

J V(a

o

~

2 -

t2 ) (b 2

-

t2 ) dt = a2 b2 gJ cn 2 u dn 2 udu. 0

[See 361.03.]

u,

gjtn 2 udu,

219.12

y=l=b. [See 316.02.]

o

219.13 where u,

PIg f1 -

219.14

P

1-

O(~ 5n 2 U sn 2 U

du '

0(2

[See 340.01.1

o

where 219.15 where R(t2 ) is any rational function of t 2 •

Integrands involving Va 2 - t 2 and Vb 2 - t 2 , (a> b > y ~ 0) g = 1ja,

sn U 1 = sin tp . b

220.00

jV

(a 2

Y

220.01

~

dt

-

t 2) (b 2 - t 2)

=gfdu

=

g~

0

=

gF(tp,k),

=

gsn-1 (sintp,k) [ef.219.00.]

", 2 udu =~L-E (~) = -g--jdn 2 a k'2 a 2 k'2 o

= a2~'2E (tp, k).

60

I


jt b

220.02

)'

dt 2 -

P

l~ V~ = =

(a 2 _b 2

b2

-

)gj'"

Gt 2

sn2 u

k'2 a2 g b2 _

[See 400.]

P II(tp, rx. 2 , k),

rx.2 = b2(P - a2)/a2(p - b2),

where

du

Pot -

P=Fb2 •

220.03 220.04

[See 315.02.]

jb~2_t2 a t

~~dt=k'2b2g 2 _

2

Y

220.05

[See 318.02.]

0

JV(a 2b

Y

j'"sdsudu.

= k 2a4 k'4 g Jsdsu nd2 u du. ~

t S ) (b 2 _t2) dt

0

[See 361.19.]

220.06

[See 320.05.]

220.07

[See 310.02.]

220.08

[See 312.02.]

y=F O•

220.09

[See 321.05.] Il,

220.10

(as.! b2 )m

j dn o

2m

u du.

[See 314.05.]

220.11

y=l=O.

[See 316.02.]

220.12

y=l=O.

[See 313.02.]

220.13 where


V/2 +a

2

and

V/2 +bD; (y > 0;

a

> b).

61


Integrands involving Vt 2+ a 2 and Vt2+ b2, (y >0; a >b) aZ - b2

k 2 = _a_ s ' g= 1fa ,

, 221.00

f

~

V(/2 + b

1

o

dt 2)

(/2

+ al)

gfdU = gUl = g tn-1 (tanlf', k) 0 = gF(If', k), [cf.222.00].

=

221.01

,

221.02

f

o

d/

P _ t2

where

'"

y

221.03

t 2 +a2 2 f dc 2 udu --dt=ga /2 + b2 .

f 0 y

221.04

f 0

221.05

221.06

[See 321.02.]

0

", 2 t2 +bd t = gb 2 f nc 2 udu. /2 + as 0

[See 313.02.]


62

y

221.07 221.08

~

j o

(t2+ a2) (t 2

= -g-jcn2 ucd2 udu.

dt

+ b2) V(t 2 + a2) (t 2 + b2)

a2 b2

0

[See 361.28.]

",

y

J

JV(t 2 + a2 )(t2 + b2) dt = a2 b2g dc 2 u nc2u du. [See361.13.]

o

0

[See 316.05.]

221.09 dt j t+a

2+ 2_ kl2jn dB u d u.

y

221.10

-2--2

'"

t b t+a

[See 315.02.]

-2--2--g

o

0

221.11

[See 318.02.]

221.12

[See 310.02.]

221.13

p+o.

where

221.14 where y

221.15

~

R(t2) dt f(t2 + a2) (t 2 + b2)

j o

= jR(b2 tn2 u) du g

I

0

where R(t2 ) is any rational function of t2 • Integrands involving

tn 2 u

a2

= t2 •

k2 tp

222 •00

{J OO

V(a2

Y

dt

Va 2 + t 2 and Vb 2 + t 2 ,

(00

=

tn u 1 = tan tp •

a2

-

a2

1

b2 •

g = --;- ,

>Y:;;;: OJ a >b)

= am u1 = tan -1 (ajy) .

+ t2) (b 2 + t2) =

j'"

g du 0

=

gu

1

=

gtn-1 (tan'" k)

=gF(tp,k).

T'

[cf.221.00].


Va2+ t2 and Vb2+12,

(00

> y:2; 0;

a> b).

63

222.01 222.02

jt2

~ p V::! :: =

f't _

Y

0

oc2 = (a 2

where

222.03

:2Usn2 u

g

+ p)/a

=

gII(1p, oc2 , k); [See 400.)

2•

dt j'"sn "'udu. jO?(t + a2)'"-======~g~ V(a2 + t2) Wi + t2) a 2

2m

2

Y

0

[See 310.05.)

222.04

222.05

222.06

j

t 2 dt +b2

y

Vtz+a2 12 +b2

I

0

'"

00

222.07

j (t 2

Y

222.08

+

t 2 dt a 2 ) V (t 2 a 2 ) (b 2

+

J~ 12 j Y

where

222.12

[See 312.02.J

0

y=j=:O.

[See 313.02.)

y=j=:O.

(See 321.02.)

0

Y

222.11

+ t 2)

+ = g] 'ne2 udu,

00

222.10

= gjen 2 udu.

2 2 jdt tz Vtb2 +at2

Y

222.09

[See 315.02.J

=g nd 2 udu.

+ 2 +

t b j -~-=g a 2 12 2

'"

de2 udu

00

tDdt

(t2

'

0

+ b2 ) V(t 2 + b2 ) (t 2 + as)

", =gjed2 udu. 0

[See 320.02.J

64


f

00

222.13

(Pl - t 2) dt (P - tl) }'Ttl al) (b B

+

+t

2)

-

gf 1 -

-

1-

"I

"

etfsnlu du etl sn2 u '

[See 340.01.]

0

where

OtB=(all+p)faB.

Ot~=(aB+Pl)/all,

222.14 where R(tB) is any rational function of tll.


1f v +

V,lI + ri and V,lI + ell; ell, ell conjugate complex numbers, (0::::: y< 00)

00

225.00

(tl

"

225.01

225.02

225.03

225.04

225.05

gfdU = gUl = gcn- (coscp, k) ...

dt

el) (tll

+ el)

=

1

0

=

g F(cp, k) ,

[Gf.263.00J.

65

225.06

225.07

[See 312.02.J

225.08

225.09 where R(t2) is a rational function of t2.

Integrands Involving the Square Root of the Cubic, Vao (t+r 1) (t+ r2)(t+r3)' Introduction. We consider here the reduction of the integraP :v

230.00

1=

S"

=fR(t) dt

:v,

V:P

'

1 On setting t=t~-I'I' (1'1 real), we may write

fp

f VaO(t~+1'2-1'1)(tf+l'a-I'I)'

R(t) d

RI (tn dtl

t=2

The integrals in 230.00 can therefore be reduced to those of 211-225. One may occasionally encounter an integral in the form

-f

Y

~

1-

Y,

t R(t2 ) dt Vao(t2+1'1) (t2+l'z) (t2+1'3) ,

where the factors under the radical sign occur in even powers of t, with an odd power of t appearing outside either in the denominator or numerator. By applying the transformation tl = T, it is immediately seen that such integrals are equivalent to those given in 230.00. In this case, we thus have

~

-f

:v

1-

:v,

R(T)dT 2Vao (T+1'1) (T+l'z) (T+ra)'

(Y= YI, Yl=Yl)·

(d. 575.10). Byrd and Friedman, Elliptic Integrals.

5

66


where P =ao(t +r1) (t +r2) (t +r3) is a polynomial whose three linear factors are all distinct. By the method of partial fractions, the integral 230.00 may be expressed linearly in terms of the general integrals y

ft;;~ y,

,

Now

230.01

and if

P is

f

y,

y

not a zero of P(t),

dt (t - p)m

_

2

lIP

Vao(Y

= 2(m -

1 1) P(P)

{2V;o~(Yl+r2) (Yl +1'3)

+ 1'1) (y+ 1'2) (y+ 1'3). + (5 (y - p)m-l

(Yl - p)m-l

_ 2m) aofY y,

dt (t - p)m-3

VP

+

230.02

+ ao(3 xfY y,

When P(P)

=

2m) dt

(I - p)m-l

0,

[3p2

+ 2p (rl + r 2 + r3) + r1r2 + r1r3 + r2r3J X

Vp } ,

[m=l=1].

67

230.03

:v

+ (3-2m) aof :V.

dt 1/ (I - p)m-2 vP

+2ao(1- m) (3P+rl +r2 +rS ) X

Every elliptic integral of the fonn 230.00 thus depends on the three basic integrals J dt/vP, Jtdt/VP and J dt/(t-P) Vp· The integral J dt/vP is finite for all values of t and is always an integral of the first kind. An integral of the type t dt/VP possesses an algebraic infinity at t = 00 and leads to an integral of the second kind, while dt/(t - P) vP is of the third kind, becoming logarithmically infinite at the point t=P as ± [In(t-p)J/VP(P). [In case P(P) =0, this latter integral is of the second kind and is algebraically infinite at t=p.] Considering a, b, c, al and bl real, one may write the radicand pet) in 230.00 in one of the following ways:

J

J

laol (t-a) (t-b) (t-c); j aol (a-t) (t-b) (t-c); Iaol (a-t) (b-t) (t-c); Iaol (t-a) [(t-al)2+b~J; Iaol (a-t) [(t-al)2+b~J; (a >b >c). The roots of the equation P= 0 are all real in the first three cases; but in the last two cases, two of the roots are complex. Reduction to Jacobian nonnal fonn for the cases when the zeros of the radicand are all real is accomplished by means of substitutions of the type

230.04 where AI,A2,As and A4 are real constants chosen so that dt/vP=gdu and g is some real constant. If two roots of the equation P= 0 are b ;;;;; y 2 t-c sn u=b=-c'

> c)

2

k2=~~,

a-c

g=V-' a-c

~

233.00

I

f

tp = amu1 =

.

SIn

-1

Y- c V-,

sn u1 = sin tp .

b-c

y

~

V(a - t)

dt

(b - t) (t - c)

=

gfdu

g F(tp, k),

=

0

c

= g u 1 = g sn-1 (sin tp,

k)

[d. 234.00].

233.01

233.02

1

Y

dt f(p-t)V(a-t)(b-t)(t-C)

c

",

du p-cf. 1-b:;::: y >c).

233.06 y

233.07

fV

(a~ ~(:- t)

-

~

dt

=

(b - e) (a - e) g fcn 2u dn 2udu.

[See 361.03.J

o

233.08

!V(a-t) (b-t) (t-e) dt ",

1

C

g!

~

= (a - e) (b - e)2

sn 2 u cn 2 u dn 2 u du. [See 361.04.J

y

233.09

~

=--g~fnd2mudu.

dt (a - t)m Via - t) (b - t) (t - c)

f

(a - e)m

[See 315.05.J

0

C

",

233.10

/I (

233.11

(b- e)m

Y=l=b.

(a- t) (b- t)

f

V(:~ (b-~ t) (t -

233.13

g c) (b -

(a -

",

fnc 2U nd 2 u du, c)

Y =l= b.

[See 361.12.J

f

~

dt (p-t)mV(a-t)(b-t)(t-c) =

(1,2

f V

y dt ---.a - t

f~V a- t

y

233.15

[See 313.05.J

o

y

Y

233.14

'

e)

t)

g

(p_e)m

C

where

o

y

=

233.12

= --g~fnc2mudu

b- e

f~cd2udu

~--g

a- e

0

[See 336.03.J

P=l=e.

(b - e)/(p - e),

b- t = (a - t) (t - c)

o

.

[See 320.02.J

~

t-e = b-~gfsd2udu. (a - t) (b - t) a- e

V

dtf b- t

=

du (1-1'J. 2 sn2 u)m'

a- t (b - t) (t - c)

[See 318.02.J

o

",

a - e fd c2 u d u, =--g b- c

o

y=l=b.

[See 321.02.J


74

233.16

Jyb dt_

t

V

t- c

J'"

-

g tn 2 udu

(a _ t)(b _ t) -

c

233.17

Y

J c

233.18

y =l= b. [See316.02·1

I

0

r.==-=.=:=tffl;=d=t=.==:=- = V(a-t)(b-t)(t-c)

f

Y

V

em

gJ [1 '"

_c_-_b sn 2 u]m c

0

dt

[See 331.03.J

g =cm

t m (a - t) (b - t) (t - c)

f [1--c -sn u] '"

du c-b

J

[See 336.03.J

Y

c

rJ.2

Y

J

c

gJ

u&

(PI - t) dt . = (P - t) V(a - t) (b - t) (t - c)

where

233.20

m·

2

o

233.19

du .

=

PI -

C

P- c

0

gJR [e -

(e -

I

[See 340.01.J

(b - e)/(p - e), ",

R(t) dt = V(a - t) (b - t) (t - c)

du

1 - oc~ sn 2 u oc2 sn 2 u

1-

p=l=e.

b) sn uJ du 2

I

0

where R(t) is any rational function of t.

Integrands involving Va - t, Vb - t and Vt - c, (a> b > y :;;;; c) sn2 u _ -

(a - c) (b - t) (b - c) (a - t) ,

cp = am u = sin-1 1

234.00

~~ a- c

(a - c) (~ - y) (b - c) (a - y)

I

I

g=

2

V-' a-c

sn u1 = sin cp .

'"

I

b

234.01

V

k2 =

=g Jdu=gu1=gsn-1(sincp,k)

o

=

[ef.233.00].

gF(cp, k),

J

dt . g . = -(a-t)V(a-t)(b-t)(t-c) a-b

Y

~

J

dn 2 udu

0

=

-g-E (u1) = -g-E (cp, k). a-b

a-b

Integrands involvingVa-t, Vb-t and Vt-c, (a>b>y:;:::c).

234.02

f Pdt-

1

b

t

a- t

(b - t) (t - c)

=

a- b

p- b g

:Y

= ;

where

234.03

V

=:

j 1=

(a - t) (t - c)

V(:~

1-

du sn2 U

(X2

0

g II(rp, oc2 , k),

oc2 =k2 (p-a)/(p - b),

y

f"'

75

[See 400.J

p=f=.b.

t) (b - t) (t - c) ",

(a _ b{(b _ c) f

dn 2 u dc 2 udu, y =f=. c.

[See 361..11.J

o b

234.04

~

dt (t - c)m V (a - t) (b - t) (t - e)

f

= -( g-fdc2mudu, y=!=c. b - e)m

f

b

234.05

dt

(a-Wn

[See 314.05.J

)dt=(a-b)gfnd2udu. f V (b - a~t t t- c

[See 315.02.J

~

y

0

b

~

dt=kf2(b-c)gfsd2UdU. fV (a - b-t t) (t - c) b

fV (a

_tt~: _

'"

t) dt

Y

[See 320.02.J

= (b - c) g fCd2UdU. 0

b

234.09

fV (a -/~bc-

~

t)

dt

= (b - c) (a - b) k'2 g fSd2und2u du. 0

:Y

b

0

= (a - b) (b - c) g f cd2 u nd2 u du.

"

0

b

234.11

fV (t Y

:~bt- t)

[See 361.19.J

~

f V (a - bt t(- c) dt

234.10

[See 318.02.J

0

y

234.08

fdn2mUdU.

o

b

234.07

'"

g

(a- t)mV(a-t)(b-t)(t-c)

y

234.06

[See 321.05.J

0

:Y

[See 361.16.J

~

dt

= (b - C)2 k'2 g fSd 2 U cd2 udu. 0

[See 361.27.J

76

234.12


f vr.-(a---:-t)-:-:(b:-------:t)-:-(c---:-t) d t '"

1

b

Y

=

(a - b) (b-

C)2

f

k'2 g nd 2 U sd2 U cd 2 U du.

[See 361.18.J

o

b

234.13

f~V t- c

~

a-t

(b'- t) (I - c)

dt= k'2gfnc2udu k2

Y

[See 313.02.J

0

b

234.14

b-t f~V t - c (a - t) (t -

~

dt=k'2 ftn2udu c) g ,

Y

b- t f~V a - t (a - t) (t -

~

c)

[See 310.02.J

g

0

fV ~

I"'dt

(a - t) (b - t) (t - c)

Y

234.17

[See 316.02.J

dt = k 2 fsn2udu.

Y

234.16

y=l=c.

0

b

234.15

y=l=c.

'

f

b

dt=gb'"

dt t"'V(a-t)(b-t)(t-c)

Y

f

'" ( 1--sn2u a k2 )'"

0

b

(1 - k 2 sn 2 u)'"

f =z;m (1~

g

duo

[See 340.04.J

dn 2"'udu ak2sn2u)"'·

0

b

[See 339.04.J b

234.18

f Y

dt (p-t)"'V(a-t)(b-t)(t-c) =

g

(P-b),,'

f

'"

dn 2 "'udu (1-rx2 sn2 u)""

0

[See 339.04.J

where b

234.19

~

_ (Pl - b) f 1 - rx~ sn2 u d f ---:------:-~====.=:'=::::;=:===c g u, (p-t)V(a-t)(b-t)(t-c) p-b 1-rx2 sn2 u (PI - t) dt

Y

0

where

".

b

234.20

f Y

[See 340.01.J

fR[b-ak 2 sn2 uJd R(t)dt V(a-t)(b-t)(t-c) =g dn2u u, 0


Integrands involvingYa-t. yt-b and yt-c. (a:::::y>b>c).

> b>

Integrands involving Va - t, Vt - band Vt - c, (a :2: y sn2u _

11' = amu

1

235.00

1

k2 =

(a - c) (t - b) (a - b) (t - c) ,

-

=

dt y(a - t) (t - b) (t - c)

j Y.

=gjU~u =

gUl

ya- c =

sin9?'

gsn-l(sinrp,k)

=

[cf.236.00J.

=gF(9?,k).

0

b

g=

snul

(a - b) (y - c) ,

c)

2

~~ a- c '

sin-l V(a - c) (y - b)

77

'"

= -g-jdn2 udu b-c

235.01

235.02

dt V j yI=p

1

t- c

(a - t) (t - b) =

j'"

b- c b- Pg

1-

du . sn2 u

0(2

0

b

=

where

gil (11',11.2 , k) ,

b- c

b-p

11.2 = (a-b)(c-p)/(a-c) (b-P), Y

235.03

o

= -g-E(ul ) =-g-E(9?,k). b-c b-c

j-~V t - c (t -

P*b.

~

t-b c) (a - t)

[See31O.02.J

=gk2 jsn Z udu 0

b

Y

235.04

~

dt = g jdc 2 '"udu, (a - t)'"Y(a - t) (t - b) (t - c) (a - b)'"

j

0

b y

235.05

j

V(a _tt)(tC_ b) dt =

(b - c) gj nd 2 u du.

jV (a -

t)/ t t-

c

)dt=(b-c)gk jsd2 udu.

[See318.02.J

0

j V(t - a)(t tY

b

~

) dt = c

(a - b) gjcd2 U du.

Y

t)/ j-~-V a- t (a - t t -

[See 320.02.J

0

b

b

[See 315.02.J

~

2

b

235.08

[See 321.05.J

0 y

235.07

y*a.

~

b

235.06

[See 400.J

~

)=gk l2 jtn 2 udu,y*a, [See316.02.J C 0

78


, 235.09

fVa

~

-/~(tc- b) dt =

(a - b) (b - c) g k 2fSd 2U cd 2udu.

b

__

"1

fVE:~tt- ~dt =

235.10

(b - C)2 g kzfsd2und2udu.

jVa-/~(t~-C) dt = b

dtf 't- c

235.12

b y

f a dt _ t

235.13

[See 361.19.J

0

b

235.11

[See 361.27.J

0

,

j~d2und2udu.

(a - b) (b - c) g

[See 361.16.J

0

V

a- t (t - b) (t - c)

=

I a -- g b fUcn 2 udu. b- c

V

~

t- c (t _ b) (a _ t)

=

b- c f 2 d -a~b g nc u U,

[See 313.02.J

y

~

JV(a -t) (t- b) (t- c) dt=g(a - b)(b _C)2 k 2J sd2 ucd2 und2 udu. b

235.15

235.16

!I(:. -~ (;~ V(:~ 'I

l

=

0

(a_b{(b_C)j~n2udc2udU,

y=1=a. [See361.31.J

o

f

'

tmdt _ bmfUI V(a-t)(t-b)(t-c) -g

f b

(1 - ~sn2Ur

dt (t-p)mV(a-t) (t-b}(t-c)

[See 340.04.J ~

=

g f dn 2m udu (b_p)m (1-a. 2sn 2u)m' 0

oc 2=(a-b) (c-p)/(a-c) (b-P),

where y

f b

du.

(1-k 2sn2u)m

0 y

235.18

[See 361.18.J

II{I-bl (1- 'I

b

235.17

Y =1= a.

0

b

235.14

[See 312.02.J

0

(t-Pl)dt (t-p)V(a-t)(t-b)(t-c) -

[See 339.04.J

P=1=b. ~

(b-Pl)gf1-a.Isn2u du b-p 1-a.2 sn2 u ' 0

where oc~=(a-b) (C-Pl)/(a-c) (b-Pl)'

oc 2=(a-b) (c-p)/(a -c) (b-P),

P=1=b.

[See 340.01.J

Integrands involvingVa-t. Vt-band Vt-c. (a>y;;;;:b>c).

f

235.19

y

79

f (1 _ ~

dt t'" V(a - t) (t - b) (t - c) =

g

1)"

dn 2 "'udu ~~ sn 2 u)'" ,

b O b

[See 339.04.]

1 y

235.20

~

R(t)dt =gfR [b-C!!.2 sn2 U ]du V(a - t) (t - b) (t - c) dn2 u '

b

0



Va - t, Vt -

sn2u=~

band

V--

Vt -

.

tp=amu1 =sm

-1

a- y --, a-b

;;;: b > c)

2 g=_.-

J.,2= a-b a-c'

a-b'

(a> y

c,

Va-c'

sn U1 = sin tp .

"1

= gfdu = gUl = gsn-1(sintp, k)

236.00

o 236.01

jl V a

t-c

(a - t) (t - b)

=

dt=(a-c)gfU~n2Udu=(a-C)gE(U1)

y

g

a-

=

(f..~=(a-b)/(a-p),

1V( y

I

a

236.04

y

236.05

o

1-

du sn2 u

0(2

p=t=a.

a

t- b ) a - t) (t - c

"1 f p

(a-c)gE(tp, k).

-g-II(tp.(f..2,k) , [See400.J a-p

=

236.03

=

0

236.02

where

[d. 235.00J.

gF(tp, k).

~

dt = (a - b) gfen 2U du.

I I abdt = VTt= ) (t t_) - c

[See 312.02.J

0 ~

(a - b) gfSn 2UdU.

[See 310.02.J

0

j~~-'::bf dt= (a-b)2gf~n2Ucn2UdU. [See 36I.01.J y

0

80


fV

____

a

236.06

(a

-/~(Ib- e)

U1

dt= (a - b) (a - c) gfsn 2u dn 2 u duo

Y

V .

a

236.07

f

:~I 1- e)

(I -

~

dt = (a - b) (a - c) gf cn 2 U dn 2 u du.

jV(a-.t) (t- b) (t-c)dt = (a - b)2(a -c) gj~n2ucn2udn2udu.

Y

0

a

236.09

[See 361.03.J

0

Y

236.08

[See 361.02.J

0

[See 361.04.J ~

=

dl

f

(I - b)m V(a - I) (I - b) (t - e)

-~--fnc2mudu y=l=b. b)m

(a -

Y

'

f

",

a

236.10

dl

g

(a - e)m

(t - e)m V(a - I) (t - b) (t - e)

Y

a

236.11

f

t

~tb

Y

V

"0/_

(I _ ab

e) =

g ftn 2u du, y =1= b. [See 316.02.J 0 ~

a-I f~V 1- e (t - b) (I -

e)

=

gk 2fsd 2 udu.

[See 318.02.J

0

a

f--~-V t- b (I -

Y

~

t-e

b) (a - t)

=.JLfdc2Udu 2 k

y=l=b.

'

[See321.02.J

0

a

236.14

[See 315.05.J

~

Y

236.13

fn:d 2m u du. o

a

236.12

[See 313.05.J

0

~

1- b f~V 1- e (a - I) (I -

e)

=gk2fcd2Udu.

Y

[See 320.02. J

0

f

a

236.15

Y

==dt====== (I - b) (t - e) V(a - t) (t - b) (t- e)

", =

236.16

f Y

(a -

g b) (a - e)

fnc2und2udu,

y=l=b.

[See 36I.l2.J

o

a ====I=m=d=t===- =

v~-~~-~~-~

am gfu'[1 __a __~ sn2 u]m du. 0

a

[See 331J)3.J


Vt-a, Vt-b

and Vt-c, (y>a>b>c).

a

dt I (t-p)mV(a-t)(t-b)(t-c)

236.17

=

".

g

(a_p)m.

"

du (1-c:t2 sn2 u)m-'

[See 336.03.J

0

where

(1.2= (a-b)/(a-p),

f

P=ta.

a

~

(PI-t)dt . (p-t)V(a-t)(t-b)(t-c)

236.18

f

Pl-~gI1-c:t~sn2u du p-a 1-c:t2 sn2 u '

=

"

[See 340.01.J

0

where

(]..2=(a-b)/(a-p),

f

236.19

81

a

f [1-

p=ta.

~

dt g tmV(a- t) (t - b) (t - c) =, am

"

du a - b sn2u]"" a

0

[See 336.03.] a

236.20

~

=gIR[a- (a-b)sn 2 u]du,

R(t)dt

I V(a - t) (t - b) (t - c) "

0


Integrands involving sn2 U

= t- a

t-b'

rp = amul

:Y

237.00

{

I

Vt - a, Vt - band Vt ~ c, (y > a > b > c)

v

=

sin-l

dt

~

It - b a

Ya- c'

g=~.-

a-c'

snul

a ,

=

V

=

sinrp.

".

gIdu = gUl = gsn-l(sinrp, k)

= g F(rp, k) .

