A Table of Elliptic Integrals of the Second Kind* Table of Elliptic Integrals of the Second Kind*...

mathematics of computationvolume 49, number 180october 1987, pages 595-606

A Table of Elliptic Integrals

of the Second Kind*

By B. C. Carlson

Abstract. By evaluating elliptic integrals in terms of standard Ä-functions instead of Legendre's

integrals, many (in one case 144