0

:Y

237.01

2

b- c

V-yy-b -

(t - a) (t - b) (t - c)

a

{

k2 =

".

t-c _~_ dn2udu=~E (t - a) (t - b) - k'2 I k'2 (Ut) 0

= Byrd and Friedman, Elliptic Integrals.

k~2 E(rp, k). 6

82


1j V y

237.02

dl

1-

P

(I -

.

1- b

a) (I - c)

a

pgj

a- b

=

a -

_

a- b

",

du oc2 sn 2 u

1-

0

-~~g

a-p

JI( tp,OC, 2 k) .

[See 400.J

where 237.03

[See 316.02.J

j V(I-Ia~

237.04

f~C2UdU.

dt = (a - b) g

a

y

237.05

",

j V (1a

I(C

a) 1 - b)

dt=(a-c)gjdC 2udu. ~

j V(t-Ia~(tb- c) dt = (a - b) (a - c) g jtn 2udc 2udu. a ",

j V(t -

la~(tC- b) dt = (a - b)2 g jtn 2u nc 2u du.

a

[See 361.07.J

0

y

237.08

~

jV(I-lb~(la- c) dt = (a - b) (a - c) gjdc 2unc 2udu. a

j~V 1-

a

b

~

(I -

t-a

b) (I - c)

=gjsn2udu.

j~V 1- c (t -

a

",

t-a

b) (I - c)

=k I2 gjsd 2udu.

[See 318.02.J

0 y

237.11

[See 31O.02.J

0

y

237.10

[See 361.13.J

0 y

237.09

[See 361.15.J

0

y

237.07

I-b j~V 1- c (I - a) (t -

",

c)

=k f2 gjnd2udu.

[See 315.02.J

0

a y

237.12

[See 321.02.J

0

y

237.06

[See 313.02.J

0

~

dt

j (t - b)m V(I - a) (t - b) (t - c)

a

=

g

(a - b)m

!cn 2m udu. 0

[See 312.05.J

Integrands involving yt- a, Yt- band Vt- e, (y >a >b >e).

, 237.13

~

dt

f

(t - e)'" V(t - a) (t - b) (t - e)

=

237.15

1 ( ' 1

bl (, - 'I "('

J'Vet -

1

=

a

237.16

(a - e)m

0

[See 320.05.]

fcn 2 u cd2 udu.

[See 361.28.]

"'01 (, - ~ (, - 'I

(a _ e)g(a _ b)

o

a) (t - b) (t - c) dt

(a _ b)2 (a _ c) g J"tn 2 u nc 2 u dc 2 u du. o

f' V(t -

I"'dl

-:;;===;=:==~==c

a

237.17

=

a) (I - b) (I - e)

,

=

a

'"

f'" g

(1.2=

,

f

d

U.

[See 340.04.]

(a-p)'"

f'" 0

cn 2m udu

(1-cx2 sn2 u)m'

[See 338.04.]

(b-Pl/(a-p), P =f= a.

(I - Pi) dt

a - Pi = --g (/-p)V(/-a)(t-b)(/-e) a-p

--:--~~==~===c=:;===;:-

a

u)

(1 - sn2 u)m

g

=

[See 361.17.] 'm

b

1 - -;sn2

itl

a

237.18

(

0

f (/-P)"'Y(t-a)(/-b) (I-e)

where

fcd2"'u duo

g

a

237.14

83

f

0

~

cx~ sn2 u d U 1-cx sn u '

1-

- - 2- 2-

[See 340.01.]

where 237.19

f 'I"'V(t-a)(/-b)(/-e) dl

a

g

f'"

cn 2m udu

=-;·0 (1- ~ sn2u)"'· [See 338.04.]

y

237.20

f V(t -

a

fR[a-bsnu uJd. g ~

R(/)dt a) (I - b) (I _ e) -

2

0

cn2

U,

where R(t) is any rational function of t. 6*

84


Integrands involving sn2 u

=

Vt - a, Vt - band Vt - c, ( y ;;;;: a > b > c) a- e

k2 =

t-c'

b-

e

2

--==0-

g=

a-c'

Va- e

~

q;

238.00

1

am u 1

=

. -1 V a - e SIn -- ,

=

00

V(t - a) (t - b) (t - e)

=gfdu =gu1 =gsn-1 (Sinq;,k)

Y 00

Y

V

~

1-

b

(t - a) (I - e)

=

gfdn 2 u du

g E(u1)

=

Y

where

0(2

b)

=gfU, 0

1 -

du !X2 sn2 u

[See 400.J

=(p-c)/(a-c). ~

_.~_t- - - - -

f

(I - e)m V(t - a) (t - b) (t - e)

=

g

(a - e)m

Y

fsn 2m udu.

~

dt = g fsd2mudu. (t - b)mV(1 - a) (I - b) (I - e) (a - e)m

f Y

f

(I -

~

dt = a)mV(t - a) (t - b) (t - e)

g

(a - e)m

f

00

Y

dt

(1..2

00

f~ t- a

Y

238.08

g

(t-p)mV(t-a)(I-b) (I-e) =

where 238.07

=

(a-e)m

[See 316.05.J

a

f 0

~

sn2m udu (1-!X2sn2 u)m'

[See 337.04.J

(p - c)/(a - c).

V

",

t- b = (I - e) (I - a)

gfdc 2 U du, Y =F a. [See 321.02.J 0

=gf"~C2UdU fOO~V~C.t(t Y

ftn2mudu, y=F a . 0

Y

238.06

[See 318.05.J

0 00

238.05

[See 310.05.J

0 00

238.04

g E(q;, k) .

=gII(q;,(1..2,k),

00

238.03

=

0

t-e f~V t- P (t - a) (t -

238.02

g F(q;, k) .

=

0

f _!-L. t- e

238.01

sin q; .

=

",

dt

f

sn u 1

y-e

a) (/ - b)

0

,

y =Fa.

[See 313.02.]

Integrands involving Vt- a,

yt=b and

00

j~V

238.09

t- c

y 00

238.10

j~ t- b

gjcn 2 udu

[See 312.02.J

0

V

_

~

=gjcd2 udU

t- a_ (t - b) (t - c)

[See 320.02.J

.

0

"1

00

j~~V t- b (t -

t-c = jnd 2 udu. a) (t - b) g

Y

[See 315.02.J

0

j

238.12

00

~

(t-Pl)dt = (t - P) V(t - a) (t - b) (t - c)

y

{~-oc~Sn2u g. 1 - oc2 sn2 u

~

dt (t - c) (t - b) V(t - a) (t - b) (t - c)

j

=-g-jsn2 usd 2 udU. (a - C)2

0

l'

(a - C)2

o

y =F a.

1J

[See 361.29.J

j"1

g

dt

OO

238.17

[See 361.25. ]

"1 = --g-jsn2 u tn 2 udu,

238.14

238.16

'

[See 340.01.J

oc2 = (p -c)/(a -c).

oc~= (Pl-c)/(a -c); 00

238.15

du

0

where 238.13

- - - - - - : : - ; : = = = = = = = - - tn 2 u sd 2 udu, (t - a) (t - b) V(t - a) (t - b) (t - c)

Y

I j

(a -

C)2

y =F a.

j;i - a) (t -

y

Y

0

[See 361.24.]

b) (t - e)V(t - a) (t - b) (t - c) "

=_~j~nz u sd2 u tn 2 U du, (a - c)

(y =F a). [See 361.30.]

o

j

~

00

dt g tmV(t-a)(t-b)(t-c) = (a-e)m

0

sn 2m udu (1-oc2sn2u)m'

[See 337.04.]

where 00

238.18

85

"1

t-a = (t - b) (t - c)

Y

238.11

Vt- c, (00 >y 2 a >b >c).

j y

R(t)dt V(t - a) (t - b) (t - e)

"1

=g{R[c+(a-c)ns2 u J du, •

0


86


Vt - Vt -

Integrands involving a, band a real, b, C complex, (y > a)

cnu=

+

k2 =

A a-I A-a+t'

A2 = (b 1 - a)2 +a~,

a~ =

+ b) -

A

a 2A'

I If

(b

_

~ ·W

Iv

=gfdu=gu1 =gen-1 (coscp,k)

+ an

!

+t -

V

a)2

[(A

+t -

".

gf

=

o

V

Y

f

+1-

(A

d2u

(I -

b1)2

a

+ a~

=

2

1-rJ. sn u

dt

VA f

a) [(I - b)2

=

du

1 - cn U 1 cn u

+

0

+ a~l [See 400.J

g II(cp, cx.2 , k).

"1

__

1-

a)2dl

all V(I -

a)2 - 4A rJ.2(1 -

g E(u1)

= g E(cp, k).

o

! a

j~n2 u du =

g

a

1-

[d. 241.00].

g F(cp, k) .

=

+ a~

(I - b)2

dl

(A

Y

239.03

T

0

a

239.02

en u1 = cos m •

".

dl (I - a) [(I - b)2

Y

1

VA'

b1 = b ~~ = b ~_~ ,

Y

a

239.01

g=

Cj

(t - b) (t-c) = (t- b) (t- b) = (t- b1 )2 +a~;

cp = am u1 = cos -1 [AA-a+y + a - y 1'

239.00

Vt -

.

[See 361.53.J

239.04

I

".

y

239.05

a

239.06

(A

+

dl a - 1)2

1-

(l-b)2+a~

Te=-I a

a

=

~ftn2udu. [See316.02.J 4A

o

b~/+af = 4~

".

f

o

sd 2 Udu.

[See 318.02.J

Integrands involving :Y

239.07

1I V(t a

I

dt

a) [(I -

bl)1

Vla-1, (Y>1).

+ a~l

~

=g(a-A)F(cp,k)+2 gAI o

239.08

IV(t -

R (I)

:Y

a

dl

a) [(I - bl )2

+ aD

87

IU~

=

g

du

1 + enu

[+

A

a

[See 341.53.]

.

+ (a - A) en U] du, + en u

1

0



Vt3 -I,

(y

>

I)

Sp'ecial case of 239:

Iv :Y

240.00

1

dl

13

-

1

"1 =gIdu=gu1 =gF(cp,k).

[d. 242.00].

0

240.01

240.02

1J

J'"

{J13+1-1?dt

:Y.

[(V3 + t- i)" - 4ot2 V3(1 - 1)] Vta -1

1

=

1+enu

~

1-1 1"+1+1 -

1

g

0

I Vt:~ :Y

1

[See 361.53.]

.

0

:Y

240.05

[See 400.]

",

1

240.04

du 1- ot"sn2 u

g II(cp, r:t..2 , k).

t-1 dt-v-I1-en~du I Vt2+t+1 3 I I (V3+dtt -1)' V 4V3 :Y

240.03

go

1

=g(1- V3)F(cp, k)

2 d sn u u.

+ 2grSI 0

[See 310.02.]

~

1

::nu . [See 341.53.]


88

J

240.06

Iffldt =

g

V13-1

J

(1-V-)mm! ~ (- 3 3

"/-Jj!(m-j)! 1=0

(1-

~= (1-

where y

J

240.08

--

(V3+1-1)

1

r Y

240.09

•

I

240.10

dl

(12 + I +

Y

R~I) dl

J Vt3 -

I

2

1'3 P

=

•

"1

p)m /-J7T~--=--1f!

du (1+ or;cn u)i '

P =1= 1 .

[See 341.05.J

1=0

0

V3 -p)/(1 + V3 -p).

V

1- 1

t2+t+1

J-

u.

tn 2 U d u.

g ---

4V3

[See 316.02.J

0 ~

V--~~= 1) 12 + t + 1

1

[See 341.55.J

~m!(or;-l)1J

_~__

dt

0

(l+cnltji

In

g

240.07

__

___ . "1 V3)IJ~

m

y

-g-Jsd2 U du. 4

V3

[S'ee 318.02.J

0

gJ~ [1.+ Vi +(l-V3)c~!:,] du, 1

0

+ cn u



lit - a, lit - b and Vi - c; a real, b, c complex (=>y:;;:::a)

en u =

I-

a- A

k2

t - a+ A '

=

A

+ bl -

A2

a

2A'

=

(bl _ a)2

+ a21,

g=V~'

(t-b)(t-c)=(t-b)(t-b)=(t-bl)2+a~, bl=b~b,

2

bj2

al

= -

(b _ 4

'

m .,-

1V

=

am u

I

=

cos -1 [--"-y_-_a_-_A--c--] y-a+A '

J

241.01

00

dI

J -------(t - a + A)2

Y

dt

(I - a) [(I - bll S

Y

COSgJ.

u,

00

241.00

cnul =

+ ail

=gJdu=gu1 =gcn-I (COSgJ,k)

V------+ (I - b l )!

t- a

af

= g F(gJ,

0

=

u, 2

k),

[d. 239.00].

gJdn udu=gE(ul)=gE(gJ,k). 0

Integrands involving Vea-1, (00 >y;;:;; 1).

1!

00

141.02

(t- a+A)ldt [(t - a + A)I - 4A a.2 (t - a)] y(t - a) [(t - bI)1 + afJ

",

l

~u

=gf

= gil (qJ ,a.2,k).

I

o1-a.snu

00

241.03

Y

"

f

00

241.04

dt (t - p) (t - a) [(t - bI)1 +aiJ

"

f

"

00

0 ~

dt

t-a

0

(t _ bl)1 + ai =

[(t - bl)1 + aiJ

f 4A

~

t- a

g

dB

s

u

d

U.

[See 318.02.J

0 00

"

u. [See 361.60.J

4A

:Y

f

1 + a.cn u

o

=-Lftn2udu. [See316.02.J (t-bl)l+ai 4A

dt f (t-a-A)I

f

(t - bI)1 + ai

V V

00

241.08

p)

",

"

241.07

+a-

(A

f1-cnud

~~-=-LJsnBudu. [See310.02.]

dt -• (t - a + A)I

241.06

".

g

a.= (A -a +P)/(A +a-p). 00

241.05

!Lf1- cnu duo A 1 + cn u 0 [See 361.53.]

=

Y

[See 400.]

~

=

dt (t - a) (t - a) [(t - bI)1 + ail

f

89

R(t)dt

11 (t _ a) [(t _ b )2 V I

+ al]I

=

",

gJR[ a+A

+ (A -

10

cn u

a)cnu]du,

where R(t) is a rational function of t.

Special case of 241:


Vta -

cnu= t-1-YJ.

k2=~,

t- 1 +

qJ = amUt 00

242.00

f

ll

=

Y3

4

cos-1 [Y -

1-

Y- 1

> y:;:;;:

I, (00

YJ],

+ YJ

I)

g=1/~/3-,

cnUt

V

=

cosqJ.

",

dt

vtS - 1

= gfdu = gUt = g cn-l (coSqJ, k) =

,,0

gF(rp, k).

[ef.240.00].

90

242.01


f

00

Y

(t -

+V t + 1) dt 2V 1 + 3) t

(tl

8 -

1

f

'"

=g

2

dn u du

= g E(u1) = g E(rp, k) .

0

242.02

00

242.03

(t-1)Vt 8 -1

Y

f

00

242.04

'"

= -L-f1 -

dt

f

V3

dt

=

(t-P)Vt8-1

0

cnu du, 1+cnu

g

(V3+t-P)

Y

'"

1 - cnu 1+a.cnu

[See 361.53.J

d U,

0

[See 361.60.J

ot={V3 -1 +P)/{V3 + 1-p).

where

f

00

242.05

f

y =l= 1.

(t - t) dt {t-t+Y3)IVtB-1

f sn ~

=

g

4V3

Y

2

udu.

[See 310.02.J

0

242.06

where

--g-ftn udu. f (t- 1-(t-1)dt Y3)2Vt 8 -1 - 4 Y3 00

242.07

'"

2

Y

242.08

0

R(t)dt f VtB-1 00

Y

=

g

fR [t + V3 1-cnu + eV3 - 1) tnt'] du, '"

0


[See 316.02J

Va-t, Vt-b and Vt-c; a real, b, c complex, (yb>c>d>y)


sn 2 u = ct.2

(a - e)~- t) (a - d)(e - t) ,

= ~~ > 1 ,

251.00

I

g=

(b - e) (a - d) (a - e)(b - d) ,

cp = am U 1 = sin -1

a- e

y) ,

dt

t

:Y

T

1

1

=gF(cp,k).

V

1f Via -

m.

".

fd n ...

b- t b -d g -,-------,,-:----,--;-::----,= (a - t) (e - t) (d - t) e- d

2U

dU

0

b-d b-d ( ) = ~-gE(Ul) = ---gE cp,k . e-d

251.02

sn U 1 = sin

y)

fdU =gu =gsn- (sincp,k)

:Y

d

2

o

f e-

t,

Via - e) (b - d) ,

V(a(a -- d)e) (d(e -_ = g

d

251.01

k2 =

Vd

t and

e- t t) (b - t) (d - t)

c-d

dt _ c _ d -

(

= (c -

f

".

)g

0

1-

du chn2

u

d) g II(cp, ct.2 , k). [See 400.]

Va-t. Vb-t. Vc-t and Vd-t. (a>b>c>d>y).

I

d

251.03

Imdt V (a - t) (b - t) (c - t) (d - t)

y

251.04

I

d

d

V

dt=(c-d)oc 2 f

dt- ( -d) -

a

g

V

b-t (a - t) (c - t) (d - t)

f

dt c- t

dt=(b-d) f

dn 2 udu rx.2 sn2 u •

g

1/

a- t V (b-t) (c-t) (d-t) =

1-

f c=.-(jg cn udu.

t)(~ =:)

fa-c dt- -t V(a -

d- t =oc2 t) (b - t) (c - t) g

(a-

",

2

[See 312.02.]

0

J~ V t

[See 339.01.]

a- d

Y

a

(c- t)

--;;-~~- gj~n2u duo

Y

[See 316.02.]

0

Y

f V

f"'sn udu.

-dtb- t

f

",

d- t c- d 2g = --oc (a - I) (b - t) (c - t) b- d

Y

f~ b- t

V

a- t (b - t) (c - t) (d - t) -

~

b- d g

dt a- t

V

b- t (a - I) (c _ I) (d _ t) =

~

b- d a_ d

g

fd

0

.

(See 320.02.]

0

d

Y

[See 318.02.]

~- d fcd2Udu

Y

f

sd2 u d U.

0

d

251.13

[See 310.02.]

2

0

d

251.12

•

[See 338.01.]

0

a

251.11

cn2 udu rx,2 sn2 U

1-

~

Y

251.10

[See 337.01.]

0 d

251.09

1-

J

Y

251.08

sn2 udu rx,2 sn 2 u •

g

~

a_t (b - t) (c - t) (d - t)

f

duo

[See 340.04.]

0

f

sn2 u )m

~

d-t (a - t) (b - t) (c - t)

f

rx,2

)m

t) (b - t) (c - t) (d - t)

V

d

251.07

(1 -

d

dt

t m V(a -

Y

251.06

c

1 - - rx,2 s n2 u

o

y

251.05

I"('

= gdm

99

2

c u

d

U.

[See 321.02.] 7*

100


f V d

251.14

-dt-

b- t

Y

d

dtf a- t

251.15

f

~

c- t d =c"-n d2 u d u (a - t) (b - t) (d - t) b- d g .

V

~

c- t - d f nc 2 u d U. =c(a - t) (b - t) (d - t) a- d g

Y

[See 313.02.]

0

d

251.16

[See 315.02.]

0

f

'"

(e-t)(d-t)dt=(c_d)2(X2 f sn 2 udu (a-t)(b-t) g (1-a.2 sn2 u)S·

Y

0

[See 362.15.]

d

251.17

~

(e-t)(a-t)dt=(a_c)(c_d)(X2 f

f Y

0

d

(c-t)(b-t)dt=(c_d)(b_d) f (a - t) (d - t) g

Y

0

d

251.19

dnsudu (1 -a.2 sn2 u)S .

[See 362.17.J ~

(a- t)(b - t) dt = (a -d) (b - d) f

f

[See 362.16.J

~

f

251.18

cn 2 udu (1 -a.S sn2 u)2 .

g

(b - t) (d - t)

cn 2 udn2 u

g

(e - t) (d - t)

Y

(1 -a.S sn 2 u)2

[See 362.20.J

0

d

- t) (d - t) dt = f l~ V(c-t)(b-t)

251.20

~

(a - d) (c - d)

(X2

f

g

Y

0

d

251.21

0

d

0

d

f

dt (b-t)m Y(a-t) (b-t)(e-t)(d-t) -

...

0

Y

[See 340.04.J ...

d

f

[See 340.04.J

g f(1-a. 2 sn2 U)m d (b-d)m (1 - k 2 sns u)m u.

Y

251.24

[See 362.19.]

dt g f"'(1-a. s sn2 U)m d (a_t)m Y(a-t) (b-t)(e-t)(d-t) = (a - d)m (1 - sn2 u)m u.

Y

251.23

[See 362.18.J

~

Y

f

sn 2 ucn2 udu (1-a.2 sn 2 u)2·

(b-t)(d-t)dt=(b_d)(c_d)CJ..2 fsn2udn2udu (e-t)(a-t) g (1-a. 2 sn2 u)2·

f

251.22

duo

dt (e-t)m Y(a-t) (b-t) (e-t)(d-t)

= --g-f(1-oc2 Sn 2 u)m du . (e-d)m

0

[See 331.03.]

Va-t, Vb-t. Vc-t and Vd-t, (a >b >c >d >y). tl

fc~e "

251.25

= (b _ d) (XI

(b - t) (d - e) (c - t) (a - t)

" d

fa~t "

251.27

d

251.28

f

dt b- t

" d

251.29

f

de b-t

" d

fa~t

251.30

y

d

251.31

f

[See 362.11.] lit

o

(c - d)2(lt2 g !~~ a- d 1 - (lt2 snDu .

o

[See 362.06.]

(c- e) (d - t) (a- t) (b - t)

(c - e) (a - t) (d- t) (b - t)

(a - c) (c b- d

d) (ltB g

!".

CdDudu 1 -'- (ltB snB u

o

=

(c - d) (b - d) g! deB u du . a- d 1 - (ltD snDu

[See 362.09.]

lit

(a - d) (b - d) g! enB u dnD u du . c- d 1 - (lt2 sn B u 0

d

t) t)

= (a _ d) (XI

"

[See 362.12.]

lit

f snB u enS u du g 1 - (ltD snDu . 0

d

f

~.

dt V(a-t) (d-t) b- t (c - t) (b - t)

[See 362.10.] ".

=

(a-d) (C-d)(ltDgfsdzUenBudu b- d 1 - (ltD sn2 u •

"

f V (! =:)

•

[See 362.08.]

o

de Via - t) (b - t) (c - t) (d - t) c- t

[See 362.07.]

".

=

(c - t) (b - t) (a - t) (d - t)

l/fa - t) (d f~ c - t V(c - t) (b -

251.33

[See 362.14.1

".

(c- e) (d - t) (a- t) (b - t)

"

251.32

dUo

(b - d) (c - d) g f dnButnZ udu a- c . 1 - (ltB snBu •

dt V(b-t)(d-t) a- t (c - t) (a - t)

f

snBudnBu 1 - (ltB sns u

o

d

251.26

!g ".

101

0

[See 362.13.]

d

251.34

{

(d - t)

"

(c - t)

dt lit

=(C-d)2(b-d)(X2 fsnBudn2udu g (1 - (lta snz U)3 o

•

[See 362.21.]

t02


251.35 [See 362.23.J d

f V

251.36

,

(a - t) (b - t) (d - t) C- t

dt ~

= (a - d) (b - d) (c - d) oc2 fsn2ucn2udn2u du. g

(1- (l(2sn2 u)3

[See 362.24.J

o d

1

fV

251.37

dt

(a-t)(b-t)(C-t) d-t

=(a-d)(b-d)(c-d)

,

g

fU~n2Udn2UdU.

o

(1 - (l(2 sn2 U)3

[See 362.22.J

d

!Vrc-(a-----,--t)-,.,.-(b---,---t)-,---(c--t-,----,)(d-t:,dt

251.38

251.39

1

= (a-d)(b-d) (c-qrx 2

f ,

d

sn 2 ucn2 udn 2 u (1-(l(2sn 2 u)'

du. [See 362.25.J

dt (P- t)mV(a- t) (b- t) (c- t) (d- t)

- --g--f

~

-

where

251.40

f

g o

(P - d)m

o

(1- (l(2sn2 u)m (1 - (l(~sn2u)m

du

oc:=(p-c) (a-d)/(p-d) (a-c),

j 1,

(P- t) V(a-

~l(~~ t~t(C

where

P=t=d.

- t) (d - t)

- PI - ~ -

[See 340.04.J

'

f

~

1- (l(~sn2u_du

P - d g 1- (l(~ sn2 u o

'

[See 340.0 I. J

oc~ = (Pl - c) (a - d) /(Pl - d) (a - c),

oc:=(p-c) (a-d)/(p-d) (a-c), d

251.41

f ,

~

R(t)dt V(a-t)(b-t)(c-t) (d-t)


=g

P=t=d.

fR[d-C(l(2 Sn 2 u 1-(l(2sn2u

0

]d

U,

Va-to Vb-to Ve-t and Vt=-d. (a>b>e~y>d).

103

Va t, Vb - t, Vc - t and Vt - d,

Integrands involving (a > b > c ;;:;: y

snSu = (a - e) (t - d) , (e-d)(a-t)

m T

k2

=

>

d)

g=

...,..(a_------:-b)7.-(c_-_d).-:(a-e)(b-d)'

2

V(a-e)(b-d)'

. -1 ~a-e) (y-d) , «= 2 d-e . rp. = am u 1 = SIn - < 0 ,sn Ut = sm (e _ d) (a _ y) a- e

1

y

~

dt = g du =gUt =gsn-1(sinrp,k) f Y(a - t) (b - t)(e - t)(e - d) f

252.00

I

0

d

= g F(rp, k) , y

-dt-

f a- t

252.01

d

V

~

b- t

- dg - -b -

(a - t) (e - t) (t - d) -

=

'V f 1

a- t dt (b - t) (e - I) (I - a)

252.02

a- d

b- d a-d

d

252.04

a-

f

f

du

1-

(XB snB

u

0

= (a - d) g II(rp, «2, k). [«2 given above]. [See 400.] ~

,

dte- 1

V

y=Fc.

[See 321.02.]

0 ~

a-I (b - t)(e - t) (I - d)

a -- dg f nc2 u d u, y=Fc. =c- d

[See 313.02.J

0

,

~

fa~' V(a-t)(b-=-~)(e-I) = :=~gfsn2udu. d

,

f V -dle- 1

d

f

~

=a-- -dg tn2 udu, (a - I) (b - t) (e - t) a- e

, f V -dlb- t

[See 310.02.]

0

1- d

y=Fc.

[See 316.02.J

0

d

252'.07

~

= (a - d) g

d

252.06

0

dt- V b- t =b-- -dg fd c2 u d u, f (a - t)(c - t) (t - d) e- d e- t

252.03

252.05

dn2 u d u

f

g E(u) = b - dd g E(rp, k).

d

,

[ef. 253.00.]

sf

~

t- d

d « = a--- g (a - I) (b - I) (c - I) d- b

0

sd2 u d u.

[See 318.02.J

104


gIcn

y

252.08

I~V a- I (a -

~

c-I = I) (b - I) (I - d)

c-d a- d

0

d

252.09

I b=tV y

dl

-'b-= ~ gI nd udu. '"

a- I

(b - I) -:-(c---I::-)(-:-1---d=--)

IY~V--c::"'-t-b- 1

(a - I) (b - I) (I - d)

2

[See 315.02.J

I"~d2Udu.

[See 320.02.J

o

d

252.10

[See312.02.J

2 udu.

=

c- d g b- d

d

0

252.11 where

rxi=a(d-c)/(a-c) d.

J

252.12

y

-.LJ"'

dl = I"'V(a-I)(b-I)(C-I)(I-d)

(1 -

sn 2 tt)m du.

(1- a()(2 sn2u)m

dm

()(2

d O d

[See 340.04.J

IV Y

252.13

(a - I) (b - I) (I - d)

dt= c-d (

)

IV Y

1-

I-d

(a - I) (b - I) (c _ I)

dt

=

(d

-

a

)

[See 338.01.J

I

'"

2 (X

g

IV

I

Y

(a -

I

Y

I) (c - I) (I - d)

dt=(b-d)

g

0

(c - I) (1- d) dt = (d - a) (c - d)

(a - I) (b - I)

dn 2 udu ()(2 sn2 u •

1-

[See 339.01.J

I

'"

(X2

g

IV Y

d

sn2 ucn2 udu . ()(2 sn 2 U)2

(1 -

0

d

252.17

[See 337.01.J

~

b-I

d

252.16

sn 2udu 1 _ ()(2 sn2 u •

0

d

252.15

cn2 udu ()(2 sn2 u .

g

0

d

252.14

I

~

c-I

I

[See 362.18.J

~

(c-I)(a-l)dt=(a-d)(C-d) (b - I) (I - d)

g

0

cn2 udu ()(2 sn2 U)2 •

(1 -

[See 362.16.]

Va-t, Vb-t, Ve-t and Vt-d, (a >b >e:::::: y>d).

,

252.18

j

(e - t)(b - t) dt

(a - t) (t - d)

=

~

(b - d) (c - d) j cn2 udn 2 udu

g

, j

dn 2 udu

g

(1 -

[See 362.17.J

~

(a-t)(t-d)dt=_(a_d)2oc2 j sn 2 udu (b - t) (e - t) g (1 - (X2 sn2 U)2 0

d

,

•

[See 362.15.J

"1

(b - t) (t - d) dt = (a _ d) (d - b) oc 2 j sn2 udn2 udu (a - t) (e - t) g (1 - (X2sn2 u)2

j

0

d

=;==~=:::::- = j '-:---.:::-::r.==.=.d;=t (a-tlmV(a-t) (b-t) (e-t)(t-d)

252.22

(X2 sn 2 U)2 •

0

j

•

~

(e - t) (t - d)

,

252.21

(X2 sn2 U)2

[See 362.20.J

(a- t) (b- t) dt = (a _ d) (b-d) j

d

252.20

(1 -

0

d

252.19

105

g

(a- dim

d

•

[See 362.19.J

j"(1 -

oc2 sn2 u) m d u .

[See 331.03.J

0

252.23 [See 340.04.J

,

j 252.24

d

dt

(e - tim V=(a=-=t)=(b=-=t)=(=e-=t)=(=t-=d)

[See 340.04.J

252.25

,

~

V(e - t) (t - d) = (d _ c) oc2 jsn 2 ucn 2 udu j~ a- t (a - t) (b - t) g 1 - (X2 sn 2 u . d

0

,

252.26

dt j b-t d

(e - t) (t - d) (a - t) (b - t)

(d - a)

[See 362.10.J

"1

ie - d) rx. 2g jSd 2 UCn 2 UdU

b- d

o

1-

(X2 sn2

u •

[See 362.13.]

106


I

252.27

y

dt

b-

t

(e-t) (a-t) (b-t)(t-d)

=

g o

b- d

d

y

~lj{c-t)(b-t)

I a - t V~ -

252.28

t) (t - d)

=

0(2 sn2 u

.

[See 362.08.J ~

0

y

[See 362.12.J ~

_dt Via - t) (b - t) _ (a- d) (b- d) I dC2U~ I e- t (e - t) (t - d) e- d g 1 - 0(2 sn 2 u'

252.29

d

0

y

J-b~ t

252.30

(a - t) (t - d) (b - t) (e - t)

= -

(a - d)20(2 g b- d

d

I '!!_y

252.31

e- t

(a - t) (t - d) (b - t) (e - t)

(a - d)2 0(2 g I

y

Ia~t

252.32

(b-t)(t-d)

(a -

t) (e - t)

V(b - tj (t-d) = (a - d) (d Iy~ e- t (a - t) (e-t) e- d

y

g

1-

0(2 sn 2 u

tn 2 udu

•

...l-

1 _ 0(2 sn2 u'

Y --r- c.

[See 362.06.J

0(2 sn2 u

•

[See 362.11.J

=F C.

Y [See 362.14.J

0

~

V(b-t) (e-t) (t-d) dt = (a-d)(d-b)(c-d) a. 2 I sn 2 ucn2 udn2 u du. a-t g (1-0(2sn2 u)3

-----

b-t

J

[See 362.24.J

",

g

0

y

0

sn2 ucn2 udu .

(1-0(2 sn2 u )3

[See 362.23.J ~

IV(a-t) (b-t)(t-d)dt=(a_d)2(d_b)a.2 I e- t g

.

[See 362.07.J

0

d

sd:~

'

b) I"tn 2 u dn 2 u du, g 1 - 0(2 sn 2 u

252.35 IV~-t)(e-t)(t-d)dt=(a-d)2(d-c)a.2

d

o

1-

o

d

y

=FC

",

d

I

U

(d-b)a. 2 Isn 2 udn 2 udu

d

252.33

I

o

=

y

[See 362.09.J

",

e- d

d

252.36

cd2 udu 1-

(b-d)(e-d) Icn 2 udn2 udu. a- d g 1 - 0(2 sn2 u

d

252.34

",

(a-d)(e-d~ I

sn 2 udn 2udu . (1-0(2sn2 u)3

[See 362.21.J

Va-t, Vb-i, Vc-t and Vt-d, (a>b>c>y:;;;:;d).

107

!V(a-t)(b-t)(c-t)(t-d)dt

252.37

1 d

",

sn2u cn2 u dn2 u =(a-d)2(d-b)(c-d)oc2g f (1-a 2sn 2 u)4 duo

[See 362.25.J

o y

~

252.38 f V(a-t)(b-t) (c-t) dt =(a-d) (b-d) (c-d) f dn 2 ucn2 udu . t- d g (1-a 2 sn2 u)3 d 0 [See 362.22.J 252.39

f

y

(P - t)m

d

dt

V(a - t) (b - t)

-g-f

...

-

(c - t) (t _ d) -

(p_d)m

0

P-4=d.

y

(Pl-t)dt Pl-d f 252.40 f --:-----:---::r==========.======-g (P- t) V(a- t) (b- t) (c- t) (t- d) P- d 0

...

1-~sn2u 2

1 - cxa sn2 u

252.41 f

V

d

U,

[See 340.01.J

oci = oc2 (P - a)/(p - d),

where

'

[See 340.04.J

where oc 2 is given above and oc~=oc2(p-a)/(p-d),

d

(1-a 2 sn2 u)m du

(1- cx~sn2u)m

P -4=d.

",

R(t)dt = fR[d-aa 2 Sn 2 u]du. V(a-t) (b-t)(c-t)(t-d) g 1-:x2 sn2 u 0

d


Integrands involving Va t, Vb - t, Vc - t and Vt - d, (a > b > c > y ~ d) sn2 U=

(b - d) (c - t) , (c - d)(b - t)

k2

_

(a - b)

(0 -

cd k2b>c>Y:2:d).

253.12

I

c

y

109

I -em (1-ex~sn2u)m

dt ImV(a-I)(b-I)(C-I)(I-d)

~

(1- ex2 sn2 u)m

g

d

0

U,

[See 340.04.] where

a~ = b a2/e;

IV

a 2 given above.

c

253.13

~

c-I dt-(b) 2 I (a-I)(b-I) (I-d) -e a g

Y

IV

0

c

253.14

IV c

(a - I) (b - I) (c -

c

dt={e-d) I cn2 udu I) g 1 - ex2 sn2 u • 0 [See 338.01.] ~

a-I

(b - I) (c - I) (I - d)

dt

=

( ) I a- e g

dn 2 udu

1 - ex 2 sn 2 u •

0

Y

253.16

[See 337.01.]

~

I-d

Y

253.15

sn2 udu

1-ex2 sn 2 u'

[See 339.0 I.] ~

(C-I)(I-d) dt= (b-e) (e-d)oc2 Isn 2 ucn 2 udu

IV (a- I) (b - I)

g

Y

(1- ex 2 sn 2 u)2 •

0

c

253.17

[See 362.18.] ~

I l~I)(a-l) dt= (a-c) (b-e)oc2 Isn 2 udn2 udu V~I)(I-d)

Y

g

(1-ex2 s n2 u)2'

0

[See 362.19.]

__ "1 t) (b tLdt = (b - e)2 oc 2 I-~~ l{(C I VTa - t) (I - d) g (1 - ex2 sn2 u)2 • Y 0 [See 362.15.] c

253.18

c

253.19

__

111

(b-t)(a-t)dt=(a_c)(b_c) I

I

(c- t) (t- d)

Y

g

253.21

[See 362.17.]

0

c

253.20

dn 2 udu

(1-ex2 sn 2 u)2'

~

(a-I)(I-d) dt=(a-c)(c-d) Icn 2 udn2 udu IV (b - t) (c - t) g (1- ex2 sn2 u)2 • Y 0 [See 362.20.]

I

Y

c

(b-I)(t-d) dt= (b-c) (c-d) I (a-I)(c-t) g o

"1

cn 2 udu --

(1-ex2 sn2 u)2'

[See 362.16.]

.110

Reduction of Algebraic Integrands to Jacobian Elliptic Functions .

J c

253.22

'Y

253.23

~

dt = _ g _ j ( 1 - I XBSnBU)'" du (a-t)"'lf(cl-t) (b-t) (e-t) (t-d) (a-c)'" (1 - kBsnBu)'" • 0

1

c

dt J (t-d) "'Jf(a"-t) (b-t) (e-t) (t-d)

Y

g (e-d)'"

j

c

dt (b-t)"'Yb:2:y>c>d).

,

254.09

J V dta- t

,

C

254.10 where

254.14

(1 -1X~sn2u)'" d (1 -1X2 sn2 u)'"

o

J

J(1~

(1 -1X2 sn2 u)'"

,

t-c dt-( (a - t) (b - t) (t - d) -·C

V

J

,

-

d)

=

(

254.19

[See 337.0 I.]

[See 339.01.]

~

b-t dt=(b-c) J cn2 udu (a - t) (t - c) (t - d) g 1 - 1X2 snB u . o

[See 338.01.]

V

'"

b-t (a - t) (t - c) (t - d)

b -c gJcn2u dUo c-d o

[See312.02.]

~

(t-c)(t-d)dt=(c_d)2oc2 J sn2 udu (a-t)(b-t) g (1-1X2 sn 2 u)2'

o

[See 362.15.] ~

(t-c)(a-t)dt=(a_c)(c_d)oc2 Jsn 8 udn2 udu

g

0

, J

snSudu 1X2 sns u •

1-

J d!l2udu a - c) g 1 _ 1X2 sn2 u . o

(b - t) (t - d)

'

U.

~

a-t dt (b - t) (t - c) (t - d)

dt J t- d

J

J

2

oc g o

IV

J

d

2

[See 340.04.]

V

J

J

dsn u)'" c

1X2

~

C

254.18

U,

[See 340.04.]

0

, 254.17

-- c'" gf.

t"'dt

dt g t"'V(a-t)(b-t)(t-c)(t-d) =-;m

, 254.16

[See 318.02.]

ti,

,

,

254.15

a- c

oc~=(b-c)dlc(b-d), and oc 2 is given above.

,

254.13

c - d 2 gJ s d2 u d U. --oc

V(a - t) (b - t) (t - c) (t - d)

c

254.12

=

0

J &

254.11

~

t- c (a - t) (b - t) (t - d)

(1 -1X2 sn2 u)2 •

[See 362.19.]

(a-t)(b-t)dt=(a_c)(b_c) J~nsudn2udu (t-c) (t-d) g (1-1X2 sn2 u)2'

o

[See 362.20.J ~

(a-t)(t-d)dt (b-t) (t-c)

Byrd and Friedman, Elliptic Integrals.

=

(

c-

d)(

) J dn2 udu a-c g (1-IXS sn2 u)2'

o

[See 362.17.] 8


114

y

(t - c) (b - t) dt I V (a-t)(t-d)

254.20

=

~

(b _ c) (c _ d) oc2 Isn 2 uc n2 udU g (1-oc2 sn2 u)2'

[See 362.18.] .

o

f

y

(b-t)(t-d)dt=(b-C)(C-d) I V (a-t)(t-e)

254.21

254.22

~

f

dt = (a-t)mV(a - t) (b - t) (t - e) (t - d)

dt --_.....

I

cn2 udu (1-oc2Sn2U)2'

[See 362.16.]

I

~

y

•

o

y

c

25423

g

...

(b-t)mV(~-I) (b-I) (I-e) (I-d)

g

(a- e)m

~

g I (b-C)ffl

= --

(1-oc2 sn2 u)m du (1 - k 2 sn2 u)m .

0

[See 340.04.]

(1-oc 2sn 2u)m du (1-sn 2u)m '

f

.

[See 340.04.]

0

C

y =1= b

y

254.24

C

dt (t - d)m V(a - I) (b - I) (t - e) (t - d)

"1

y

254.25

Ib~t

(t-e)(a-t) _ (b - t) (I - d) -

y .

254.26

254.27

I)(t I ~V(at- d (b - t) (I -

I

y

e)_

d)

dl

(a - t) (b - t)

t- d

(t-e)(t-d)

(a-e)(c-d) b- d

=

dt V(t-e)(b-t) I a- t (a - t) (t - d)

o

254.29

It~d

(a -

g

[See 362.11.]

(a - e) (b - e) g I c- d

"1

cn2 udn2 udu 1 - oc2 sn 2 u .

o

[See 362.12.] ~

=

(b-c) (e-d)oc 2 g ISd2UCn2UdU a- e 1 - oc 2 sn2 u . 0

[See 362.13.]

"1

(I-e)(b-t) =(b-c)oc 2 Isn 2 ucn 2 udu. t) (t - d) g 1 - oc2 sn2 u

(a -

y

dt V(b-I)(I-d) I a- t (a - I) (t - e) C

.

sn2 udn 2 udu 1 - oc2 sn2 u .

c) oc2 I

[See 362.10.]

o

254.30

=l=b

[See 362.14.]

C

y

,y

"1 o

y

254.28

Itn 2 udn 2 udu g 1 - oc2 sn 2 u

(b - e) (e - d) a- e

gf"1 o

cd2 udu oc2 sn 2 u •

1-

[See 362.08.]

Va-t, Vb-t, Vt-c and Vt-d, (a >b::::: y >c >d).

f

254.31

y

dt

b- t

(a- t) (t- d) (b- t) (t- c)

(c - d) (a - c) b- c

fa~t

(t-c) (t-d) (a-t) (b-t)

(c - d)B(X2 g a- c

y

254.32

f

254.33

254.34

y

dt b-t

b-

d

o

f'" o

[See 362.07.J

1 - (X2 snB

o

y

"'

f

0 y

(a-t) (t-c) (t-d) dt b-t

= (c _ d)2 (a _ c) oc2

y

f V(a-t) (b-t) (t-d) dt t-c

g

f

[See 362.23.J

'" sn2 u dna u du

o

= (a - c) (b -c) (c - d)

(1 - (X2 sn2 U)3

f

•

[See 362.21.J "'

g

fcn2udn2udu. (1 -(X2sn2 u)3

0

c

254.37

y =f= b. [See 362.06.J

fV(b-t)(t-C)(t-d) dt=(c-d)2(b-c)oc2 fsn2ucn2udu. a-t g (1-(X2sn2u)3

c

254.36

y . [See 362.09.J

sdBu du 1 - (X2 sn Bu .

c

254.35

=f=b

dc 2 udu

1 - (X2 sn2 u'

"' tn2 udu u'

(c- d) 2 g f

(t-c) (t- d) (a-t) (b-t)

gf"'"

11S

[See 362.22.J

'"

y

(a-t) (b-t) (t-c) dt= (a-c) (b-c) (c-d)oc2" fsn2ucn2udn2u du. t-d g (1-(X2 sn2 u )3

o

c

[See 362.24.J

y

1

JV(a-t) (b-t) (t-c) (t-d) dt

254.38

f( '"

C

=(a-c)(b-c)(c-d) 2 OC2

g

o

254.39

f c

where

y

dt (P-t)mV(a-t) (b-t) (t-c) (t-d)

sn 2 ucn 2 udn 2 u 2 2 )4 1-(X sn u

=

-g-f

'"

(p_c)m

0

oc~=(b-c) (P-d)/(b-d) (P-c),

f'"

PI-C p-c g

o

duo

[See 362.25.J

(1-(X:sn 2 u)m du,

(1-(X3sn2~t)m

[See 340.04.J p=f=c.

1-r4sn1 u 1-(X:sn2 u

du, [See340.01.J 8*

116


where

oc~

=

(b - c) (PI - d)/(b - d) (PI - c),

oc~

=

(b - c)

(P - d)/(b - d) (P - c),

y

254.4lf

~

R(I)dl. V (a-I) (b-t) (I~c)(t-d)

c

P =t= c.

= fR[c-a. 2 dsn 2 u]d g 1-a.2 sn 2 u u, 0

where oc2 is given above and R(t) is a rational function of t.

Va - t, Vb - t, Vt - cand Vt - d,

Integrands involving (a > b > y ;;;;: c > d) sn2 u =

J~

-

c) (b- t)

k2 = . (b

0-~0-~'

0d).

(b _ ) 2 c oc g

y

b

255.28

j

dt

1-

y b

j

255.29

y

d

dt ~t)(b-I) t- c (I - c) (t - d)

b

255.30

j

dt t- d

y b

jt~c

255.31

(t - c) (b - .1) (a - t) (t - d)

y

(a - t) (b - t) (t - c) (t - d)

(a - t) (t- d) (b - t) (t - c)

1-

b-

r u,

g o

b

0(2 sn2

1-

o

y =1= c . [See 362.06.J

u'

0(2 sn 2

• u

[See 362.07.J

(b _ d)

oc 2

",

dc 2 U du

1 _ 0(2 sn2 u '

j t~c

[See 362.09.J

g

sn2 udn2 udu . 0(2 sn2

1-

u

0

[See 362.II.J

(b-t)(t-d) = (a-b) (b-
j

=l=c

,y . [See 362.14.J

a- c

1 _0(2 sn2 u

o ~

b

j V(a-t) (t-c) (t-d) dt= (a-b)(b-c)(b-d) jcn2 u dn2 u du . b- t g (1- 0(2 sn2 U)3 Y

0

[See 362.22.J

b

~

j Vb-t) (I-c) (t-d) dt= oc2(a_b)(b_d)(b_c)jsn 2 u cn 2 u dn2 u du. a- t g (1 - 0(2 sn2 U)3 Y

255.36

[See 362.13.J

tn 2 udu 1-

(a - b) (b - d) g j b- c

y

255.35

1 - 0(2 sn 2 u

(a-b)20(2 g j"' sd2 udu b- d

•

", sd 2 ucn2 udu

j

o

y

255.34

cn2 u du 0(2 sn2 u

[See 362.10.J

d

(a-b)2 g a- c .

u

o

(a-b) (b-C)0(2

b

255.33

sn 2

o

l~ t) (t- d)_ = j~ a - t V~- t) (t - c)

255.32

j ",

119

0

[See 362.24.J

b

~

I

jV(a-t)(b-t)(t-C) dt=(a-b)2(b-c)oc2 j t- d g

Y

sn2 ucn2 udu. (1 -

0

0(2 sn2 U)3

[See 362.23.J

b

fV(a-t) (b-t) (t-c) (t-d) dt

255.37 •

",

Y

= (a _b)2 (b -d) (b -c)

oc2 g j o

~u~u~u

(1 -0( 2 sn 2 u )4 du.

[See 362.25.J

120


255.38

f

f

b

Y

dt = (p - t)m V(a- t) (b - t) (t- c)(t - d)

where

g

(P - b)1n

o:i=(p-a) (b-c)/(P-b) (a-c),

~

(1-a; 2 sn2 u)1n du (1 - a;:sn 2 u)1n .

0

[See 340.04.J

P-4=b.

b

~

255.39 fV(a - t!J? - tL(t - -d:)-dt t- c

=

(a _ b)2 (b _ d) 0:2gf sn 2 udn2 udu • (1 -

Y

255.40

0

f

sn2 U)3

[See 362.21.J

~

b

Y

a;2

(Pl-t)dt (p-t)V(a-t)(b-t)(t-c)(t-d)

where

Pl-b gf 1-IX~sn2u du p-b 1-lXisn2 u '

=

0

[See 340.01.J

o:~ = (PI - a) (b - C)/(PI - b) (a - c),

o:i =

(1' - a) (b - c) /CP

-

P-4= b.

b) (a - c),

b

~

R(t)dt fR[b-aIX 2 Sn 2 u]d f V(a-t) (b-t)(t-c)(t-d) =g 1-a;2sn2u U,

255.41

Y

0



Va-t, Vt-b, Vt-c and Vt-d,

(a ;;;; y

sn 2 U =

(a - c) (t - b) (a - b) (t - c) ,

k2 _ -

> b> c>

d)

(a - b) (c - d) (a - c) (b - d) ,

k2
e>d). a

~

sn2ucn2udu • sn 2 u )3

257.UfV(a - I) (I - b) (t - d) dt = (b - a) (a _ d)2 oc2 f I-e g Y

257.25 f

1~-t) (I-b) (I-e) dt=(d-a) (a-b) (a-c) oc2 f sn2ucn 2udn2 udu •

V-~

I

g

t- d

j

o

1-

0(

[See 362.25.J

sn u)

~

dt = g f(1-C(2sn2U)mdu. (t-c)mV(a-I)(I-b)(t-e) (t-d) (a-e)m (1-k 2 sn 2u)m

Y

0

a

257.28f

[See 340.04.J ~

dt = g f(1-oc2Sn2u)mdu. (t - d)m V(a - t) (t - b) (t - e) (I - d) (a - d)m

l

Y

f

Y

[See 331.03.J

0

a

f"I(1-C(2sn2U)mdU g (a-b)m o (1-sn 2 u)m '

dl (t-b)mV(a-I)(t-b)(t-e)(t-d)

y =f= b.

a

~

[See 340.04.J

257.30 f~V(t- c) (I - d) = (a - e) (a - d) f- dC2U~U_~, 1- b (a - I) (t - b) a- b g 1 - C(2 sn 2 u Y

0

"1

a

f~ t- d

(I-e)(a-I) = (c-a)oc 2 fsn2udn2udu. (I -

g o

b) (t - d)

Y

a

257.32f~V'(t - c) (a- t) t- b

(I -

=

b) (I - d)

1-

0(2

sn 2 u

257.33f~V(I- c) (t- b) t- d (a - t) (t - d)

=

[See 362.14. J

"1

(a - b) (a - e) g f cn 2udn 2 udu a- d 1 - C(2 sn 2 u . 0

y

a

257.34 f~V(a - t) (t - b) = t- c (I - c) (t - d)

[See 362.12.J

"1

(d_- a) (a - b2C(2 gf sd 2 ucn 2 u du a- e

1-

Y

0(2

0 a

257.35f~V(a-I)(t-b) 1- d

(t - c) (t - d)

[See 362.II.J

"1

0 a

=f= b. y [See 362.09.J

(a - d) (a - e) g f tn 2 udn2 udu b- d 1 - C(2 sn2 u .

Y

y

sn2 11)3

[See 362.U.J

"1 s~2ucn:u~n24u duo

a

257.31

0(2

V(a-t) (t-b) (t-c) (t-d) dt

Y

257.27f

(1 -

0

= (a-b) (c-a) (a-d)2oc2gf

257.29

[See 362.23.J

~

Y

257.26

(1-0(2

0

a

127

sn2 It

•

[See 362.13.J

~

=

(b-a)oc 2 fsn2ucn2udu.

g

0

1-

0(2

sn 2 u

[See 362.10.J


128

"

257.36f~ t- c

,.

"

257.37f~ t- b

,.

"

257.38f~ t- c

,.

(a-t) (t-d) = ~- d)2 0(2 g f'" sd2 udu . (t - c) (t - b) c- a 1 - 0(2 sn2 u

[See 362.07.J

o

(a-t) (t-d)

=

(t - b) (t - c)

(t-b)(t-d) (a - t) (t - c)

(a- d) 2 g f b- d

=

o

",

tn2u~

1 - 0(2 sn2 u '

Y=l=b.

[See 362.06.] ~

(a-b) (a-d)gf cd2 udu . a- c 1 - 0(2 sn2 u

[See362.08.]

0

"

257.39f

,.

dt (P - t)m V(a - t) (t - b) (t - c) (t - d)

where

(1~=(P-d) (a-b)/(a-p) (b-d),

257. 40 f

"

:Y

where

257.41 f

=

(Pl-t)dt (p-t)V(a-t)(t-b)(t-c)(t-d)

~

Pl-a gf 1-0(~sn2u du, p-a 1-O(fsn2 u 0

(1~

= (PI - d) (b - a)/(Pl - a) (b - d),

(1~

= (P - d)(b - a) /(P - a) (b - d),

"

p=l=a.

[See 340.01.]

P =l= a.

~

R(t) dt = gfR [a - 0(2dsn2U] du V(a-t) (t-b)(t-c)(t-d) 1-0(2sn2 u '

,.

0

whereR(t) is a rational function of t.

Integrands involving (y 2

(b-d) (t-a)

_

sn u - (a-d) (t-b) ,

258.00

,I.

Vt-a, Vt-b, Vt cand Vt~d,

>a>

b > c > d)

k 2 = (b-c) (a-d)

(a-c) (b-d) ,

f{! = am"t = sin-1

g=

(b-d) (y-~ (a-d) (y-b) ,

2

V(a-c) (b-d) ,

sn"t = sinf{!.

~

dt _ du - u - sn-1 sin k f V(t-a) (t-b) (t-c) (t-d) - g f - g 1- g (f{!,)

"

0

= g F(f{!, k) .

Vt-a, Vt-b, VI-c and VI-d, (y>a>b>c>d). y

258.01

{

dt t-c a-c t- b V(t-a) (t-b) (I-d) = a-b

f

~

129

a-c

2

g f dn udu = a-b g E(u1 )

a

0

a- c

= a-b gE y

258.02

{

(

)

fP,k .

~

t- b

fV(t-a) (t-d) (t-c)

dt

a

b du =(a- )g.f1-tX2 Sn2 u

=

0

(a-b) gII(fP,oc2 , k),

[See 400.]

where oc2 is given above.

",

y

t- b = 258.03 ft~c -:------:-:-.,----:-:-:-----::(t-a) (t-c) (t-d) a

f

y

258.04

dt t_ c

V

a (t-b) (I-c) (t-d) I-

a

=

y

t- d

a

o

f

t-a (t-b) (t-c) (t-d)

=

u,

y

f

dt t- c

11 t- d V(t-a) (I-b) (I-c) =

[See 320.02.]

0

f

dt t- b

t-d (I-a) (t-b) (t-c)

-;-:-----;-;~;_:__;:_--,-

=

a-d

f

~- g a- b

u,

en2 U d u .

[See 312.02.]

o

~

t-a =oc2 fsn2udu. f~V t - b (t-b) (t-c) (t-d) g

[See 310.02.]

0

a

f

y

a

[See 316.02.]

~

a - d f d2 d a _ c g e u U.

y

dt t- d

f'"

t-b a-b g -;---;-;.,---,--;-:-----::= ~(I-a) (I-c) (I-d)

a- d

ne 2 U d u .

f~ 11 t- d

[See 313.02.]

o

y

258.10

[See 318.02.]

0

a

258.09

d U.

a-b gftn 2 udu. b- d

y

258.08

~

a- b 2 d2 a _ c oc g s u

a

258.07

[See 315.02.]

0

258.05 f~V

258.06

a-b g f ndu 2 du . a- c

~-

~

t-c

VTt= a) (I -

b) (t - d)

a Byrd and Friedman, Elliptic Integrals.

= a-c gfde2udu.

a- d

[See 321.02.]

0

9

130

Reduction of Algebraic Integrands to Jacobian Elliptic Functions. 1

~

t"'dt J (1-/XB snBu)'" J V(t-a) (t-b)(t-c)(t-d) = a"'g (1 -/X!snau)'" dUo [See340.04.J

258.11

0

4

at~ = (a - d) b/a (b -d).

where

f

g

dl

1

258.12

t"'}'(I-a) (t-b) (t-c) (t-d) = a'"

t-c J11 VTt= a) 1

"'(1-/Xl snB u)"'du

0

4

258.13

f (1 _

(I - b) (t _ d) dt

dnludu 1 _ /XI snBu •

(a - c) g

1

'"

t- d dt= (a -d) J (t-a) (t-c) (t-b) g

JV

J

1

~

t-a dt-( (t-b) (t-c) (t-d) - a-

b) ats g

snBudu 1 _ /XI sna u .

1

J

'" (t-c)(t-d) ( ( ) Jcnaudnaudu 6 (t-a)(t- b)dt= a-c) a-d g (1-/XI sna u )1. [See3 2.20.] 0

4

1

~

(t-c)(t-a) dt=(a-b)(a-c)at2 JsnBudnludu (t-b)(t-d) g (1-/Xa snl u)8'

J

0

4

1

J

[See 362.19.J

(I-c) (t-b) dt- ( b) ( ) J '" dn1udu (t-a)(t-d) - aa-c g (1-/XBsnBu)B' [See 362.17.]

4

0

1

258.19

J

(t-a) (t-b) d t (t-c) (t-d)

=

(a _ b)2 at! J '"

g

snBu du . (1-/X2 sn2 u)B

[See 362.15.]

0

4

1

~

(t-a)(t-d)dt=(a_d) (a-b)at2 Jsn 2 ucnl udu. (t-b) (t-c) g (1-/Xa snB u)a

258.20 J

o 1

258.21 JV(t-b)(t-d) dt-(a-b)(a-d) J (t-a) (t-c)

4

[See 337.01.]

0

4

258.18

[See 338.01.J

cnBudu /XI snl u .

1-

0

4

258.17

[See339.01.]

0

258.14 JV

258.16

[See 340.04.]

J'"

=

4

258.15

/X1b snlu)'" • a

-

g

0

[See 362.18.J

~

cn2 udu

(1-/Xa snB U)2

•

[See362.16.J

V'-a. V'-b. V'-G and V'-d. (y>a>b>G>d).

I

y

258.22

I

~

dl C (I-G)"'V(t-a)(t-b)(t-c) (I-d) = (a- c)'"

I

y

258.23

C (a - d) '"

I

y

dl (l-b)"'V(e-a)(t-b)(l-c)(I-d)

=

C (a- b)'"

258.26

dt

t- b

-etlsn1u)'" du

f"

I'" 1- ocllsnSu "'du (

).

[See 331.03.]

(a-c) (a-d) cIen1udn1udU a- b 1 - etl sna u •

~

dt t-b

258.28

I"

.

(I-c) (I-a) = (a - b) (a - c) etlCI sn1udnaudu (I-b) (I-d) a- b 1 - eta snl u •

de t-d

I"

dt 1- d

I t~ 11

c

'"

258.31

258.32

I.. "t~

c

~

(t-c) (I-a) (I-b) (t-d)

(a- b) (a- c) Citnaudni udu b- d 1 - a l snl u .

(I-b) (I-d) (t-a) (I-G)

(a- b)(a - d) CI cdS u du • a- G 1 - a l snl u

~

o

'"

(t-a) (t-d)

I"t~

(I-a)(t-b) (I-G) (t-d)

/I

c

[, See 362.08.]

(t-a) (I-d) _ (a - d) (a - b) et· CI sda u enD u du (I-b) (I-c) a- c 1 - eta snD u .

I t ~b

..

[See 362.14.]

o

o

"

[See 362.09.]

o

A

258.30

[See 362.11.]

(t-c) (t-b) _ (a - b) (a - c) c i de· u du (t-a) (I-d) a-d i-al sn1 u·

/I

258.29

[See 362.12.]

o

o

258.27

.

[See 340.04.]

'"

(t-c) (t-d) (I-a) (t-b)

U.

(1 - sna u) '"

0

f"

d

[See 340.04.]

0

A

258.25

1(1 ~

dt --:(t-----=d)-::"'::-Vi7(t=-=a=)(::=I-===b)=;:(I=-=C::=)(::=,-===='""d) -

A

258.24

(1 - etlsnBu)'" (1 - klsnlu)'"

0

A

131

[See 362.13.] [See 362.10.]

(t-b) (t-c)

=

(a-b)D et2 CI a- c

o

'"

sdludu 1- eta snl u

•

[See 362.07.] 9*

132

Reduction of Algebraic Integrands to Jacobian Elliptic Functions. y

258.33

!~

t-d

a y

258.34

(t-a) (t-b) (t-c) (t-d)

=

",

(a - b)2 g! tn2 u du . b-d 1-oc2 sn2 u

[See 362.06.]

0

~

(t-b)(t-c)(t-d)dt=(a_b)(a_c)(a_d) !Cn2 Udn 2 udu.

!

g

t- a

a

!

y

258.35

[See 362.22.] ~

Vt-a) (t-c) (t-d) dt=(a-d) (a-b) (a-c) oc2 !Sn 2UCn 2udn 2udu. t-b g (1-oc2 sn2 u)3

a

0

y

258.36

(1- oc2 sn 2 U)3

0

[See 362.24.]

~

(t-a) (t-b) (t-d) dt = (a _ b)2 (a _ c) oc2 ! sn 2 u cn2 u du t-c g (1-oc2 sn2 u)3·

!

0

a y

[See 362.23.] ~

(t-c)dt= (a-b)2(a-c)oc 2 !Sn 2 Udn 2 udu. 258.37 ! V (t-a) t(t-b) - d g (1-oc2 sn2 u)3 a 0 [See 362.21.]

I

!V(t-a) (t-b) (t-c) (t-d) dt

258.38

a

y

258.39

! a

".

2 udu =(a -b)2( a -d)(a _ c) oc2 g!Sn2 uCn2 udn 2 2)4 • [See 362.25.] (l-ocsnu

o

~

dt g ! (1-oc2 sn 2 u)m d (p-t)mV(t-a) (t-b)(t-c) (t-d) = (p-a)m (1-oc~sn2u)m U, 0 [See 340.04.]

where

258.40

oc~=(P-b) (a-d)/(p-a) (b-d),

!

y

a

p=4=a.

f

".

t) dt _ PI - a 1 - oc~ sn 2 u d --g U, (P _ I) V(t-a) (I-b) (I-c) (t-d) P - a . 1 - oc~ sn2 U 0 [See 340.01.] (PI -

-:----::---.:F'===~~=;===;=;==~ -

where oc; = (P - b) (a - d)/(P - a) (b - d), oc~

258.41

!

=

(PI - b) (a - d)/(PI - a) (b - d).

y

a

P =4= a

~

R(t)dt V(t - a) (t - b) (t - c)

(t _

ell

=g!R[a-boc2 Sn 2 u]du, 0


1-

oc2 sn! u

Va-t, Vt-b, Vt-c and Vt-c, c,ccomplex, (a:;;;:y>b).

133

Vt=c

Integrands involving Va t, Vt - b, and Vt - Cj (a, b real, a ~ y > b, c, C complex)

V(a-t) (t-b) (t-c) (t-c)= V(a-t) (t-b) [(t-bl)2+a~]; (c - C)2

2

cn u --

a1- - - - 4 - '

+

= (b _ b )2

B2

I

cnu

1

Y

I

259.00 ( f V

= eoscp

2

aI'

(a - t) B - (t - b) A

g

= _1_

b

y) B - (y - b)A] (a - y) B + (y - b) A •

1

=

gf"~U = g U

1

dt

259.02

+ (t -

b) A J2

(t-c) (t-c) (a-t) (t-b) =

f a~

t

t- b _ (a-t) (t-c)(t-c) -

Y

259.03

b

-g-fdn2UdU (a - b)2 0

E(u 1)

(a!.- b)2

o

In

.

.

~ ~m-1 (ex - CX2l' m! X ~ (m-j)!j! 1=0

(a!.- b)2

y

E(cp, k) .

=l=

a. [See 361.53.]

+

g(aB bAlm X (A - B)m

f

U1

0

du

(1+excn4'

[See 341.05.]

oc= (A -B)/(A +B), OC2= (bA -aB)/(aB +bA).

where Y

dt f --:--~:::-::;:===========(t-p)mV(a-t)(t-b)(t-c)(t-c) b

(A+B)mg [A(b-p)-B(a-p)Jm X

-

259.04

=

~

tmdt V(a - t)(t - b)(t - c)(t - c) -

k) .

~

=

B gf 1- cnu d A 1 +cnu U,

b

f

= g en -1 (cos cp, k) = g F(cp,

0

Y

2

cp = amu = eos-I[(a-

,

dt (a-t) (t-b) (t-c) (t-c)

[(a - t) B

+ aI'

4AB'

Y

259.0 I

a - bI )2

k 2 = (a - b)2 - (A - B)2

VAB'

b

f

(

A2 =

(a- t) B+ (t- b)A '

[See 341.05.]

oc= (bA -aB +PB -pA)/(aB +bA -pA -PB),

where

f

OC1 = (A - B)/(A

",

Y

259.05

b

+ B).

dt (t - c) (I - c)

(a-t) (t-b) = (a-b)2gfsd2udU. [See 318.02.] (t-c)(t-c) 4AB

o

134


f

:Y

259.06

(a-t) (t-b) = (t-c) (t-C)

dt [(a-t)B-(t-b)A]B

b

4AB

[See 316.02.]

o y

259.07

", -g-ftnZ U du.

f

~

R(t)dt

=gfR[aB+bA+(bA-aB)Cnu]dU A B (A - B) cn u '

Y(a-t) (t-b) (t-c) (t-C)

b

+ +

0

where R(t) is a rational function of t. In the special case when a = - b, and c = of this appears in 259.75.


Vt t',

C, see 213. An example

-

(0< y ::.s;; I)

Special case of

259: dt f yt(1-ea) y

259.50

o

y

259.51

'"

=gfdu=gu1=gcn-1(cOS1p,k)= ~~F(1p,k), r3 0

[d. 259.00]. _ _ ~."1

Y

dt V'+t+t2 - g f dn2 udu -g 1 2 ( ) E(u1 )- - ~rE(1p,k). [H(3-1)t] t1-t r3

f

o

0

259.52

g

259.53

(V3-1)"'~

(m-i)!i!

1=0

f

o

259.54

y

(HV3)mg

dt

f

=

(V3 -1 )/(V3 + 1),

0

0:1

=

(p -

0

du [1-(Y3-2)cnu]i·

[See 341.05.]

X

(p - 1 - p V3)'" m . . "1 ~ (1."'-1((1.1 - (1.)1 m! du XL! (m-i)!i! (1+(1. cnu)i ,

(t - P)"'Yt(1 - t8 )

i=O

0:

."1

.

m

~(-1)"'+I(J-V3)lm!f

[See 341.05.]

1

1-

PV3 )/(1 - p - pV3)·

Vt-a. Vt-b. Vt-c and Vt-c; e.c complex. (b y ~ I)

".

dt

"

Z-

~ 1)1

= ""tl:-'-_-----c-+-:-'-:-, t(t -1)

t

y =1= 1.

".

~fCd2UdU. 4

[See 321.02.]

[See 320.02. J

o

00

261.58

" 261.59 f "

[See 315.02.]

dt (2t - 1)1

00

".

R(t)dt

Vt(t-1)(tl-t+1)

=fR[' + 2dSU] du 2

0


'

Vt4 + 2 b t + a

Integrands involving (Here all the zeros of

t4

2 2

+ 2b + a 2 t2

+a

y2

00

f :Y

Vt4 +

00

>y

~ 0)

are complex)

2a2

2 '

qJ = am ~ = cos -1 [ y2 -

263.00

(0 < b < a;

k2=aB-~

cnu=t2 -a 2 t2

4

4,

2

+ aa2 ] ,

g=1/2a, '

cn ~ = cos qJ.

",

dt 2b2 t 2

+ a'

= gfdu = g~ = g cn-1 (cos qJ, k) = g F(qJ, k), 0

[ef.225.00].

263.01

263.03

[See 310.02.J

263.04

[See 312.02.J

263.05

[See 341.55.J

263.07

263.08

[See 321.02.J

142


JV"+ co

263.09

",

JR[asnU]d

R(t)dt

y

a' =g

2b1 tl+

0

i-cnu

U,

where R(t} is any rational function of t.

V + I, (00 > y ;;;;:, 0)

Integrands involving t4 Special case of above:

tl- 1

cnu=-z--' t

+1

cn~=coslp,

11' = am ~ = cos-1 [~: ~ :] •

co

263.51

k2 =1/2,

J{:++1;.

...

dt=

~Jdn2udU= ~ E(u1 } = ~E(lp,1I2/2).

Y

0

~ II(lp,oc 2 , V2/2).

263.52

[See 400.]

263.53

[See 310.02.]

263.54

[See 312.02.]

263.55

[See 316.02.]

f tV" + co

263.56

Y

263.57

",

~f1-cnu dU=~ln[ 2(i+dnuJ ] =-.!.In[t+J9B] .

dt

1

2 o

snu

2

cnu1 +dnu1

2

Vy'+1-1

[See 341.55.]

f

2 f de u du,

00

263.58

",

Vt4+ 1 (t 2 _ 1)2

-

1

dt -

f

00

263.59

2

y =1= 1.

[See 321.02.J

...

V

1 fSd 2 U duo -8

t = (1'+1) 1'+1

y 00

263.60

2

0

y

".

f R(t)dt

VI' + 1

y

= ~fR[ 2

[See 318.02.J

0

snu_]dU, en u

1-

0


V~ : ~:,

tn u =

1

g = -+ -k', 2a

264.00

f

rp=am~=

I

dt Vt 2 + 2b2t 2 a

y

~ e)2'

t an

c= :

-1 [1V- -+- ,

+ a'

(t S

+ 2aet + as) dt

= -L k'

f

dn 2 udu

0

..

'" ~ (_1)"'+l m I2'

ga ~

1=0

(m-i)lil

:'

f (1 + "1

0

E(rp, k) .

du Vk'tnu)i'

[See 342.05. J

...

a

(a-t)"'dt

(a

y

".

= { , E(u1) =

=

f 264.04

,

[d. 263.00J.

0

f (t2-2aet+a2lVt4+2b2tS+a4

y

S

b

t nUl =tanrp.

a- y] k' a y

m

f

Va22~2

=gJdu=g~=gtn-l(tanrp,k)=gF(rp,k).

264.02

264.03

~ ;;- =

...

a

y

264.01

k 2 = (1

+ t)m VI' + 2b2 tS + a4

a

dt

(t - p)m

VI' + III

~

X

~ 1=0

2h2 t 2

=gk''''ftn2mudu.

+ a4

ml (2a)i (m-i)!j!(p-4

[See 316.05.J

0

f (1+cx.tn4' ".

0

du

[See 342.05.J

144


a=(a+p)Vk'/(p-a). p+a.

where a

~

(a-t)2dt =(1+kl)gfsn2udu. f (t 2- 2act + a2) Vt' + 2b2 t 2 + a'

264.05

Y

a

f

264.06

(a+t)2dt 2act + a2) VI' + 2b 2t2 + a'

(t 2 -

Y

~

=

(1

+h /)gfcn2udu h' . ~

(a+t)2dt f 2 (t + 2act + a2) Vt' + 2b2 t 2 + a'

=

g(1

+ kl)fCd2UdU.

Y

= f -:;;=.:=R=(~t)=.dt=~ VI' + 2b t 2 + a'

gfR [a

2

Y

[See 320.02.]

0

",

a

264.08

[See 312.02.]

0

a

264.01

[See 310.02.]

0

0

1 - Vk'tn

u ] du

1 + Vh' tn u

'


Integrands involving Vt 4

tnU=(1+V2):~:,

Special case of above:

g=

2-

Y

264.52 f

y

=

"1

dt =gfdU = gu1 = Vt'+t

am U1 = tan -1 [(1

k'=V1-k 2 ,

+ V2) : ~ ; 1'

(2- V2)tn- 1 (tan V', k) = (2- V2)F(V', k). [d. 263.50].

0

1

y

264.53 f

V'

k2=4(3V2-4),

tnu1 = tanV'. 1

264.50 f

V2,

+ I, (O:;;;:y b) 2

cn u T

"

V[( a -

cos-1

dt

t) (t - b)]3

V

", = gfdu =g(K + u1) -K

f "[ + -2-V(a 1

271.04

t) (t -

a-b

b

b)]

=

g [K+ cn-1 (cosq;, k)J

= g [K + F( g;, k)J .

dt r[(a-t)(t- bl]3

dt

+ 21X2V(a- t) (t- b)JV[(a -

[(a - b) (i-1X2)

b

t) (t - b)]3

'"

= ~g-f d2U 2 = ~b II(rx. 2 , k) + ~b II(g;, rx. 2 , k). a-b i-IX sn u aa-

f "( 4

f

~

2t-a-b)2dt _

= 2(a - b)2g sn 2 udn 2 udu.

y

-K ~

V[(a - t) (t - b)]3 y

b

[See 312.02.]

=Va-bfcn2udu.

(a-~)-2v(a-t)(t-b) dt=(a-b)gfsn2udu. [See3IO.02.]

b

f

[See 361.02.J

-'K

~

dt V(a-t) (i-b)

f

[See 400.]

r

-K

V[(a - t) (t - b)]3

b

271.07

cn U 1 = cosq;.

-K

y

271.06

Va-b'

f

b

271.05

2

g=--

", = 2g dn 2 U du = 2g [E + E(g;, k) J.

f" 271.03

,

4 (a - y) (y - b) (a _ b)2 '

b

271.02

k2=1/2

2V(a-t)(t-b) a-b'

= amu1 =

m

271.01 { f

=

4

R(t) dt

y[(a-t)(t-b)]3

=

-K ~

gfR(t [a -K

+ b + V2(a -

b) snudnuJ) du,

where R(t) is a rational function t. 1

There are other cases in 575.14 and 575.15 which may also be reduced.

Integrandsinvolvmg Y(1_t 2 )3 and Y1-t~ (1;:;;;y>0).


Vi -

cn 2 u =

1p = am u1 =

271.51

1

y

t2) 3

k2

t2,

Vi -

and

Vi - t

f f

o

271.57

dt (1 - t2 )3

Y

= 1/2, g = V2 y2; en u1 = cos 1p ,

COS -1

= gfdu =V2~ = V2cn-1 (cos1p, k) = V2F(1p, k), 0

V?{ en ",

dt ~r;--;z. V1 -

y 1-

t-

--

~

• 0

2

u d u.

[See 312.02.]

hf'"

lf1="t2 dt= 1V2 sn 2 udu.

• V'~ V(1 - t 2 )3

f y 1-111="&2 o

1+ V1-t2

[See 310.02.]

0

dt

Y(1- t2)3 =

1

V2

f"'d 2 d 0

s

u u.

271.58

271.59

> 0)

[See 361.02.]

o

271.56

(I :::: y

4---

271.54

271.55

2,

",

fv

o

V(l -

149

[See 318.02.]

[See 320.02.]

f y• t2 d t o

Y1-

t2

V-f'"sn 2 ucn 2 udn 2 udu.

=2 2

0

271.60 where R(t) is a rational function of t.

[See 361.04.J

150


Integrands involving V[(t - a) (t - b)]3 and (oo>y>a>b)

=

en u

a- b- 2 a- b

rp= amu1

f"V[(t -

272.00

4

a

f

272.01

=

V(~Jt--=-li

+ 2 V(t -

cos

=

a- b+ 2

= g

dt a) (t - b)]8

V(y -

a) (y - b)

'" fdU = g en

-1

a) (t - b),

g=Va~b'

1/2,

-1 [a - b - 2 V~(y---a)-(y---b) ]

, en U 1 = cos q;.

(cos q;, k)

= g F(rp, k).

0

"

a

k2

b) ,

a) (t -

if (t -

(2t-a-b)2dt [(21 - a - b)2+4(a - b) V(t - a) (t - b)]

=g

V[(t- a) (t -

b)]8

f ",dn udu=gE(u 2

1)

=gE(q;,k).

o (I

f"

[(2t-a-b)2+4(a-b) V~----ant=-~J dt [(2t-a-b)2+4(a-b) (1-21X2) V(t-a) (t-b)] l![(t-a) (t-b)]3

272.02 jlZ = g

f "-'

o

r

f

" [(2t-a- b)2

_d2U _ 2-

i-IX

+ 4(a -

sn u

= g II(q;, oc2 ,

dt b) V(t- a) (t- b)] y(t- a) (t - b)

f

272.03 jlZ

= 8(a-b) g

272.04

f" IZ

+ 4(a -

",

V~t=b)J dt

b) V(t - a) (t - b)] =

[See 310.02.]

sn 2 udu.

o

[(2t - a - b)2 - 4(a - b)

[(2t - a - b)2

[See 400.]

k).

g

V[(t -

f'"en udu. 2

a) (t - b)J8

[See 312.02.]

o

[See 361.53.]

V(t


, 272.06 J a

1)3 and

Z-

t

Vt

Z -1,

>y >1).

(00

151

~

=

(2t-a-b)Z (t - a) (t - b)

8( g

a-

b) Jsd 2 udu. [See 318.02.] 0

,

~

272.07 J (2t-a-b)C4(a-b) V(t-a) (t-b) dt =gJcd2 Udu. [See320.02.] a

(2t-a-b)Z V[(t-a) (t-b)]3

0

,

J Vet - a) (t - b) dt

272.08

"

'11,1"1

= ;

[F(tp. k) - 4 J 1::nu o

,

0

[See 341.53 and 341.54.]

~

= gJR[(a+b)(1+cnU)+2(a-b)dnU]du

272.09 J . a

+ 4 J(1+~:U)I].

R(t)dt V[(t-a) (t-b)]3

2(1+cnu)

0

•

where R(i) is a rational function of t.

Integrands involving cnu

Special case of above:

=

1-Vtl -1 1+Vtl -1

2

_1)3 and k2

•

J ~~ J 1

=

1/ 2.

Vt

2

-I, (00 > y > I)

cn~

=

cosV'.

1[1- Vyc1].

V' = amu,. = cos-

,

272.50

V(t

1+V y l-1

~

f(tI-1)8

=

du

= u,. = cn-1 (cos V'. k) = F(V'. k).

0

272.51

272.53

lSee 310.02.]

152

272.54


J

Y

",

~ = ~4 JSd 2 UdU. t2 t2-1

1

272.55

[See 318.02.]

0

J yt Y

J

...

dt 2

-i =

1

i-cn U ,+cnu

d

[See 361.53.]

u.

0

"tdt

272.56 J .

V(t 2

-i)8

",

=2J dnudu i+cnu

1

=

0 Y

2snu1 i+cnu1

=2y 2-1 y .

......

272.57 JVt2-1dt=[F(1p.k)-4 J ,::nu 1

0

+4 J

(1:C:U)2j.

0

[See 341.53 and 341.54.] [~ee

312.02.]

[See 320.02.]

27260 •

r,

•

R(t)dt

-

y(t2_1)8 -

1

J'" R[

2dnu 1+cnu

0

jdU,


273.00 J b

273.01

,

",

y[(t Y

dt

b)2

J a + V(t b

+ as]8 + +

= gJdu

b)2 a2 [(t - b)2 a2]&

V

= gUt =

g cn-l (cosa>b; n=3. 6, 8).

_~ - 2

275.06

157

m

m,\,ml(_1)i(1+V3l 3 m-3 i +l (-1) .LJ i!{m-i)! X i=O

3m-3i+l.

'\'

.LJ

X Y

275.07

1=0

",

[See 341.55.]

(V3-3)1(3m-3i+1)!f du 7!{3m-3i+1-i)1 (1+cnu)i' 0

1

R {,) d, _ 3ftR (V1-t 3 )dt t

-V1- a--

- of lfT2-1 -2y

where R(.) is a rational function of •. Integrands involving y(t-a) (t-b), (Z >a >b; n

V(t-a) (t-b) dt

=

V(a-b)2j4.,

=

n (a-b) _ ,n-l d. 4 V1+,n '

y

=

f

.

V(t-a) (t-b)

f

n=3,6,8.

=

gl

dt

..

\I(t-a) (t-b)

f ,n-2 d, . V1+,n

0

=

-3 4

y

V4(a-b) f

,d, ---= V1+,3

0

j

X [f \

d tL o V1-t~

z

275.52

f

\I(t-a) (t-b)

=

~V4(a-b)4

f a

_tl d tl

4

1

V1-t~ -y _

r

V4(a-b)

X

[See 244.05.]

,'d, V1+,6

[See 578.02.]

2V4(a-b)6 f ~

[See 584.04.J

2

•

0

Y

Z

275.53

= -3

y

dt

a

+ .n],

y

dt

z

275.51

3,6,8)

= t [a + b + (a-b) Vi

V4 (Z -a) (Z -b)j(a-b)2,

z

275.50

t

=

dt

\I(t-a) (t-b)

=

0

V1+,8

158


I" z

275.54

y

R(t) dt

V(t-a) (t-b)

a

glI1~ R (~[a+b + (a-b) V1+-rnl) d-r,

=

v1 +"r"

0

2

where R (t) is a rational function of t.


y(l- 't"n)"-l,

tn =4-r"(1--rn ),

11-tn=(1-2-r")2,

1/V2

276.00

I V(1~;)n-l

(0;;:;: Y
0)

",

2 {)

= k'fdu = k' u 1 = k' sn -1 (sin "P, k) = k' F("P, k), [d. 160.02.J

0

282.02

282.03

'" +n sin2f}df}=k' J~nd udu. JV1 o 2

2

[See 315.02.J

0

[See 318.05.J

282.04

282.05

'" cos2m{)d{) Vi + n sin o f

---1

2

2 {)

'" =k'fcd2mudu.

[See 320.05. J

0

See 510 to 532 for additional integrals involving trigonometric integrands.

282.06 f o

282.07 f o

> 1).

165

f{J=f=n/2.

[See 316.05.J

Integrands involving V1- n 2 sin2 D, (n2

· ·

~

tan2m Dd{} =k'2m+lftn2mudu, Vi n 2 sin2{}

+

0

dD

(1 - 1X2 sin2{}) Vi 1

+n

2

sin2 D

= k'f~ 0

dn2u~

1 - 1X2 sn2 u '

[See appropriate case in 410 and 430.J where

(/,.2

282.08 282.09 f o

·

R (sin2{}) dD =

V1+ n

2

sin2{}

= (/,.~k'2 + k2.

", k'fR(k' 2 sd2 u) d u, 0

where R (sin 2 0.) is any rational function of sin 2 0.. Integrands involving VI-n2sin2&, (n 2 > I) sn 2 u=n2 sin 2 0.,

283.00

·

fv o

283.01

f o

283.02

f o

where

283.03 283.04

(J =

·

1 -

am~

d~2 . n

S1ll2 {}

[d. 162.02J.

cos2 {} d{} n 2 sin2D

~

= kfdn 2 U du = k E(u1 ) = k E({J, k) . 0

dD = (1-IX~ sin 2 D)V1- »2 sin2D

·

nsinf{J:S;: 1.

'" = k U1 = k sn-1(sinp, k) = kF({J, k), = kfdU

~

kf 0

2

2

~u

2

i-IX sn u

JV1- n 2sin 0. dO. = k J~cn u duo

o

snu1 =sin{J,

0

V1-

·

k=1jn,

= sin-l [n sin f{J],

=

kll({J, (/,.2, k),

[See 410 and 430.J

[See 312.02.J

0

[See 310.05.J

166

Reduction of Trigonometric Integrands to Jacobian Elliptic Functions.

[See 314.05.]

283.06

[See 318.05.]

283.07 f

'"

",

V1-:::2~n2D d19=kfcd 2 UdU.

o

[See 320.02.]

0

[See 313.05.]

283.09

283.10

('J..2

[See 410 and 430.]

f'" o

where R

= ('J..~/n2.

R (Sin2 0)dO V1-n2sin20

(sin 2 19)

",

=kfR(k2Sn2u)du, 0

is any rational function of sin 2 19.


V(I -

n~ sin 2 &) (I - n~ sin 2 &),

(I >n~>n~>O)

", =gfdu=gu1 o =gsn-1(sin"P,k)=gF("P,k). 284.01

f'"

o

dO 1- n~sin20

n~ sin 20

",

1--~-.~=g

1- n~sm20

fd n 2 u d u=g E() u1 =g E("P, k) .

o

=

g II("P, ('J..2, k).

[See 400.J


Va2 sin2D-b2,

[a >b, n!2

> rp:::: sin-l (b/a)J.

167

284.03

[See 315.02.J

284.05

=

g

f~n2mudu. [See 312.05.]

o

284.06 [See 318.02.J

284.07

=

"1 -( g 2-)mftn2mudu. 1-n 2

o

[See 316.05.J

where R (cos 2 fJ) is any rational function of cos 2 fJ.

284.09

fV

"1

tp

o

R (sin2D) dD

(1-n2 sin2D) (1-n2 sin2D) 1

2

=

gfR [ 0

2sn2 U2 2 1-n2+n2sn u

1du,

where R(sin 2 fJ) is any rational function of sin 2 fJ.

Integrandsinvolving Va2sfD.2&--b2, [a> b, n/2 2 D) 2 sn ~ U -_ a (1-sin a 2 _ b2

1p

fv a sm D 2

tp

285.01

k2a2 -b2 -a-2

,

cp ~ sin-l (b/a)]

g -1/ - a,

= am u1 = sin-1 [(cos !p)/kJ, sn U 1 = sin 1p. "1

",/2

285.00

'

>

.dD 2

b2

=gfdu=gu1=gsn-1(sin1p,k)=gF(1p,k). 0

168


[See 400.J where

[See 310.05.J

285.03

f Va n/2

285.04

UJ

Sin2 & COS2& d& k2fsn 2 u ~7=~:==== = g 2 sin2& - b2

'P

dn 2 u d U.

285.05

[See 314.05.J

f Va ~

285.06

f

~

2

r1 . .

sin 2 0 - b2dO = (a 2 - b2) g cn 2 u duo

in', _ b'~:V."in" _ b'

j

[See 312.02.J

0

'P

285.07

[See 361.02.J

0

~ - ;-(a-; -2- -'!."- :b~2)mi~C2mu du rp=l=sin-l(bja).

[See 313.05.J

[See 318.05.J

:rr/2

285.09

f

'"

(1 -

0I:~sin2&)m d& = g(1 - OC~)mf(1 - oc2 sn 2 u)m du,

Va 2 sin2&- b2

'P

oc 2 = oc~ k2/(oc~

where

285.10

:rr/2

f 'P

[See 331.03.J

0

- 1)

",

R(sin2&)d& 2 sin2&-b 2

Va

=gfR (dn 2u)du, 0

where R(sin2D) is any rational function of sin 2 D.

Vcos 2aD,


(0< atp ~ 11;/4).

169

Integrands involving Veos 2a 50, (0< a cp ::;;; 1t/4)

sn 2 u = 2sin2 aoO,

k2 =1/2, g=1/aVi,

P= amu1 = sin-1 [Vi sin a 00] ' 286.00

cos 2a D

o

286.01

'"

rp

f ~ rp

f

=

aD _ doO Vcos2aD COS2

o

gfdu = gUl = gsn-1(sinp, k) = gF(P, k). 0 '"

= g f dn2 Udu =g E(~) = g E(P, k). 0

rp

286.02 f

286.03

~

dD (1 - 2ct2 sin2 a D) Vcos 2a D

o

snu1 = sinp.

=f

d2u

2

1 - ct sn u

0

f sn

f _===rp

o

sin2"'aDdD g = ---m Vcos2aD 2

'"

2",

U

=gII(P,rx.2,k). [See 400.]

d U.

[See 310.05.]

0 '"

rp

286.04 !Vcos2aoOdoO=g!cn2 udu. o

286.05 f

rp

o

sin2 aDdD cos2aDVcos2aD

'"

=~ftn2udu, acp=f=n/4. 2

f

cos2 '" aD Vcos 2a D rp

o

286.08

f o

tan2 '" a D dD = J_ Vcos2aD 2'"

gfnd2 "'udu.

=

f sd '"

0

du.

f

[See 318.05.]

~

g ne2 '" U du, a cp o

(cos 2aD)'" Vcos 2aD

~ cos2 aD doO= g f

2 '" U

0

dD

rp

[See 315.05.]

0

rp

o

286.09 f

~

dD

o

[See 316.02.]

0

rp

286.06 f

286.07

[See 312.02.]

0

'" C

d2 U d u.

=f= n/4. [See 313.05.]

[See 320.02.]

170


286.10

'"

I

'"

~) df}=gIR(dn 2 u)du, cos 2a f}

o

0

where R (cos 2 a f}) is a rational function of cos 2 a f}.

286.11

'"

I o

2 V(Sin af}) cos 2a f}

",

df}=gIR[(sn 2 u)/2]dU. 0

V- cos 2a B-, (n/2 > acp ~ n/4)


sn 2u = 2 cos 2 af},

[V2 cosalJ?] ,

1p = am u1 = sin-1 I v

df}

- cos2af}

= gldu = gUl = gsn-1 (sin1p, k) = gF(1p, k).

'"

287.01

sn u1 = sin1p.

'"

n/2a

287.00

k 2 = 1/2, g = 1IaV"2,

0

n/2a '" 2udu=gE(U1 ) =gE(1p,k). sin2af} df} =g/dn I V- cos 2af}

I! '"

0

n/2a

287.02

287.03

(1 -

2oc2

cos2 : : )

V-

cos 2af} = g

n/2a cos2m a f} I

V- cos 2a f}

df}

o

0

V- cos 2af}df} =

'" gJ cn 2 u duo

[See 312.02.J

0

n/2a cos2af}d{} I cos2af}V-cos2a{}

=

'" _L/tn2udU, 2

alJ? =Fn/4. [See 316.02.J

0

n~a

'"

[See 310.05.J

2m

n/2a J

I

[See 400.]

",

'"

287.06

oc2 sn! u

= ~/sn2mu du.

'"

287.05

du 1-

= g ll(1p, cx.2 , k).

'"

287.04

I'"

(cos

~

df} 2af})m V-cos 2a f)

=

(_1)mglnc2mudu, 0

alJ? =F n/4. [See 313.05.]

Integrands involving Vsin 2af}. (0< arp::;;' n/2). 10/2 a

tI,

j --:i=CO~t=2~a~f}=d=f}~ = .!..jsd2 U du.

287.07

V-cos2af}

'P

tI,

df} j sin2m a () V- cos 2alJo

= gjndz,,'udu.

n/2a

[See 315.05.]

0

'P

287.09

[See 318.02.]

2· 0

10/2 a

287.08

171

tI,

R(cos2alJo) dD=gjR[(sn 2 U)/2]du, V-cos2alJo

j

0

'P

where R (cos 2 a D) is any rational function of cos 2 a D. ...

n/2a

287.10

R(sin2alJo) dD=gjR(dn2u)du. V-cos2alJo

j

0

'P

Integrands involving Vsin 2aD, (0< a rp ::::'1'&/2) 2 snu-

1

2sinalJo + cosav .D.+'sma f}'

k 2 -1/ 2,

-V2,/ g-2a

. -1 ~ 2sinarp . A =am = SIn , snut=sinA. ut 1+cosarp+smarp ...

'P

287.50 j

dlJo =gjdu=gu1=gsn-1(sinA.k)=gF(A,k). Vsin 2aD

o

0

'P

287.51

j

(1

(1 +cos alJo+sin alJo) VSin 2a IJo

o

281.52

j o

+ cosalJo)dlJo

...

=gjdn2Udu=gE(ut)=gE(A,k).

j '"

'P

(1 + cos a f) + sin a 1Jo) dlJo [1 +cos af}+(1-2cx~) sin alJolVsin 2af}

(1 +cosalJo-sinaO) dO (1 + cosaD + sinaD) Vsin2alJo

j o

=gjcn2itdu.

o

gll(A,rxtk) . [See 400.] [See 312.02.]

0 ...

'P

287.54 j

du

g 1-cx~snBu o ...

'P

287.53

0

sinalJodlJo =.!..jsnzudu. (1 +cosaD+sinaD)Vsin2alJo 2 0

[See 310.02.]

172

Reduction of Trigonometric Integrands to Jacobian Elliptic Functions. ~

287.55 f o

287.56 f

~

sin a f}df} (1 +cosaf)-sinaf})ysin2af}

=~ftn2udu, aqJ=f=n/2. 2

0

[See 316.02.]

",

~

(1 + cos a f}) df} =gfdc2udU, (1 + cos af) - sin a f}) ysin 2af}

",

o

1

~

f 287.57 o

[See 321.02.]

0

sin af}d~ = 2g k 2 f sn 2 udu ysin 2a f}

+ [CII(A,

aqJ=f=n/2.

2k'2 gF(A, k)

+

0

(X2,

k)

+ CII (A ,

[See 310.02 and 400.]

k)] g,

(X2,

where

111

qJ

287.58 fR(Sinaf}) d{}= fR[ ysin2af}

o

"1

g

4sn2 udn2 u ]du= f 4dn4 u+sn4 u g

0

0

R [2Sn 2 udn2 U]du, 1 +cn4 u

where R (sin a {}) is a rational function of sin a {}.

Integrands involving Va

+ b sin 50, (a > b > 0) 2

g=--

Ya+b'

I: \>

=

am U 1

=

.

SIll

-1

n/2 71 al >0).

[See 331.03.]

0

~

"1 J Vcos2 {} d{} =4gJ sn 2 uen 2 udu. a + b sin 8",/2

288.04

[See 361.0 I. J

0

~

"I

",/2

d8g J nd2m udu. - = -------m (a + bsin8-)mVa+ bsin8(a+b)

J

288.05

"'/2

J

288.06

1

"1 d{) = 2 gJ en 2 u d u.

"1

(1

1!

"/2

[See 312.02.J

0

"/2

J ~

288.08

+ sin8-

Va+bsin8-

~

288.07

(1-sin{})d8sin 8-) Va b sin{}

+

+

=gJtn 2udu,q;=l=-nI2. [See316.02.J 0

"1

tan 2 8- d8-

Va + b sin8-

"1

"I

!J ns 2 une 2udu - g J ne 2 u du + gJtn 2 udu, o

0

q; =l= - n12.

0

[See 361.10, 313.02, 316.02. J

"1

"/2

288.09

[See 315.05.J

0

~

JR (sin 8-) d-& =gJR(1-2sn2u)du, Va + bsin80

~

where R (sin {}) is a rational function of sin {}.

Integrands involving 2

_

sn u-

b (1 - sin {})

a+b

tp = am u1 =

'

. SIll -1

k2

(1 - sin m) r

> q; :;;;: -

+b

a+b

-

.

'

-n 2

b'

sn~ =

sin tp,

(a)

. - 1 -SIll b

.

J V d8-. = gJdU = gU1 = gsn-1(sintp, k) = gF(tp, k). a+bsm-& 0

~

",/2

288.51

V2

a

"I

,,/2

288.50

(b > la I> 0)

---u;-' g-

_

Vb

Va + b sin &,

J ~

(V + sin~) d8a+bsm8-

"I

= 2g J dn 2 u du = 2g E(u1 ) = 2g E(tp, k). 0

174


I!

~

288.52

288.53

(a + b -

d{} b sin (}) Va + b sin {}

~2 b + ~2

sinm{}d{}

Va+bsin{)

~

:n/2 f cos 2 {} d{} Va+bsin{}

rp

2

f rp

=

'"

4k 2g f sn 2 u dn 2udu.

",

=

rp

g

(a + b)m

fnc2mudu,

b >0).

df} . =-g-Jdn 2 udu=-g-E(u1 )=-g-E(A,k). {a-bcosf}!Va-bcosf} a-b a-b a-b

J o

291.02


[See 310.02.J

0

J•

0

X

291.08 J o

tan 2 f}df} 4{a- big Va-bcos{} =Ta+b)ctt X

"1

[-F(A'k)+(2-rx.DII(A,rx.~'k)-J{ {1-2ct~)2d~2]' sn u o

•R

(cos f}) df} Va-bcosf}

1-(Xl

[See 400.J

"1

=g J R [1 -{b-ak 2 sn 2 u)nd 2 u 1du. b 0

where R (cos D) is a rational function of cos D. Byrd and Friedman, Elliptic Integrals.

12

178



P

P = Vb 2 + c2 ,

r .

tpl = sm

I Vb

p'

=

-1 b

p

am u 1 = sin -1 =

cos

p'

-1 C •

V

; sin tp -

tpl -

""2 S. tp < tpl·

1-

~

2 '

; cos tp ,

n

df} =gIdu=gu1 =gsn-1 (sinr,k)=gF(r,k). sin f} + c cos f} 0

...

~

I(p+bsinf}+ccOSf})df} 2pgIdn 2udu=2pgE(u1 )=2pgE(r,k). Vb sin f} + Ccos f}

292.01

0

q>

292.03

k2 =

Vp'

...

q>1

q>

g = 1f2

~ sin {} - ~ cos {}

sn 2 u = 1 -

292.00

Vb sin {} + c cos {}

I

~

(P -

q>

...

= g pI sn 2 u du.

bsin f) - Ccos f}) df} Vb sin f} + Ccos f}

...

~

292.04

[See 310.02.]

0

[See 312.02.]

JVbsin{}+ccos{}d{}=gp J cn 2 udu. 0

q>

...

...

~

sinf}df} =~Icn2udu - V2cg Isnudnudu. 292.05 I Vb sin f} + Ccos f} P P 0

q>

0

[See 312.02, 360.02.] q>1

292.06 I q>

292.07

l

I q>1

q>

f}df} Vb sin f} + Ccos f} cos

csi~f}-bcosf}

Vb sm f} + Ccos f}

"1

gpC I cn 2udu+ V2:b o

f snudnudu. "1

0

[See 312.02, 360.02.J

d{}= -gpVl. j~nudnudu=gpVl.cnUr 0

=

-

gp Vl. + gV2P Vbsintp

0

+ ccostp.

Integrands involving Va+b sint)+c cost). (0< I al< Vb l +c2). ~

~

292.08 f V ~ (cost)) dt)

bsmt)+ccost)

~

= gfR[~ (c cn2u P

0

179

+ b }'2snudnu)] du,

where R (cos t?) is a rational function of cos t?

Integrands involving Va

+ bsint? +e cost?, (0< Ja J< Vb 2 + e2 )

sn 2 u=-1-[p-bsint?-ccost?], g=V2, k 2 = a+ p , a+p

P=Vb 2+C2,

P

y=arn~=sin-l Va~p

2P

(p-bsinlji-ccoslji),

ljil = sin-l (bJP) = cos-l(cJP); ljil - cos-l (- alP) 9'1

"1

293.00 fra+ .dt)

a+b smt)+ccost)

~

=gfdU =

~Iji 0)

=gjdu=gul=gsn-l(sintp,k)=gF(tp,k).

+ ccos1}

0

+ bsin{} + ccos{}d{} =

+ p)j'dn2u du = g(a + P) E(ul ) = g(a + P) E(tp, k).

'1'.

[2P - exBp

+ =

g(a

dD

ex2

(b sinD

2~

I

+ c cosDlJ Va + b sinD + ccosD

",

o

1-

:2~n2 u

=

;p II(tp,

(1.2,

k).

[See 400.}

Integrands involving Va+ b sin#+ c cos#, (a > Vb 2 + c2 >0). ~

~

(b cos # - c sin #)2 df) = 4 g p2Jsn 2 ucn 2 u d U. J Va + bsin# + csin#

294.03

~

~

(P + bsin# + ccos#) df} = 2 g pJcn 2 u d U. J Va+bsin#+ccos#

J 294.05 1

'P'(b

cos # - c sin#) d#

~=:=:~=====;;:

294.06

Va +

b sin # + c cos #

2P

= -

J

U d ' g sn u cn u u

0

2

= -

J 'Pl

'P

[See 312.02.]

0

'P

'P

[See 361.0 I.]

0

'P

294.04

181

[Va + p - V-a-+-cbc-s-'--in-cp-+-c-co-s-cp].

'Ill

R(cos#)d# =gJR[c(a+p)dn2U-2~psnucnu-ac]du, Va+bsin#+ccos# P 0

where R (cos f}) is a rational function of cos f}.

J

'P,

294.07

tp

R(sin#) d#

Va+ bsin&+ ccos#

'"

gJ R [b(a+ p) dn 2u+;:psnucnu-ab ]dU, o

where R (sin f}) is a rational function of sin f}.

Reduction of Hyperbolic Integrands to Jacobian Elliptic Functions. In addition to the algebraic or trigonometric forms given in the foregoing sections, elliptic integrals encountered in practical problems may also involve hyperbolic integrands. The reduction of some important cases follows 1 :

J JY

295.00

o

J

tn-'(sinhtp,k)

tp

=

df}

Y1+k'· sinh·f}

0

Jy

df}

kB+k'· coshlf}

o

J

R (coshf}) df}

Yk· + k'i coshl f}

Jy

df}

J

J

R(ndu)du.

[O q; >0).

> a > 0;

Integrands involving Vb cosh 19 -a, (b sn2u=b(coshf}-1) b cosh f} - a '

00

[See 313.02.]

~

(b.!.a)mfcn2mudu. [See312.05.] 0

",

2 f R(COShf}) d{} =gfR [b - a:n u] du, Vb cosh f} - a b en u o 0

where R (cosh 19) is a rational function of cosh 19.

186

Reduction of Hyperbolic Integrands to Jacobian Elliptic Functions.

Integrands involving sn U =

sinhf} coshf} 1,

+

'!jJ =

k2

Va + b cosh ff, a- b

=

(0 < if>
- Va + b] .

cosh f} df} J 297.33 J -;I==::===:='" = g [nc 2 U + tn 2 UJ d U. Va + b coshf}

297.34

[dcu1 -

k

Va + b cosh f}

=

g

J"'R [ 1 + sn2 U ] du,

0

cn2 U

where R(coshff) is a rational function of coshff.

Integrands involving Ya- b cosht? [a >b >0; 0< 4>< COSh-l (a/b)].

187

Integrands involving Va -b cosh'!?, [a> b > 0; 0< 4)< cosh- 1 (a/b)] sn2 U

a - b cosh t? a-b

_ -

•

k2

_

-

a- b a+b

g -_ -2-

--.

Va+b'

m

_

'VI -

CO

sh- 1 ( a) -

b '

1jJ=am~=sin-IVa-bCOSh4>, sn~=sin1jJ. a-b

. 'I>

299.01

dD-

'I>

299.00

\

JU'

=g du=gu1=gen

-1 (eos e, k) =gF(e,k).

0

u.

~

J

(Vb 2 - a2 + a sinhD-__ + b coshD-) dD2 dnuu 2 d =g (a sinhD- + b cosh 6-) Va sinh D- + b cosh DJ

~

0

= 2gE(u1} = 2gE(e,k).

JVa sinh 19 + b eosh 19 dB> = g Vb 'I>

299.02

Jne u du. ~

a2

2 -

'1>.

[See 313.02.]

2

0

'I>

299.03

_

dD- . =--g--Jen2m udu. J (a sinh 6- + bcoshD-)m VasinhD-+ b cosh DV(b 2 - a2 )m

'1>.

299.04

0

[See 312.05.]

J 'I>

".

(a sinh D- + b cosh 6- - V~) dD- = gVb 2 - a2 J tn 2 udu. Va sinh D- + b cosh Do [See 316.02.]

'1>. 'I>

~

(a sinh D- + b cosh D- + Vb 2 - a2 ) dDV--- J = 2g b2 - a 2 de2 u du. 299.05 J --'----:;r=:::=:==;;==;:=='=='-=:=;;-----'-Va sinh D- + b cosh D'1>. 0 [See 321.02.] 'I>

299.06

J '1>.

_

~

R(sinhD-)dDVa sinh D- + b cosh D-

=gJR[b V2tnudcu-anc2 U]dU, Vb 2 - a2 0

where R (sinh 0) is a rational function of sinh 19.

299.07

J 'I>

~

R (coshD-) dDVa sinh D- + b cosh D-

~

dO

=

gJR [bnc 2 U - a V2tn U dCU] du, Vb 2 - a2 0

where R (eosh 0) is a rational function of eosh 19.

Table of Integrals of Jacobian Elliptic Functions. In the foregoing sections, where elliptic integrals having diverse algebraic, trigonometric, and hyperbolic integrands were reduced to those involving elliptic functions, it is seen that certain standard integrals constantly recur. The following tables of integrals I give explicit evaluation to those Jacobian normal forms, to which specific reference was made in each formula of Item Nos. 200-299.

Recurrence Formulas for the Integrals of the Twelve Jacobian Elliptic Functions. Integrals of odd powers of the twelve Jacobian functions, it is to be noted, are expressible solely in terms of Jacobian elliptic functions and more elementary functions, while the evaluation of integrals of even powers requires in addition the two functions E(u) and u.

A =f m

t m dt

V(1-t2)(1-k2t2)

=f sinmtp dtp

V1-k 2 sin 2cp

=fsnm u du

[t = sin q:> = sn u].

310.00

Ao = J du = u = F(q:>, k),

310.01

Al = J sn udu =-i- In (dnu-kcnu) = - ~ cosh-I [(dnu)/k'].

310.02

A2

=Jsn 2 udu = ~2- [u - E(u)] ,

310.03

A3

=Jsn3 udu =~ [kcnudnu + (1+ k 2 ) In (dnu - kcnu)].

[q:>=amu].

[E(u)

=

E(q:>, k)J.

2k

1 With all the following indefinite integrals, the constant of integration is to be understood. Moreover, for brevity we write E(u) for E(am u, k).

Table of Integrals of Jacobian Elliptic Functions.

192

310.04

{A4

310.05 A

J sn 4udu=

=

_

2",+2-

3~4

[(2+k 2) u- 2(1+k2)E(u) + + k 2 sn u en udnuJ.

sn2m-lucnudnu+2m(1+k2)A2m+(1-2m)A2m_2 (2m+1)k2 •

_ sn·'" u cn u dn u + (2m + 1) (1 + k 2 ) A' m+1 - 2mA''''_1 310.06 A 2m+32(m+1)k2

B

m

-J

=

t"'dt -J drp V(t2 -1) (t 2 - k 2) sin"'cp V1- k 2 sin2rp

=Jns"'udu,

[t = 1/sinp = nsuJ.

311.00 Bo= Jdu=u=F(p,k),

Bl =Jnsudu=J~

311.01

snu

=

[p=amuJ. In [~_u_]. cnu

+ dnu

311.02 B2 = J ns 2 udu = u - E(u) - dnuesu,

311.03 Bs =Jns3 udu = ~ [(1 2

311.04 { 311.05 311.06

+ k 2 ) In (~_u_) cnu + dntt

[E(u)

=

E(p, k)J.

esu dnunsu].

B =Jns4udu =

B B

4

i

[(2

+k

_

u - 2(1

2m(1

2m+2 -

+k

2)

E(u) - dnuesu (ns2u

+ k2) B 2m + (1 -

+ 2 + 2k2)J.

2m) k 2 B 2m - 2 - en u dn u ns2m+lu 2m+ 1 •

(2m +1) (1+k 2) B 2m + 1 - 2m k 2 B 2m - 1- cn u dn u ns2m + 2 u 2(m 1)

+

2m+3 =

Cm -- - J

2)

tmdt - J eos"'cpdrp - Jen'" u V(1- t 2 ) (k'2-+ k 2 t2) V1- k2 sin2rp

d u,

[t = cos P = en u J.

312.00 Co = J du 312.01

=

u =F(p, k),

C1 = J enudu = [eos-1 (dnu)J/k = [sin-1 (ksnu)J/k.

[p=amuJ.

t 93

Integrals of the Twelve Jacobian Elliptic Functions.

2

312.02 C = J cn 2 udu =

;1 [E(u) -

[E(u) = E(rp, k)].

k'2 U ],

312.03 C3 = J cn3 udu = 2~8 [(2k2 - 1) sin-l (k sn u)

312.04

I

C4

+ ksnu dnu].

= J cn4 udu t

=

3/i'

[(2 - 3k2 ) k'2 U

+ 2(2k2 -1) E(u) + k2 snu cnu dnu].

312•05 C2m+2 = 2m{2/iI-t) CI ", +

(2m -t) /i'l C sm (2m

+ t) /i

-.+ snu dnucnl"'-lu

l

312.06 C2mH = (2m+t) (2/iI-t) Clm+1 + 2m/i'· Cam- 1 +snudnucnamu 2{m

D =j

'"

V(tl -

=j

tmdt

t){/i l + /i'S tl)

+ t) /il

dfll

V

cos m tp t - /ilsin·fII

=jnc"'udu

'

[t = 1/cosrp = ncu].

313.00 Do = J du = u = F(rp, k), 313.01 D1=

[rp

j ncudu= j -cnu- = -Ii' 1n [/i' sn cnu+ dn du

u

t

u ]

.

313.02 D 2 = Jnc 2 udu= /i~1 [k'2 u -E(u)+dnutnu], [E(u) = E(rp, k), rp

313.04

I

D4

= Jnc 4 udu =

3

/it"

[k'2(2k'2 - k 2) U

+ (2 -

4k2

= amu].

=

am u].

+ 2 (2k2 -1) E(u) +

+ k'2 nc2u ) tnudnu].

313.05

=

313.06 D

2m/iID. m _ 1 + (2m+ t) (t- 2/iI)D."'+1 +tnudnunc l "'+1 u 2{m t) /i'. • Byrd and Friedman, Elliptic Integrals. t3 2 ... +3

+

t94

G",


=

-f

t"'dt =f(1 - kl V(t - tl) (t l - h'l)

sin2 tp)("'-1)!2 dtp =fdn"'udu,

[t = V'--1-----=k:-72 ---: si---=nl:-tp = dn u]. 314.00

Go = J du = u = F(tp, k),

314.01

G1 = J dn udu = amu = sin-1(sn u).

= amu].

[tp

314.02 G2 = J dn 2udu =E(u) =E(tp, k),

[E(u) =E(tp, k), tp = amu].

=1 [(1 + k'2) amu + k 2snuenu].

314.03

G3

=J

dn 3 udu

314.04

G,

=J

dn'udu =

1 [k 2 snu en u dnu -

k'2U + 2(1

+ k'2)E(u)].

314.05 G2",+2 =

hldnz",-lusnucnu+ (1- 2m) h'zG.",_,+ 2m(2-hl) G. m

314.06 G2",+3 =

hI dnl"'usnucnu- 2mh'IG.",_1 + (2m+ 1) (2- hI) Glm +1 2(m t) .

2m

+1

.

+

315.00

10 = J du = u = F(tp, k),

315.01

I1-

315.02

1 2 = Jnd 2udu= h~. [E(u) -k2 snuedu],

-f n

-f- - _ - a n

d U d u-

du

dn u

-

1 t

h'

_1[h'SnU-Cnu] h' sn u + cn u .

[E(u)

315.03

13 = Jnd3u du = ~

1

2h

= amu].

[tp

[(2 _ k

2)

=

E(tp, k), tp

= am u].

tan-1 (h:snu - cnu) _ h snu+cnu

- k' k 2 sn u en u nd2 u] .

315.04

315.05


j

14 = fnd4udu

=

~ [2(2 - k 2 ) E(u)3k

- k'2 U

1

-

k 2 snu edu (k'2 nd 2u

_ 2m(2 - k2 )I2m + (1 - 2m) I 2m (2m

2m+2 -

315.06 12 ... +3 -_

(2m

+ 1) (2 -

316.00

10 =

316.01

T Jl

316.02

12 = f tn 2 u du

f du

195

2k 2)].

k2 snucnund2m +1 u

2 -

+ 1) k i2

•

k 2 sn u cnund2m +2 u

k 2) I 2m + 1 - 2m I 2m - 1 2(m 1) k'D

+

= u = F(cp, k),

[cp = amu].

sn u = J tnudu = J --du cnu

= k~2

+4 -

=

J seudu

1 [ dn u + k' j . = ---In 2k' dnu - k'

[dn u tn u - E(u)] , [E(u) = E(cp, k), cp = am u].

316.03 316.04

1a =

ftn 3 u du

= -~4k'8

[2k' de une u - (1

+ k'2) In (dnU + k')j. dnu-k',

+k'2)E(u)1 14=ftn4udu=~[2(1 - k'2 + tnu dnu (k'2 ne2u 3k

U

2 - 2k'2)].

+ (1- 2m)]2m-2 -2m(1 + k'2)]Dm

316.05

T _ J2m+2 -

tn2m - 1 udcuncu

316.06

];

tn2m udcuncu-2m]2m_l-(2m+1)(1+k'2)]2m+l 2(m + 1) k'2 •

317.00

Lo = f du = u = F(cp, k),

317.01

II =Jesudu =J cnu du = In[~- dnUj.

_

2m+3 -

snu

(2m

+ 1) k'D

[cp

.

= amu].

snu

13*


196

317.02

L2=

Jcs2udu=-dnucsu-E(u), [E(u)

317.04

317..05 317.06

l

L4 =

E(cp, k), cp

=

=

am·u].

Jcs4 udu 1

= 3[2(1 + k'2) E(u) - k'2 U + dn u csu (2

+ 2k'2 -

ns 2 u)].

L

_ 2m+2 -

L

_ -2mk'2L 2m _ 1 - (2m +1) (1+ k'2)L 2tn +1 - dcuncucs2tn +2 u 2m+3 2(m + 1) .

M = f

(1-2m)k'2L2m_2-2m(1+k'2)L2m-dcuncucs2m+lu 2m + 1

tmdt = f V(1 - k'2t2) (1 + kl t2)

'"

sin"'!pd!p (1- k2 sin2 tp),m+l)/2

= fSd"'udu

'

[t = (sincp)/V1 - k 2 sin2cp = sduJ.

31S.00

Mo =

Jdu = u

31S.02

M 2=

Jsd2udu= k2~'2 [E(u) -k'2 u - k snucduJ,

31S.03

Ma =

[cp = amuJ.

= F(cp, k)

2

[E(u) = E(cp,k), cp=amuJ.

Jsd u du 3

d]

) _ _1_[( k 2 -1(kk'(1+cnU))_kk'd - 2k8k'8 2 -1 tan k2cnu-k'2 C un u .

31S.04

31S.05 31S.06

f M4 = Jsd u du = 1 4

M.

+ k'2(2 - 3k 2) u- k 2 snucdu (k'2 n d 2u + 4k2 - 2)J.

3k:k"

[2(2k2 - 1) E(u)

_ 2m(2k2-1)M2",+(2m-1)Mltn_2-cdundusd2m-lu

2m+! -

M2m+a=

(2m

(2m

+ 1) (2k2 -

+ 1) kl k'2

+ 2mMsm - cd u nd u sd2tn U 2(m+1)k2k'2

1) Msm+l

.


319.00

No

=

J du

=

u

=

F(91, k),

319.01

N1

=

J ds u du

=

J dn u du snu

319.02

N2

= J ds 2 udu = k'2U - dnucsu - E(u) ,

197

[91=amu].

=

In [

sn u ]. 1+cnu

[E(u)

=

(E91, k), 91

=

am u].

319.03

N3=Jds3udu=~[(1-2k2)In( snu )-cnuns2uJ. 2 1+cnu

319.04

N4

319.05

o

. m

= J ds 4 udu =

+

+ 2(2k2 - 1) E(u)dn u csu (ns2u + 2 - 4k 2 )].

[(2 - 3k2) k'2U -

N. = k2k'Z(2m-1)Nzm_z+2m(1-2k2)Nsm-cnundZudszm+lu 2m+2 2m + 1

=-J

tmdt =J cosmrpdrp V(1-t 2 )(1-k2 t 2 ) (1_k 2 sin2 rp)(m+1)/2

[t =

Jcdmudu,

(cos 91)/V1 - k 2 sin 2 91 = cd u].

320.00

0 0 = J du = u = F(91, k),

320.01

0 1 = Jed u du = J cn u du = ~ In [ 1 +

320.02

O2 = J cd2 udu

[91=amu].

dnu

=

iz- [u -

0 3 = Jcd3udu =_1_[(1 8 2k

k sn u ] . dnu

k

E(u)

+ k2snucdu], [E(u)

320.03

•

=

E(91, k), 91

=

am u].

+ k2) In(1 +k snu)-kk'2snund2u]. dn u


198

320.04 320.05 0

Jcd udu = -;[(2 + k 2) u 3k

=

04

-

2m+2 -

+ k 2 ) E(u) + + k 2 sn u cdu (2 + 2k2 - k'2 nd

4

2(1

321.01 321.02

P2 =

321.03

Pa =

321.04

u)J'

2m(1 +kS)Osm+ (1-2m)Osm_s-k'ssnundsucdsm-lu (2m + 1) k S

Jdu = u =F(cp, k), dnu [1+snU] . P.1= fd CU dU= f --du=ln cnu cnu

321.00

2

Po =

[cp

=

amu].

Jdc U du = E(u) + dn u tn [E(u) = E(cp, k)J . Jdc udu ~ [(1 + k2)ln e~n:~) + k'2 snu nc2u]. 2

3

2£,

2t -

=

{P = Jdc udu=+[(2+k2)u-2(1+k2)E(u)+ 4

4

+ (2 + 2k2 + k'2 nc2U) dnutnu].

321.05 p.

_ 2m(1 +kS)Psm + (1- 2m.) kSPsm_s+k'Ssnund2udc2m+lu

321.06 P.

=

2m+22m+3

2m+ 1

(2m+1) (1+k 2) P2m+1-2mk2P2m_l+k'ssn1Indsudc2m+2u 2(m 1)

+

Additional Recurrence Formulas. a m

=

f

(a1+t)mdt tl) (1 - kl t S)

V(1 -

=

rV

•

(a1 + sintp)mdtp =f(a1 +snu)mdu, 1 - kl sins tp

[t = sinlj? = snu].

330.00 a o = 330.01

al

=

Jdu u =F(Ij?, k), J(a + snu) du ~ [a1ku + In (dnu =

l

=

[Ij? = amuJ.

kcnu)].

Additional Recurrence Formulas.

330.02

{

CT2

=

J(a + snu)2du = k~ [(1 + k2a~) u 1

+ 2a

1

330.03

{

CTa

=

J(a + sn U)3 du = 1

2

1k8

[2a 1 k (3

199

E(u) + k In (dn u - ken u)].

+ a~ k2) u -

+ kcnudnu + (6a~k2+k2)In (dn u-ken u)],

6a1 k E(u) +

[E(u)

= (Etp,k)].

[tp=amu]. where

t is defined in 361.58.

1

-j ( +dusnu)8 --

CT_ 2 -

330.51

~

1 [( k2 2 - 1 - k 2) (Z X al -1 )( a 8k8 -1 ) 2 al l

X II(tp, 1/aL k) - E(tp, k)

+ (1+kl-2afkl)0'-1 _ al

I'm =j

(1-a.1 tl)mdt

¥(1 - tl) (1 - kl tl)

=j(1_a.8 sinBtp)m klsinltp

Vi -

+ (1 -

k2a~) F(tp, k)

-+-

cnudnu ] a1(a1 + snu) .

dtp=j(1-oc2Sn 2u)m du ,

[t = sin tp = sn u] .

I'o=J du = u =F(tp,k) ,

331.00 331.01

{I'I =

J(1 -

oc2 sn 2 u) du =

331.02

{I'2 =

J(1 -

ocl sn2 U)2 du =

l

l'm+a

331.03

[tp=amu].

k~

[(k 2 - oc 2) U + oc 2 E(u)] , [E(u) =E(tp, k), tp = amu].

3~4

[(3 k' - 6oc2 k 2+2oc'+k2 oc')u+

+ 2(3 oc2 k 2 - oc' - k 2 oc')E(u) (2m

1

+ 5) kl [2 (m + 2) (3 k2 -

oc2

-

+ oc'k2 snuenudnu]. k 2 oc2 ) I'mH +

+ (2m + 3) (2oc2 k 2 + 2oc2 -oct - 3 k 2) I'm+! + 2 (m + 1) X X (oc2 -1) (k 2 - oc2) I'm + oc'(1 -oc2 sn 2 u)m+Isnu enudnuJ.

200


T", =f [1 - OCn1 - t2)]'" dt=f (1-oc~COs2tp)'" drp =f(1-OC~cn2u)mdu, y( 1-t2) (1-k2 t2) Y 1 - k2sin 2 tp [t = sin rp = sn u] .

332.01

T",= f(1-oc~cn2u)"'du=(1-oc~)"'f(1-oc2sn2u)"'du=(1-Ot~)"'Y""

where oc2=ocU(Ot~-1), and y", is given above.

T

m

=f[1-OC!(1-k2t2)]m Y(1-t 2)(1-k2t2)

dt=f[1-OC~(1-k2sin2tp)]'" Y1-k 2 sin2 tp

= J(1-oc~dn2u)mdu,

drp

[t=sinrp=snu].

332.51 Tm = f(1-oc~ dn 2 u)mdu=(1-~~ynf(1-Ot2Sn2u)mdu=(1-0t~)"'Y"" where OC2=OC~ k2/(oc~-1), and Ym is given above.

U =f [1-(1+OC 2)t2]mdt = f m (1- t2)"'Y(1- t2) (1 _ k2 t2)

= J (1- oc2tn 2u)"'du,

334.00

Uo = J du = u = F(rp, k),

334.01

U1

=J

(1 - oc2 tan2 tp)'" drp 1 - k 2 sin2 tp

y

[t

=

=

sin rp

sn u] . [rp = amu].

(1 - oc2tn 2u) du = k~2 [k'2 U

+ oc2E(u) -

oc 2dn u tn u],

[E(u) =E(rp, k)]. U2

334.02

1

=J

=

3;'4 [k'2(3 k'2 - oc4) U

+

+ 2oc2 (3 k'2 + oc2 + k'2 oc2) E(u) - oc 2(6k'2 + 2oc2 + 2k'2 Ot2 - k'2 Ot2 nc 2u) dn u tn u].

U",H =

334.03

(1- Ot 2tn 2u)2du

(2m

~ 5) k'2 [- 2(m + 2) (oc 2 + k'2 oc 2 + 3k '2) Um + 2 +

+ 2(m + 1) (1 + Ot2) (Ot2 + k'2) U", + + (2m + 3) (oc4 + 2oc2 + 2oc2k'2 + 3k'2) U"'+l + + oc tn u dnunc u (1- Ot2 tn 2u)"'+l]. 4

2

Additional Recnrrence Formulas.

= 336.00

f

(1- a 2 sn2 u)m

Vo

=

du

201

[t

=

sin!p

Jdu = u = F(!p, k) ,

=

sn u J.

[!p = amuJ.

336.01

[See 400.J

!

2(m+2)(1~a2)(k2-a2) [(2m+1)k 2 V m + + 2 (m + 1) (oc2 k + oc2 - 3k Vm + +

Vm+a=

336.03

2

2)

1

+(2m+3)(oc4_2oc2k2_2oc2+3k2)V +a'snuCnUdnU]. m+2 (1- a2sn2u)m+2

[t=sin!p=snuJ.

=

337.00

Wo

337.01

TXT -

337.02

w.2 =

YYl -

Jdu = u = F(!p, k), f

[!p

sn2 u2 du2 -- 21 [/l( !p, oc 2 , k) - F( !p" k)J 1-asnu a

sn'udu 1 f ( 2 2 )2 = 4 1-asnu a

[

TT

u - 2Yl

+ V.2J·

337.03 sn2m udu 337.04 Wm = f (1-a2sn2u)m

where

v;. is given by 336.

m

= --~ ~ (- 1)m+i m! (l(zmL.J

i=o

(m-i)!jl

V. l'

= amuJ. [See 400].

Table of Integrals of Jacobian Elliptic Functions,

202

[n >m; t = sintp = snuJ,

f

337.51

1" 2m .LJ m

sn2mudu

(-1)m+im! (m-f)!f!

(1-1X2 Sn2 u ) i i = ;

Vi +n -

m,

[n >m; see 336J.

i=O

[nm; t=sinp=snu].

hI'" a,2'"

L'"

(a,2 - hl)i ml hli(m-j)!jl

i=O

Vn-"'+i' [n >m; see 336].

[n< m; t

=

sinp = snu].

(a,S - hl)i m! hli (m - j) Ii! Ym-n-i

339.75

+

hIm a,sm

m ~

L.J

;=m-,,+1

+

(a,2-hS)im! (m - i) !il V,,_m+;'

hlj

[n< m; see 331 and 336J.

[t = sin p = sn u] .

340.00 Zo =

Jdu = u =F(p, k),

[p=amuJ.

340.01

f

Z 1 --

Additional Recurrence Formulas. 1- 2 IX~ sn. U dU B

1-

IX

2 at12) II( tp, at,

1 [(at 2 _____

sn" U

205

IX"

k)

+ at1 u]. 2

[See 400.]

3~i O.02

Z 2 -_f{1-IX~Sn2U)2d _ 1 [" + 2at12{ at2{ 2 2 )2 U - ---.- at1 U 1-lXsnu

IXt

S)V,1

at1

+(2 at -

2)2V,] 2·

at1

[See 336.]

340.04 Z", =

f{

{1_ot2sn2u)m 2 )'" 1-otsnu

!

du

ot2m

= ot~m

l:'" i=O

{ot2 -otDl m ! 21., ( I ")' Vi' ot1l.m-l.

where Vj is given in 336. For the special case when at2 = 1, Vi becomes D2i (see 313); and when at 2 = k 2 , Vi reduces to 12i (see 315).

(n>m)

340.51

[see 336].

(ncosq.>(1-k2Sin2fP)mdfP

'II

Vi -

m

Vi -

k 2 t2

= J amu sn u en u dn 2m udu

'110= Jamuenusnudu= ;2

348.50

[t

k 2 sin2 fP

= sin
k 2 •

361.55 J~- = -; [u - E(u) - csu (dnu=f 1)]. 1±dnu

361.56

l

k

J ~~ ± tnu = ~lU+II(fP.2.k) ± 1

±

2

t

1

Vl12+1- V2 dnu]

V2(1 + k'2} n Vk'2 +

t

+ )i2dn u

'

[fP=amuJ.

216

361.57


r

.

du

1

-;------,--- = dnu±cnu

361.58 J

du

1+O(snu

~ [CS k

u =f ds U] . (oc2=F1,k2),

[II(q;,oc 2 ,k)-ocf],

=

[See 400; q;

=

am u],

where

1--

1

2V (1_0(2) (0(2-k2)

~n~

[2(1-0(2)(0(2_ k2)+(1-0(2 sn2 U)(2k 2-0(2_0(2 k 2)]

V

20(2 (1- 0(2) (0(2 - k 2 ) cn u dn u

,

if (1-oc 2)(oc 2 -k2) >0;

1 { u - -1- [II( q;,---, 0(2 361.59 J -11+cnu du=k ) -oc/1 ]} , + 0( cn u 0(. 1 + 0( 0(2 - 1

oc 2 =F 1, q; = am u ,

where 11 is given in 361.54.

361.60

f

1-

c~~du

1 + 0( cn u

~{-u + _1_ [II(q;,~, k)' -oc / 1 0(. 1- 0( 0(.2 - 1

=

oc2 =F 1, q; = am u.

361.61

361.62

=

±

1 + 0(1 cn u 1 + Ill; cn u

du

J cnudu 1±cnu

J

E(u) _ =

J'},

[See 361.54 for II'}

snudnu .

1±cnu

~ {oc1 u 0(.

+ ~1 _- 0(20(1 [II (q; , ~, k) 0(2 _ 1

oc 2 =F 1, q;

=

am u.

OC I

]} ,

1

[See 361.54 for 11']

1_ [II( k) -oc /1 ]}, - ~{u + _ 361.63 J 1cnudu + 0( cn u - 0(. 0(.2 - 1 q;, ~ 0(.2 - 1 '

oc 2 =F 1 , q; 361.64

J1+:~nu =~0(.2

= am u.

[See 361.54 for 11']

[u+oc 2II(q;, 1 +oc2 , k) +oc (oc 2 + 1) 12], q;=amu,

where

1 [V~- V1+~dnltl n

Vk' 2 + 0(2 +

V1 + (1

0(.2 dn u

'

+ oc2) (k'2 + oc2)

> O.

Integrals Involving Various Combinations of Jacobian Elliptic Functions.

361.66 361.67 361.68

f

tnudu

1+odnu =

f f •

1

1+0(.2 [atU-at

361.72 361.73

362.01

362.02

1

k'2

+ 1)/2J·

± sn It

=~[u-II( ,at2,k)+at/L

snudu

1+O(.snu

IP

0(.

0(.

[IP = amu; see 361.58 for

+ (at -

ocl ) [II (IP, oc 2, k) -

fJ.

oc I]}.

[See 361.58; IP = am u. J

u du =fk'2u± (d nU=f1)csuJ. f 1±dnu=7l2[±Eu dn

1

( )

f f -l-----;-dn-u1 f k2 cnudu

snu

1 -- dn u =

cn u dn u d It

sn u d u

1-

r

_

dnu -

ns2udu 2

2

.1-lXsnu

f -f~:;s~:u =

362.03

dn u - 1 .

= -

1

1-0(.2sn2 u

362.06

f

1-0(. sn u

362.07

f

1-0(.2sn2 u

cs2udu

1

-k-2s-n-u

d

[1 +

. ~ n u + k sn u sm- l (k sn uL .

1 - cn u

n cnu+dnu .

= U - E(u) +

oc 2

II

(u, oc 2)

-

dnucsu;

[IP = amu, sinlP (1-

(k 2

f

362.04

(at 2

+ cnUdn_~].

snudu = _1_[± E(u)

1 ± sn u

1+O(.snu

361.71

1P,1 +at2,k) -

[See 361.64; IP = am It. J

361.69 f-~±-~sn u du =~ {atl u

361.70

II(

217

_

=

snuJ,

~2) k'2 [k' 2U- E(u) -oc2k'2 II(u, oc 2)+dnu tn uJ . ~2) k'2 [k 2 E (u)

u - E(u)

+ (oc

2 -

-1X 2 k' 2 II (U,1X2) -

1) II(u, oc2 )

-

k4 sn u cd uJ.

dn u cs u.

362.05 tn2;d: = sd2udu =

21 '2[E(u)-k'2 (0(. -1)k 2 12 '2 [ (IX - k ) k

u +k'2II(u,1X2)-dnutl1u].

E(u) + k'2 JJ(u, oc 2) + k2 snucdttJ.


218

362.09 J1~::~'h = 362.10

{f

362.11

f

362.12

{f

(1.8 ~ [E(u)-k'2U

SnBucntu du =_1_[_ et2 E(u) 1-(1.BsnBu k l (1.'

snBudnlu . z • du 1-(1. sn u

2 2)

+ (et -k

+ (k + et

1

2

(1.

2

k'2) U +

+ k 2 (et2 -1) II(u, et2)].

= -, [k 2 U - et2 E(u)

CnlUdnlu 1- (1.1 snl u

II(u, et2)-dn U tn U].

+ (et

2 -

.

k 2 ) II (U, et2 )] •

----du = :' [et2 E(u)

+ (et

2 -

1) k 2U

+ (et

2 -

1) (et2 - k 2) II(u, et2)].

Integrals Involving Various Combinations of Jacobian Elliptic Functions.

362.19

362.22

362.23

{f

{

If f

+

snludnlu d U -I I )1 1- a. sn u

+

1 [ 2E(u )+("2+k2 '( I ) - oc oc - 20c2k2) u 2a. a. - 1 , (2oc2 k2 _ oc' _ k2)IJ(U, oc2) a. snuc:u.dnU]. 1-a.snu

+

[See 336.] cnBudnBu d (1- a.. snBU)3 U

X [k 2IJ(U, OC2)

1 a.' X

=

+ (k

2

OC2

+ OC2 -

2k2) V2 + (OC2 - 1) (oc2 - k 2) Va.] [See 336.]

snBucnBu (1-a.snu I B )8 du

I I

=

1

4"" [ -

a.

IJ(u, oc2) + (2 - oc2) V2+ (oc2 -1) Va]·

f snBucnlud~~du = ~ [-k u + (3 k2 -

362.24

f

362.25

362.27

f

219

{1-a.ISnSU)8

a.'

+ (2oc2k + 2oc2 2

2

[See 336.]

oc2 k 2 - oc2 )IJ(u, oc2) +

3k 2 - oc') V2 + (oc2 - 1) (oc2 - k 2 ) Va]. [See 336.]

snl ucn1u dn1u du (1 - ctl snl u)'

= ~ [_k2 IJ(u, oc2) + (3 k2 ct'

+ (2oc2 k2 + 2oc2 -

sn1udnBu du (1-a.lsn1u)'

3

k2 -

oc')

Va +

oc2 k 2 - oc2) V. + 2

(oc2 - 1) (oc2 - k2 ) V,].

[See 336.]

=~ [- k 2 V.2 + (2k2 a.'

oc2) v.: a

+ (oc2 -

k 2) V.]. ,

[See 336.]


220

363.01 363.02

f f

du

1+ksn2 u dU 1-ksn2 u

=_~+-1-tan-l[(1+k)tnunduJ. 2{1+k)

2

=~+ 2

du

363.03 If {1+ksn 2 u)2

=

tan-1 [(1-k)tnunduJ.

1

2{1-k) 1

2{1+k)2

{E(u)+k(1+k)u+

+ (1 + k) tan -1 [(1 + k) tn u nd u J + du

363.04 If {1-ksn 2 u)2

=

1

2{1-k)2

+ k (k -

{E(U)

1) u

k sn u cn u_ dn 1 + k sn 2 u

U} .

+

+ (1 - k) tan -1 [( 1 _ k) tn u nd u J _ k sn u cn u dn U} 1-

364.01

l

where

364.03 where

f R2 (sn u) en u du

2 f R2 [ 1 :~2 t2]

=

1

+d~2 t2 '

snu = 2t/(1 + k 2t2).

t = (snu)/(dnu + 1),

f R3 (sn u) dn u du = 2 f R3 [1 ~ t2] 1~ t2 ' t = (snu)/(enu + 1),

snu = 2t/(1 + t2).

364.04 JR 4 (snu)enudnudu= J R4(t)dt, where

365.01

t J R(enu) sn u dn udu

=

= -

snu. J R(r) dr,

where

r = enu.

365.02 J R(dnu) snuenudu

= -

where

k~ fR(x) dx,

x = dnu.

365.03 J R(tn u) neu deu du = J R(z)dz, where

365.04 J R (am u) dn udu where

.

J R(snu, enu, dnu) du = J Rl(snu) du + J R 2(snu) enudu + + J R 3(snu) dnudu + J R4(snu) enudnudu,

where R, R 1 , R 2, R 3, and R4 are rational functions.

364.02

ksn 2 u

z = tnu. =

J R(tl) dt1 , t1=amu.

Integrals of the Jacobian Inverse Elliptic Functions.

221

Integrals of the Jacobian Inverse Elliptic Functions. 390.01 (

fSn-1(y,k)dY= fF(sin-ly,k)dy 1

--

= y sn -1 (y, k) - k In I k V 1 - y2 - V 1 - k 2 y2] .

390.02 fcn-1(y,k)dy=ycn-1(y,k)-! cos- 1 [Vk'2+k 2 y2].

390.03 390.04 390.05 390.06 390.07 390.08

f dn- (y, k) d Y = Y dn (y, k) - sin- [V(1 - y2)/k2]. f tn- (y, k) dy = Y tn- (y, k) - ;, In [k'V1 + y2' + Vi + k'2 y2]. f nc1(y, k) dy ync-1(y, k) - ;, In [k' VY2=1 + Vk 2+k'2 y2]. f nd-1(y,k)dy=ynd-l (y,k) - ;, tan- 1[Vk'2(y2_1)!(1-k'2 y2)]. f dc- (y, k) dy = ydc-1(y, k) -In [V y2-1 + Vy2 - kill. 1

-1

1

1

1

=

1

(f

sd -1 (y, k) d Y = ysd-1 (y, k) - k~' tan-1 [Vk 2 (i- k'2 y2)/k'2(1 +k2 y2)].

390.12

f ns- (y, k) dy = yns-1(y, k) + In [~ + Vy2 - k 2]. f ds- (y, k) dy = y ds- (y, k) + In [Vy2-k'2' + V~j· f cs- (y, k) d y = Y cs- (y, k) + In [V 1 + y2 + V y2 + k' 2]. f cd (y, k) d y Y cd (y, k) + ! In Ik V1 - y2 - Vi - k2 y2] .

391.01

fysn- 1(y,k)dy= 2~2 [(k2y2-i)sn-l(y,k)+E(sin-ly,k)].

390.09 390.10 390.11

1

1

1

1

-1

1

=

-1

391.02 fycn-l(y,k)dy= 2~2 [(k2y2-k'2)cn-l(y,k) +E(cos-1 y,k)]. 391.03

fydn- 1(y,k)dy= ~ {y2dn-l(y,k)+E[sin-1V(1-y2)/k2,kj}.

391.04 (f

ytn-1(y, k) dy

= ~ 2k

{k'2 y2 tn-l (y, k) +

+ E [sin-1 Vy2j(i

_ _ _~

+ y2), k]- Y V(i + k'2 y2)/(1 + y2}.

222


392.01 Jymsn-1(y,k)dy= m~1 [ym+1u_ Jsn m+1 u du ]; [u = sn-1 (y, k)].

392.02 Jymcn-1(y,k)dy= m~1 [ym+1u_ Jcn m +1 u du ]; [u

392.03 J ymdn-1(y, k) dy

= m

= cn-1 (y, k)].

~ 1 [ym+1u - J dnm+1u duj; [u = dn-1(y, k)].

392.04 J ymtn-1(y, k) dy = m ~

1

[ym+!u - J tnm+! u duj; [u

= tn-1 (y, k)].

Elliptic Integrals of the Third Kind. Introduction. Definition l . The incomplete elliptic integral of the third kind in Legendre's canonical form is defined by

II (q;, (1.2, k) ==J"

400.01*

o

_-;=d=t==== 1-t2) (1-k2 t2)

(1_OC2 / 2)

V(

'" =J~-2==II(Ul ,(1.2), 1-oc sn U o

[y

= sin q; = sn ul , t = sin,!? = sn u;

When q; = 71:/2, Y = 1, u l = K. the integral is said to be complete. In addition to the argument q; and the modulus k, this integral depends also on the parameter (1.2. [Wefind it convenient to write - (1.2 in the integrand of Il(q;, (1.2, k), while in much of the literature -(1. or +n is written.] It is necessary to consider various cases according to the value of (1.2.

Possible cases. Case I: 0rp >sin-1 (

jJya 'P

512.02

R (sin2-&) d-& 2

1

sin2-& + b 2 cos2-& =

o

b

'P

JVi -

_ 1J'PR(Sin2-&Jd-& 1/ [- b V1+ n 2 sin 2-&

2;

[2

k 2 sin2-& '

see 281.01 with k2 =

b2 - a 2 - b2--

See 280 to 299 for other such integrals_

]

,

2_ _b 0] -

2 a2 , a > b2-,see 28209 . WI-th n - -b-' ->

o

1

)-1

R (sin2-&) d-&

[0b >a 2

P

yoc2 + {32

Single Integrals.

JVa cp

512.03

o

241

cp

= ~J R (sin2D) dD . 2 cos2D - b 2 sin2D a V1-n2sin2D R (sin2D) dD

o

[See 283.10 with

f

cp

513.01

. Va o

Va

b2 cos 2!}

2-

JV1 rp

R (sin2D) dD

1 b2

2 -

R (sin2D) dD + 2 sin 2!}

o

n

[a2 > b2 ; see 282.09 with n2 =

JVa "'/2

R (sin2D) dD

513.02

cp

513.04

cos2 D

2

2

o

2;

see 314 with

cos!}

f

R (sin 2D) d!}

• Va o

2

+ b2 cos2D

JVa

Va

2

+ b2

2

Va

cp

JV1-

o

R (sin2D) dD k 2 sin2!,} •

[See 281.01 with k 2

=

b2/(a 2 + b2 ).J

J b V

1 2 -

2

J

a2

o

R(sin2D) dD 1 - n 2 sin2D '

> b2 ; see 283.10 with

rp

J Va

sin2m-lDdD - 1 -- 2 + b 2 cot2D b

-:;;==c=::::::;c=~

0

-2~2l. a - b

n2 =

Jsn u du, [b >a , U1

sin 2m Ddf}

V1 -

k 2 sin2D

-

-

1 b

2m

2·

2

0

see 310.05 with k 2 = (b 2 - a2 )lb 2 , am u1 =r:p, sin t) = sn u J

514.01 cp

=

2,

cp

R(sin2!}) dD 2 cos2 D - b2

[

o

2

cp

1

cp

o

k=a Ib

n/2>r:p>cos-1(ajb),

snu=-k-'

cp

513.03

2 -b 2

u,

iJR(dn u)du, [b >a

_b_2_.]. a 2 _ b2

~r b.

o

sin2mD dD [a 2 V1+n2sin2D

> b2 ; see 282.04 with n 2 = (a 2 -

b2 )

Ib 2] •

514.02 [See 285.05 with a2 = 0(2 + [12, b2 = [12, nl2 > r:p > sin -1 ( Byrd and Friedman, Elliptic Integrals.

f3

Voc 2+ f32

16

).J

242 Misc. Elliptic Integrals Involving Trigonometric and Hyperbolic Integrands.

515.01

!

~

~

R(cosiJ)diJ

=_2_!R(1-2Sn2 U)dU,

VI + 2alcosiJ+ a~

o

[1 + a~ > 2al

;

1+ a l

0

see 289.09 with k 2 = ( 4a l_)2' I+a l

snu! = sin (rpJ2), 2sn 2 u = 1 - cosO].

!V ~

515.02

o

~

R(cosiJ)diJ =_2_!R[2al-(I+atk22sn2u]du, I-2a cosiJ+a2 I+al 2adl- k sn u) 1

1

[1 + a > 2a 2 l

0

l ,

see 291 .08

'th k2 -_

WI

(I

+4aal l )2 '

2adl - cosq;) ] k 2 (1- 2a l cosq; + ail .

2

T~~(al+blt)~~:):;21)

sinS fll

[sin 2 0

519.01

where R is a rational function and

(I-I) I '

= t. See 250.J

Single Integrals.

J o tp

COS2m{}d{}

;-;====O==~=

520.01

V1-k2sinl8

4m2

Js"1 n u d u )J"1sn u d uJ",sn udu+ .. · to m+1 terms,

8 F( rp, k) -4m -

21

243 +4mB (4m B-2B

2

41

0

(4mB- 28) (4mB- 4B)

6!

4

0

6

[rp =

o

am~;

see 310.]

521.01

521.02

J o

=- J 1

rp

d8 ~/_ fcos8

3

2

[

dt

2 rp, t Y = 3VCOS

lr;--;B'

"

v1-t3

0::;;:

Y
a 2 ; see 295.30 with k 2 = a 2jb 2 • J t 2m dt

+=1)=(t2=+=~~2=)

-;:=(1=2

o

.

+ {J2

()(2

[See 221.09.J

J

cosh rp

cosh21n-l{) d{}

1

V()(2+ ~Ztanh2{)

V()(2

o

+ {J2

I

t 2m dt

V(t 2 _ 1) (t2 _ _ {J2_) . ()(2 {J2

+

f "2

[See 216.06.J

sinh ll

(Jj

R(t)dt Vial +blt) (a 2+b 2 t) smh 2 CPI 1

525.08

[sinh -& 2

f

(1+ 1) t

.

t. See 250.J

=

coth ll 1].

...!!....-K(1!oc) , 2cx

dl}drp k 2 sin rp sin I}

1-

= V3nK2(k) , k2=~ 4 •

V(1 + sinl}) (1 +sinrp) =2nK. sin rp sin I}

0

II

n/2 n/2

o

=

dl} drp dtp 3-cosl}cosrp-cosl}costp-cosrpcostp

n/2 n/2

o

531.12

drpdl} cx2 (1 - cosl}) (1 - cosrp)

0

nn n

531.10

•

0 0

0

(k 2 cos2 1) + k'2 cos 2 rp) dl}drp V(1 - k 2 sin2 1}) (1 - k'2 sin2rp)

If V

=

E K'

+ K E'- KK'= n/2.

1 n/2

531.13

o

0

1-

dtdl} = 2G, (G = t 2 sins I}

JJV1 - t sin f)dtdf) 1 n/2

531.14

2

o

2

= G

0.91596559; Catalan's constant),

+ 1/2,

[d. 615.01].

[d. 615.02].

0 1 n/2

531.15

ff Vcos

o

0

dtdl} = n 2/4, + t 2 sin2 1}

2 1}

[d. 615.03].

248

Misc. Elliptic Integrals Involving Trigonometric and Hyperbolic Integrands.

JJVcos 2f} + t2sin2f}dtdf} =7r}j8, o 1,,/2

531.16

0

fI

1 "/2

531.17

•

1

t

o 0

II

I

[d. 615.04].

dtdf) t 2 sin2f)

VI -

n -2

0

dt -=nIn2-2G t

[d. 615.05].

'

In!2

531.18

(1

o 0

dtdf) 12 sin 2f)

+ I) VI -

II Vl1

531.19

'P

J

!tV1- t 2 sin 2f}dtdf}

531.20

00

\

[d. 615.09].

1 - cosrp sinrp

tdtdf) /2 sin 2f)

o 0

= n2j8,

[d. 616.03].

=~[. sin2rp~ 1- cosrp] , 3

(j~j'!'

531.21

0 0

[d. 616.04].

smrp

t dt df) ------=------:--:-::-V=== = ( 1 - oc II (rp, oc 1) - oc II (rp, oc 0) + (1- (X2 sin2f)) 1-/2 sin2f) + (1- cos rp) /sin rp -In (tan rp + sec rp). 2)

2,

2

2,

[See 112.02, 111.04, and d. 616.05.]

II k

532.01

o

00

0

II k

532.02

0

t3 dtdf) = cosh2 f) cosh2 f) - t 2

V

In (1 + V2) In (1

532.03

=E -k'2K.

00

lI o

Idtdf) Vcosh2f) - /2

I

~ l(2 - k 2) E - 2k'2K]. 3

+ V2)

cosh f) cosh rp d f) d rp

V

(1 - k 2 sinh f) sinh rp) sinh f) sinh rp (1 - sinh f)) (1 - sinh rp)

o

0

I

I V(I-

=2nK.

In (1+ V2) In(l + V2)

532.04

l

o

532.05

0

I

I

In (1 + Vii)

In (-1 + V2)

cosh{}coshrpd{}drp sinh2rpsinh2f))(I- sinh2f)) (1- sinh2rp)

In (1 + Vii)

In(-l + V2)

X V(1 -

I

In (1 + Vii)

=K2(V2/2).

~hf)~hrp~htp

(1 - sinh f) sinh rp sinh tp) X

In (-1 + Vii)

df)d"rpdtp = sinh2f)) (1 - sinh2rp) (1 - sinh2 tp)

4nK2(VZ/2).

Elliptic Integrals Resulting from Laplace Transformations. Finding the Laplace transform 1 of products of Bessel functions 2 often leads to the evaluation of elliptic integrals. We shall give here, however, only a short table of such integrals. 00

560.01

fe-Pllo(rt)lo(st)dt= o

l~ Q_!(Z)

:n: vrs

=

K

Vk _

:n: rs

fdU = 0

kK , :n:Vrs

where Qy (Z) is Legendre's function 3 of the second kind of degree y, and k 2 =2/(Z+1), Z= (P2+ r2+ s2)/2r s.

l

00

560.02

1

f e-

PI

11 (r t) 11 (s t) dt =

_1_

Q~ (Z) =

K _k_3_

fsn 2 U cd 2 U du

:n:VrS:n:VrS 0

0

=

_1_

k:n:VrS

[(1

+ k'2) K -

2EJ.

A function f (P) is called the Laplace Transform of g (I) if 00

f(P) =frPlg(t)dl.

o

2 For the definition of Bessel functions, see, for example, N. W. McLAC.'ILAN'S Bessel Funclions for Engineers, Oxford University Press, 1946. 3 See A Course of Modern Analysis by E. T. WHITTAKER and G. N. WATSON, Macmillan, New York, 1943.

250

560.04

Elliptic Integrals Resulting from Laplace Transformations.

If

1

00

k 2Y + 1

=

e-Ptfy(rt)f,y(st)dt= nY~ Qy-i,(Z)

o

nYrs

fK

sn 2Y ucd2Y udu,

0

± ir ± is) >OJ

RP. (P

[y~O;

where k 2 is as in 560.01 and y is an integer; K

fsn2Yu cd2y u du o

k: y L L (~)(~) y

=

K

y

(_1)i+Y+.l (k'2V-.lf nd 2Y - i -

i=O.l=O

A udu

0

with the integral on the right given by 315.05. "'/2

00

561.01 f e-Pt[Jo(rt)J2dt = o

2

n yp2

f

+ 41'2 0 Yi -

d{) = k 2 sin2 {)

kK nr'

[P>O; r realJ,

[See 806.01.J K

00

561.04 fe-Pt{[Jo(rt)J2-[J1 (rt)J2}dt =~fcn2udu=-2- [E - k'2 KJ, nr

o

00

561.05

nrk

0

fe-Pt{[Jo(rt)J2 o

K

+ [Jl(rt)J2}dt=~fsn2udu=-2[K -EJ. nr nrk 0

K

00

561.06 fte-pt fo(rt) fl(rt) dt o

=

~fsn2udu = ~2 [K - E]. 2nl'·

2nl'

0

K

00

561.07 fe-PI {[Jo(r t)J2 - 2r t fo(r t) II (r t)} dt o

= ~fdn2 u du =~ E. nr nl' 0

Laplace Transforms.

251

K

00

561.08 ft2e-PtUdrt)]2dt= 2~:nfcd2UdU= 2~:n [K-E]. o

0

f

00

561.09

t2n e- pt Un (r t)]2dt

=

(k: t'+1

K

r~ :nV~2) f cd 2n U du,

o

[See 320.05. ]

0

where

r is the gamma function

562.01

fte-pt 10(rt) 11(rt) dt =

and k 2 is given in 561.04.

00

:

:n"P(P -

o

2

4,,)

[P2E

+ (4r2 -

P2) K]

where k=2r/p, p2=j=4r2, and 1"/ is the modified Bessel function of the first kind of order y. 00

562.02 fe-PI 10(t)!!.!...-

]It

o

562.03

0

_

-

e- pt t2

o

0

=

k2 =

. V

SIll -1

[k

=

2r/p; Re. (P) > IRe. (2r)I]·

2"

00

fJ =

2r/p.]

[Jl(t)]2dt =-I)-+-~f(1+ cos{)) VP2+2 -2cos{)d{). 2 2:n 0 [See 290.58.]

Je- pt 10 (rt) II (s t) dt 563.01 1 where

=

2pE

00

f

[k

:n(P2 - 41'2) •

K J e- pt [Io(rt)]2dt = ~, o :np 00

562.05

tn dt

je-Pt {[Io(r t)]2 + 2r t Io(r t) II (r

1

562.04

2K _, V:n(P + 1)

=

-

1 [

2

1

_1 {n

s:n

=

~ [1- Ao(fJ, k)] s

- 2 [E F(fJ, k')

+ K E(fJ, k') -

K F(fJ, k')J},

p2 + ,,2 - 52 ] + -::;:=::;::::=:~==:;::==:::::o V (p 2 + 1'2 + 2)2 - 4 1'2 52 ' 5

[S2_PLy2+ Y(P2+ y2+ s2)2_41'2s2] [r-p 2-s2+ V(P2+,,2+ S2)2_4!'2 S2] • [p2+,,2_S2+ V(P2+ y2+ s2)2-41' 2s2] [p 2_ y2+ sB+ V(P2+,,2+ sB)B_41'2s2]

Hyperelliptic Integrals. Introduction. Definitions. If

P (1') = a o 'rn and R1 ['r, integral 1

+ a1 'rn - 1 + a2 'rn - 2 ••• + an = a o(1' -

VP (1') 1 is a rational function of

l'

r1 ) (1' - r2)

and of

...

(1' - rn)

VP (1'), the general

575.00 is called a hyperelliptic integral when n is greater than four. If the degree of P (1') is equal to 2P + 2, one can always obtain by means of a rational transformation (e.g., 'r=r1 +1/t) an equivalent integral in which the radicand is of degree 2P + 1. Since the resulting integral will lead to the same sort of transcendental functions as the original one, the number p may be used to designate the class of the hyperelliptic integral. (Integrals with p = 1 are elliptic integrals, while those where p = 0 are elementary.) Special cases of hypereUiptic integrals may appear in some trigonometric or Jacobian form such as

-f

H 2-

575.01 or

575.02

R2(sin2D) dD V(1 - 0(1 sin2f}) (1 - 0(2 sin2f}) (1 -

O(a

sin2f})

,

-f Vi

H 3-

Ra(sn2 u) d U. +snu

Reduction of Hyperelliptic Integrals to three Types. It is only necessary to consider the integral

575.03

H=fR(T)dT VP(T) ,

1 This is a special case of an Abelian integral, that is, an integral of the form R (T, z) dT, where R (T, z) is a rational function of T and z, and z is an algebraic function of T defined by I (T, z) = o.

f

Introduction.

253

since the general integral 575.00 can always be written as the sum of an integral of a rational function and of an integral of the form given in 575.03. The degree of the polynomial P is taken to be 2P + 1, i.e.,

P (.)

ao(. - rl )

=

(. -

r2) ...

r 2P +1)

(. -

- a0 .2P+l -

+ a1 .2P + a 2 .2p-l ... + a2P+l'

The function R(.) may be broken into partial fractions so that

j

°

f R(T)dT =fCA +A .+A VP{T) 1 2

575.04

+ Elf

I

+ Cf

l

I

.2+'''J~+

dT (T-OC)VP(T) dT (T-O(1)VP{T)

VP{T)

+ E 2f

dT ('r-oc)2VP(T)

+Cf

dT (T-OC 1)2VP(T)

2

+ ... + + ... + ...

,

where A o , AI' A 2 , .•• , E l , E 2 , E 3 , ••• are constants and ct., ct.l , ct. 2 , •.. are poles of the rational function R(.). Any hyperelliptic integral can thus be expressed linearly in terms of two general non-elementary integrals given by

I

f

TidT

575.05 I'; = VP (T) Am (., {J) = f

Cm,

i are integers, including zero]

dTV ' (T - fJ)m P (T)

with

When {J is a root of P (.) = 0, CIne has

Am (.,{J) =

575.07

l

l

J{T-fJ~:VP(T)

=

(1-2~)PI(fJ)

X

2P

X[VP(T)_ (T-fJ)m

+;:.;.

,,(2m- i -1)pU+I)({J)f

2{j+1)!

dT (T-fJ)m-iVp(T)

1 ,

where P(j)({J)=-!-P(.). If, however, {J is not a zero of P(.),

Am(., {J)

575.08

dT1 =

[{T2!;)~1 +

2{1 - : ) P{fJ) 2P+l

+"

~

1=1

(2m -: i J!

-

2) pli) ({J)

f

dT. (T - fJ)m- 1 PiT)

V

l'

(m=l=1).

254

Hypert'lliptic Integrals.

I

For I';, we have

I';

575.09

=f V~~:) X

=

.

(2i - 2; + 1)

ao X

.

[2P +1 ~ ~. 21:'-2PVP(1:)-~(2t-2p-1+1)ai

The integral I'; may be divided into two types:

f

dT YP(T) ,

and the p integrals

f

TP dT YP(T)-'

f

f

TdT YP(T) ,

TP+1 dT YP(T) ,

f

T2 dT YP(T) , ... ,

f

..

T'-1 dT YP(T)

1 .

p integrals of the form

fTP-ldT YP(T)

f -rP_+2

,

__

dT f_T'~ l~ YP(T) , ... , YP(T)·

Integrals of the first set, which are everywhere finite on the two-sheeted Riemann surfaces, are called integrals 01 the lirst kind, while those of the latter set, which possess only algebraic discontinuities at the point 1: = 00, are referred to as integrals 01 the second kind. There exists another type of hyperelliptic integral called an integral 01 the third kind. Such integrals have a logarithmic infinity. If fJ is not a zero of P (1:), the integral

f

dT

(T - fl) YP(T)

is of the third kind. [If the polynomial P (.,;) is of degree integral

2P +~, the

f TPdT

YP(T)

is also an integral of the third kind.]

Hyperelliptic Integrals which Reduce to Elliptic Integrals. For the evaluation of hyperelliptic integrals, one must usually resort to direct numerical integration or to the use of complicated series expansions. Many hyperelliptic integrals which occur in physical and geometric problems, however, may be reduced to elliptic integrals or the sum of elliptic integrals and can therefore be evaluated more simply. Some important examples follow. An example of a hyperelliptic integral which readily reduces to elliptic integrals is

575.10

h1=f Ya

R(T)=dT=~==-cc-

o(T2

+ r1) (T' + r2) (T2 + ra)

255

Hyperelliptic Integrals which Reduce to Elliptic Integrals.

.2

By writing R(.)=2R1 (.2)+2.R 2(.2) and making use of the substitution = t, we immediately obtain

hI =j

575.11

Rl (t) dt

Vao(t

+j

+ "I) (t + "2) (t + "3) t

R (t) dt 2

Vao(t

+ "I) (t + "2) (t + "3)

,

where the two integrals on the right are treated in 230 to 267. Three special cases of 575.10 are the integrals h4

575.12

=jR(T) dT V1 +-r6

•

[See 576, 577 and 578.J The more general integral n

575.13 where

h

=

JR(.) IT (. i=1

m

ry'i d.,

L fli is an integer but eachfli is not an integer, furnishes numerous

i=1

other examples of integrals which are reducible to simpler integrals 1. For any arbitrary integers nand n', the following integrals reduce to those of class p = 1, i.e., to elliptic integrals:

h7

=

fR(.) ~-V(.

-

rl )2 n

(. -

r 2 )m-J!n (.

for m

=

3, 4, 6, 8;

for m

=

3,4,6,8,12;

-

r3 )md.,

for m=3,4,6,8,12;

575.14

for m ~2 =

fR(.) 2!"y' (. - rl )2n

(. -

r 2)2 n

(. -

=

3,4;

r3 )m-2n (. - r4 )m

for m

=

2n

d.,

3,4,6,8,12.

1 See H. LENZ: Zuruckfuhrung einiger Integrale auf einfachere. Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse. No. 10. 1951. pp. 73-80.

Hyperelliptic Integrals.

256

The following integrals may also be reduced to elliptic integrals: h13 =

h14 =

575.15

h15 = h16

=

JR(o) V~do.

JR [0, mya

m

JR(-r) V(am-o

m)

(b m _ -rm) d-r,

JR[-r, Vam--rm, j/bm--rm]d-r,

_f R (-r) V

h17 -

for m = 3, 4, 6, 8 ;

-om] do,

2

(T 2 -

a 2 )m (2 T -

(T2 -

b 2 )2n

2)2n C

d

-r ,

for m

=

3, 4;

for m = 3,4; for m = 2, 3, 4.

The integrals 575.10 and 575.12 are special cases of the more general integral 1

which can be reduced to elliptic integrals if the six branch-points r1 , r 2 , r3, ... , r6 form three pairs of points of an involution.

Table of Integrals. The remainder of this section will give a short table of some special cases of the integrals mentioned in the above two pages. The integrands of the integrals in this table will not be reduced to Jacobian elliptic functions as was done in the previous sections, but will only be transformed to the simpler algebraic forms of those in 211 to 279. Integrands involving

-r2 = t

,

d -r

VI -

= ~

2Vt'

-r6, (0 < Y;;:;: l) y = y2

1 L. KOENIGSBERGER: Uber die Reduktion hyperelliptischer I ntegrale aUf elliptische. Journal flir die reine u. angewandte Mathematik, Vol. 84,1878, pp. 273-294.

Integrands involving V-rG- 1, (1 ~ Y1].

E)2/k3 dk = ;71;2/8 - 1/2 .

jP(q;, k) dk o

616.02

=

_1 [24 (In 2)2 - ;71;2].

0

1

616.01

+ m + 1.

0

=

[d. 531.19].

cosCP]

,

[d. 531.20J.

1 -: coscp _ smcp

1+Gts~ncp -oc2 II(q;,oc2 ,O),

1 - Gtsmcp

A short lO-place table of (m + 1)

1

f

o

[d. 531.21 and see III.OIJ.

k'" K dk for m= - .9(.1) 2 may be found

in E. L. KAPLAN: Multiple Elliptic Integrals. Journal of Mathematics and Physics, Vol. 29. 1950. pp. 69-75. 18*

276

Integrals of the Elliptic Integrals.

With Respect to the Argument. 630.01

f V1 -

o

630.02

f

F(fJ,k)dfJ = k2 sinB fJ

[F(tp,k)]I, 2

Oy>1].

Derivatives.

286

732.06 ~nd-l(y, k) dy

732.07 ~dC-l(y, k) dy

=

1 1 ¥(1 - k'2 y2) (1

732.09 ~ns-l(y, k) dy

732.10 ~dS-l(y,k) dy

732.12

cd -1 (y, k)

ddy

[0 < y< ;, ],

,

1

[y >

¥(ya _ 1) (y2 _ k2) ,

1

d

732.11 dy" CS-1 (y, k)

+ kl y2)

¥(y2 _ k'l) (y2

+ k2)

+ y2)(yl + k'2)

1>k].

[y >k' >0].

,

1

¥(1

1],

[oo>y>1].

¥(y2 _ 1) (y2 _ ka) ,

d

732.08 dy"sd- 1 (y, k)

[;, >y>

1 , ¥(y2-1) (1-k'2 y 2)

[oo>y>O].

,

1 ¥(1 - y2)!1 _ kl y2) ,

[0< y< 1].

With Respect to the Parameter. Differentiation of the Normal Elliptic Integral of the Third Kind.

733.00

733.01

1

~II(U'OC2'k)= OVl on on X

=

VB-VI

01:2

=

1

201:2 (1- 01:2) (01:2

[oc2 E(u) + (k2 -oc2) U+ (oc4 _ k2) II(u, oc2, k) _

02

onzII (u,oc 2,k)=2

CJ3 II(u,oc,k 2) 733.02 ~3 -6 un

where is given by 336.

f f

-

k 2)

X

c:

01:4sn u u 2dn u] , 1-01: sn u

[oc2 sn4 udu

(1-0I:2 Sn2 u)3

2

=a1[Vl -2V2

= n].

+1S].

sn 2udu 6 V; V. TT (1-01: 1)4 =s[-V1 +3 2-3 3+~4]' sn u 01: [oc 2 =n]. 8

With Respect to the Parameter.

287

Differentiation of Other Elliptic Integrals. [oc 2 =n].

734.02

8 f an

(1-et2 sn2 u)3

734.03

8 f en

(1-et2Sn2u)m

735.01

en

735.02

en

735.03

en

735.04

8 f1-et~Sn2ud en 1 - eta sna u

8 f 8 f

8 f

du

du

snaudu

1- eta sna u

cn 2 udu

1 - eta sna u

dn 2 udu

1 - et2 sn2 u

3

=

W, -

Va)

et2

m(Vm+l - Vm)

=

[See 336.]

eta

sn'udu

[See 337.02.]

=

fsn2ucn2udu (1- et2 sn2 u)2 .

[See 362.18.]

=

fsn2udn2udu (1- et2 sn2 u)2 .

[See 362.19.]

=

f

(1-eta Sn2 u)a·

-f U -

sn2 udu

(1- et2 sn2 U)2

-0.2f 1

sn'udu

(1- et2 sn2 U)2

•

[See 362.15 and 337.02.J

Miscellaneous Integrals and Formulas. While most of the integrals in the previous sections have at least one variable upper limit, both of the limits of integration of the integrals given here are fixed. b

K

In(t)dt 80001{f • V(a2 - t2) (bl _ t2)

o

" I b

800•02 f {

o

800.03

f

b

V(a S

f

800.04

a

a

0

k 2 =b2/a 2, (a>b>O). K

In (t) dt

+ tl) (bl- tS)

k k , . = -fIn (bcnu) du =-b [K In (Va"b) - n K /4J, b 0 k2 = b2/(a2 + b2). K

In (t) dt V(a l - t 2) (t 8 -

K

= ~fIn (a dn u) du = ~fln (b nd u) du bl )

a0

= :

a0

[KIn(fab)]; k 2 = (a 2 _b 2 )/a2 , a>b>O. K

00

o

=~fIn(bsnU)du=~[Kln(fab)-nK'/4];

In (t) dt V(a S t 2 ) (t2

+

+ bl)

= ~fIn (b tn u) du a

0 K

= ~fln (acsu) du a

o

= ~KIn(fab), a

where

!V 00

800.05

{

(t S

K

In (t) dt -

al) (t 2

-

bl)

~f In (ansu) du =~[KIn(fab) + n K'/4]; o

a >b >0, k 2 = b2/a 2,

289

Miscellaneous Integrals and Formulas.

where

1= - 2KIn e

[F(A, k)]

V2k' Kin

= 1/;F({3, k')

'

if 0< rx.2< k2. ,

+ KIn (k'/k) + nf' _n[F;~,k')J2 _ _ 2K In

e [F({J, k'l] V2k K'in

'f k 2 O, a>b]

where

j P"'-l(X) = 1

n

810.00 =

o

n d{}

[x + ~ Xl - ' 1 cos{} ].-m 1

K

~ (x+ VXS-1)m-ijdnSmudu, [misaninteger;see314.05.] o

where k S =2[xYXS -1+1-X2 ], (x >1); and P"'-l(x) is Legendre's function.

811.01

j

.00

K=_1_ 4in

-loo

r(t+t)r(t+t)r(-t) (_kB)1 r(1

+t)

dt

,

where the path of integration is indented so as to separate the poles at t=O, 1, 2, ... from those at t= -t-n, (n=O, 1,2, ... ); and is the gamma function .

r

811.02 E

= __ 1

j

• 00

4in -tOO

r(t

+ t) r(- t + t) r(- t) (- kl)t r(1 +t)

dt

,

Miscellaneous Formulas.

295

where the path of integration is chosen so that the poles at t = - t - n, t=t-n, (n=O, 1,2,3, ... ), lie on the left of the path and the poles at t = 0, 1, 2, ... lie on the right.

1('[ ~

812.01

)

as

'~

Z2

+ b2 + c2

=

v;~~

(a >b >c)

F( 0;

Iql < 1J.

Developments of Jacobian Elliptic Functions.

307

00

2K

:lfU (1 +ql"')8 1- 2ql'" cos (:lfu/K) +q4'" tnu=-tan. :If 2K!l[(1- ql"')8 1 2ql'" cos (:lfu!K) +q4"'] ,

909.03 {

+

[Re. (K'/K) > 0;

Iql < 1].

Products for the other eight Jacobian elliptic functions may be obtained by taking the reciprocal and quotients of the above.

Other Developments.

910.01 { snu =

f

2:K

csc

2~ [u -

(2m - 1) iK'],

[II m (u/K) I< 1m (iK'/K)].

... =-00

910.02

f

{enu = 2~~

(-1)"'csc

2~ [u - (2m -1) iK'], [ilm(u/K)I 3)·

(A I)

Reduction to Jacobian form: - The integral (A I) may be written

J y

U =3

2

dt

V(t - 3) (t -

1

+ V3) (t ~ 1 - V3) (A 2)

314

Appendix.

+ V3.

V3, the evaluation of the

Using 23S.00, with a = 3, b = 1 c= 1integral is found immediately to be

V

K

u =---===-

1

2+V3

V2+V3

sn-1(sin1'

, E

Z(A,k) =E(A,k) -KF(A,k).

Theta Functions.

n

V

2 (k l - ocl )( 1 -

319

~ln# (w) =

O(2)

oc2 OW

0

V(kl -

KZ({J. k) o(

2)( 1 -

o(

2) oc2 '

[d. 414.02], 2V(k 2

1053.03

ocn -

0

0(2) (1 -

0(2)

OW

In#d w)= K

+

ocKZ({J.k)

V(1 _ 0(2) (k2 _ 0(2) ,

[d. 414.01]. ~Vk2 -

0(2 ~ln#2(w) = 1-0(2 OW

20(

K

+

(0(2 - k 2 ) KZ({J.k) • yO(I(1-0(1)(kl-0(2)

[d. 414.04],

where sin p = (l.Jk,

Z (P, k)

w

=

nF({J.k) -~'

= E(P, k)

n 2y0(2(1-0(2) (0(2-k2)

-

E

K

0< (1.2< k2;

F(P, k) .

~ln&(iw)=~-

ow

k 2-0(2

0

nAo(V'.k) 2 VO(l (1-0(1) (oc2-k2) ,

[d. 410.02], 2 VO(l (1-0(2) (0(2_ kl) ,

[d. 410.01],

1053.04

~l~- ~ln#2(iw)= 2

where

V~2)

. V

0(2 0(1-k2'

Sln"P =

Ao ("P, k)

OW

=

W =

~ [E F ("P, k') n

n(k l -oc2)Ao(V'.k) k2)

2 Vocl (1 _ 0(2) (0(2 -

n F (V'. k') 2K'

0

[k 2 = b2/a 2, a2 > b2 > y~ > y~J

=

a [k2 1'sn2 u du - j'ns 2 u dU]

=

a [u - E(u) - u

1'1

= a =

"1

+ E(u) + dn u cs uJ~:

[dn U 2 cs U 2 - dn U 1 cs u1J

V(b 2 - y:) (a 2 - Y~)IY: - V(b 2 - Y~) (a 2 - Y~)/Y~.

Written in the Jacobian forms to which they reduce, five other special cases are

k'2 f sc 2 udu - f cs 2udu =dc u ns u 1060.04

k 2k'2 fsd 2udu + f ds 2udu = - cdunsu, f dc 2 u du - k 2 f cd2udu = k'2 tn u ndu, f dn 2 u - k'2 fnd 2 udu = k2 snucdu, k 2f cn2ud~t + k'2 fnc 2udu = tnudnu.

An alternate form of the pseudo-elliptic integral 1060.02 is given by

1060.05

p = ~f Sa

2

Vt (1 -

R(t)dt t) (1 - k 2 t) ,

where R(t) is a rational function of t and satisfies anyone of the relations

R (t)

+ R (1/k2 t) = 0;

R (t)

+ R [( 1 -

k 2t) / k 2(1 - t) ]

R(t)

+ R [(1 -

= 0;

t)/(1 - k 2t)]

=

o.

A special instance of this occurred in the Introduction.

Tables of Numerical Values. The following short, 6-place tables give values for the elliptic integrals of the first and second kind and for some functions which are useful in the numerical evaluation of elliptic integrals of the third kind. A comprehensive survey of other tables, as well as of all known errata, may be found in Alan Fletcher's Guide to Tables of Elliptic Functions, Mathematical Tables and Other Aids to Computation, Vol. III, No. 24, The National Research Council, Washington, D. C., 1948.

Values of K, E, and q.

323

Values of the Complete Elliptic Integrals K and E, and of the Nome q. E

f

"/2

=

vr-1---k::-:2-S1:-'n-=-2o--=-d 0- ;

q=

e-(1fK'jK).

o sin-Ik

K

E

q

sin-Ik

K

0° 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° 11° 12° 13° 14° 15° 16° 17° 18° 19° 20° 21 ° 22° 23° 24° 25° 26° 27° 28° 29° 30° 31° 32° 33° 34° 35° 36° 37° 38° 39° 40° 41° 42° 43° 44° 45°

1.570796 1.570916 1.571275 1.571874 1.572712 1.573792 1.575114 1.576678 1.578487 1.580541 1.582843 1.585394 1. 588197 1·591254 1.594568 1.598142 1.601979 1.606081 1.610454 1.615101 1.620026 1.625234 1.630729 1.636517 1.642604 1.648995 1.655697 1.662716 1.670059 1.677735 1.685750 1.694114 1.702836 1.711925 1.721391 1.731245 1.741499 1.752165 1.763256 1.774786 1. 786769 1.799222 1.812160 1.825602 1.839567 1.854075

1.570796 1.570677 1.570318 1.569720 1.568884 1.567809 1.566497 1.564948 1.563162 1.561142 1.558887 1.556400 1.553681 1.550732 1.547555 1.544150 1.540522 1.536670 1.532597 1.528306 1·523799 1.519079 1.514147 1.509007 1.503662 1.498115 1.492369 1.486427 1.480293 1.473970 1.467462 1.460774 1.453908 1.446869 1.439662 1.432291 1.424760 1.417075 1.409240 1.401260 1.393140 1.384887 1.376504 1.367999 1.359377 1.350644

.000000 .000019 .000076 .000171 .000305 .000477 .000687 .000935 .001222 .001549 .001914 .002318 .002762 .003245 .003769 .004333 .004938 .005585 .006272 .007002 .007775 .008590 .009449 .010353 .011301 .012295 .013335 .014421 .015556 .016740 .017972 .019256 .020591 .021978 .023419 .024915 .026467 .028077 .029745 .031474 .033265 .035120 .037040 .039028 .041085 .043214

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°

1.854075 1.869148 1.884809 1.901083 1.917998 1.935581 1.953865 1.972882 1·992670 2.013267 2.034715 2.057062 2.080358 2.104658 2.130021 2.156516 2.184213 2.213195 2.243549 2.275376 2.308787 2.343905 2·380870 2.419842 2.460999 2·504550 2·550731 2.599820 2.652138 2·708068 2·768063 2.832673 2·902565 2.978569 3·061729 3.153385 3.255303 3·369868 3·500422 3.651856 3·831742 4.052758 4·338654 4·742717 5.434910 00

E

q

1.350644 .043214 1.341806 .045417 1.332870 .047696 1.323842 .050054 1.314730 .052495 1.305539 .055020 1.296278 .057633 1.286954 .060338 1.277574 .063139 1.268147 .066039 1.258680 .069042 1.249182 .072154 1.239661 .075380 1.230127 .078725 1.220589 .082194 L211056 .085796 1.201538 .089536 1.192046 .093423 1.182589 .097465 1.173179 .101672 1.163828 .106054 1.154547 .110624 1.145348 .115393 1.136244 .120378 1.127250 .125595 1.118378 .131062 1.109643 .136801 1.101062 .142837 1.092650 .149197 1.084425 .155917 1.076405 .163034 1.068610 .170595 1.061059 .178656 1.053777 .187285 1.046786 .196568 1.040114 .206610 1.033789 .217549 1-.027844 .229567 1.022313 .242913 1.017237 .257940 1.012664 .275180 1.008648 .295488 1.005259 .320400 1.002584 .353166 1.000752 .403309 1.000000 1.000000 21*

324

Appendix.

Values of the Complete Elliptic Integrals K, K', E, E' and the "/2 Nomes q and q'. K=f

df}

Yl-k2sin2f}

o k'

K

0.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 ·31 .32 .33 .34 .35 ·36 .37 ·38 ·39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 ·50

1.570796 1.574746 1.578740 1.582780 1.586868 1.591003 1.595188 1.599423 1.603710 1.608049 1.612441 1.616889 1.621393 1.625955 1.630576 1.635257 1.640000 1.644806 1.649678 1.654617 1.659624 1.664701 1.669850 1.675073 1.680373 1.685750 1.691208 1.696749 1.702374 1.708087 1. 713889 1.719785 1.725776 1.731865 1.738055 1.744351 1. 7 507 54 1.757269 1.763898 1.770647 1. 777 519 1.784519 1.791650 1.798918 1.806328 1.813884 1.821593 1.829460 1.837491 1.845694 1.854075

,/?,'2

K'

,

f"/2

K'=K(k'); E= V1-k2sin2DdD, E'=E(k') q= e-"K'IK; 0 q' = q (k'). K'

E

E'

1.570796 1.000000 3.695637 1.566862 1.015994 3·354141 1.562913 1.028595 3.155875 1.558948 1.039947 3.016112 1.554969 1.050502 2.908337 1.550973 1.060474 2.820752 1.546963 1.069986 2.747073 1.542936 1.079121 2.683551 1.538893 1.087938 2.627773 1.534833 1.096478 2.578092 1.530758 1.104775 2·533335 1.526665 1.112856 2.492635 1.522555 1.120742 2.455338 1.518428 1.128451 2.420933 1.514284 1.135998 2.389016 1.510122 1.143396 2.359264 1.505942 1.150656 2·331409 1.501743 1.157787 2·305232 1.497526 1.164798 2.280549 1.493290 1.171697 2.257205 1.489035 1.178490 2.235068 1.484761 1.185183 2.214022 1.480466 1.1.91781 2.193971 1.476152 1.1.98290 2.174827 1.471818 1.204714 2.156516 1.467462 1.211056 2.138970 1.463086 1.217321 2.122132 1.458688 1.223512 2.105948 1.454269 1.229632 2.090373 1.449827 1.235684 2.075363 1.445363 1.241671 2.060882 1.440876 1.247595 2.046894 1.436366 1.253458 2.033369 1.431832 1.259263 2.020279 1.427274 1.265013 2.007598 1.422691 1.270707 1.995303 1.418083 1.276350 1.983371 1.413450 1.281942 1.971783 1.408791 1.287484 1.960521 1.404105 1.292979 1.949568 1.399392 1.298428 1·938908 1·394652 1.303832 1·928526 1.389883 1.309192 1.918410 1.385086 1.314511 1.908547 1.380259 1.319788 1.898925 1.375402 1.325024 1.889533 1.370515 1.330223 1.880361 1.365596 1.335382 1.871400 1.360645 1.340505 1.862641 1.355661 1.345592 1.854075 1.350644 1.350644 00

I

K

E'

I

E

q

q'

.000000 .000628 .001263 .001904 .002551 .003206 .003867 .004536 .005211 .005894 .006585 .007283 .007989 .008703 .009425 .010156 .010895 .011643 .012401 .013167 .013943 .014728 .015524 .016329 .017146 .017972 .018810 .019659 .020520 .021393 .022277 .023175 .024085 .025009 .025946 .026898 .027864 .028845 .029842 .03085.4 .031883 .032929 .033993 .035075 .036175 .037296 .038436 .039597 .040780 .041985 .043214

1.000000 .262196 .227935 .206880 .191496 .179316 .169208 .160554 .152981 .146244 .140173 .134646 .129571 .124880 .120517 .116439 .112610 .109002 .105589 .102352 .099274 .096338 .093533 .090848 .088271 .085796 .083413 .081117 .078902 .076761 .074690 .072685 .070741 .068854 .067023 .065242 .063510 .061825 .060182 .058582 .057020 .055496 .054008 .052554 .051133 .049742 .048382 .047050 .045745 .044467 .043214

1.00 ·99 .98 ·97 .96 ·95 ·94 ·93 ·92 ·91 ·90 .89 .88 .87 .86 .85 .84 .83 .82 .81 .80 ·79 .78 ·77 .76 ·75 ·74 ·73 ·72 ·71 .70 .69 .68 .67 .66 .65 .64 .63 .62 .61 .60 ·59 .58 ·57 .56 .55 ·54 ·53 ·52 ·51 ·50

q'

q

k'

k"

Values of the incomplete elliptic integral of the first kind.

325

Values of the Incomplete Elliptic Integral of the First Kind, F(q;, k).

-J V1'P

F(

-

o

df}

k2sin2f}'

'P

sin-1k 0°

0° 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° 11 ° 12° 13° 14° 15° 16° 17° 18° 19° 20° 21 ° 22° 23° 24° 25° 26° 27° 28° 29° 30° 31° 32° 33° 34° 35° 36° 37° 38° 39° 40° 41° 42° 43° 44° 45°

gJ,

k)

0.000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 ,000000 .000000 .000000 .000000 .000000 .000000 .000000 ,000000

I

I

5°

I

I

15°

i

0.174533 0.261799 .261800 .174533 .261803 .174534 .261807 .174535 .261814 .174537 .261822 .174540 .261832 .174543 .261843 .174546 .261857 .174550 .261872 .174554 .261888 .174559 .261907 .174565 .261927 .174571 .261949 .174578 .261972 .174585 .261997 .174592 .262024 .174600 .262052 .174608 .262082 .174617 .262113 .174626 .262146 .174636 .262180 .174646 .262215 .174657 .262252 .174668 .262290 .174679 .262329 .174691 .262370 .174703 .262411 .174715 .262454 .174728 .262498 .174741 .262542 .174754 .262588 .174767 .262635 .174781 .262682 .174795 .262731 .174809 .262780 .174824 .262829 .174839 ,262880 .174853 .262931 .174868 ,262982 .174884 .263034 .174899 .263086 .174914 .263138 .174930 .263191 .174945 .263244 .174961 .263297 .174976

0.087266 .087266 .087267 .087267 .087267 .087267 .087268 .087268 .087269 .087269 .087270 .087270 .087271 .087272 .087273 .087274 .087275 .087276 .087277 .087278 .087279 .087281 .087282 .087283 .087285 .087286 .087288 .087289 ,087291 .087292 .087294 .087296 .087298 .087299 ,087301 .087303 .087305 .087307 .087308 .087310 .087312 .087314 .087316 .087318 .087320 ,087322

0° I 5° sin-1k - - - - - -

10°

I

10°

I

15° 'P

2eo

0.349066 .349068 .349074 .349085 .349100 ·349118 ·349141 .349169 .349200 .349235 .349275 .349318 .349366 .349417 .349472 .349531 .349594 ·349660 .349730 .349803 .349880 .349960 .350044 .350131 .350220 ·350313 .350409 ,350508 .350609 .350712 .350819 .350927 .351038 .351151 .351266 .351382 .351501 .351621 .351742 .351865 .351989 .352114 .352239 .352366 .352493 ,352620

I

20°

I

i

I

25°

I

0.436332 .436336 .436349 .436369 .436397 .436434 .436478 .436530 .436591 .436659 .436735 .436819 .436910 .437010 .437116 .437230 .437351 .437480 .437615 .437757 .437906 .438062 .438224 .438393 .438567 .438748 .438934 .439126 .439324 .439526 .439734 .439946 .440163 .440384 .440609 .440838 .441071 .441307 .441546 .441788 .442032 .442279 .442528 .442778 .443030 .443282 25°

30°

0·523599 .523606 .523626 .523661 ·523709 ·523771 ·523847 .523936 .524038 .524155 ·524284 .524427 .524583 .524751 .524933 ·525128 .525334 .525554 .525785 ·526029 .526284 ·526551 .526829 ·527118 .527418 ,·527728 ·528049 ·528380 ·528720 .529070 ·529429 .529796 ·530172 ·530555 .530946 ·531344 ,531749 .532160 ,532577 ·533000 .533427 .533859 .534295 .534735 .535178 .535623

I

30°

-

Appendix.

326

f V1 rp

F(gJ, k)

=

dfj

h2 sin 2 fj

o sln-1 1

Handbook of Elliptic Integrals for Engineers and

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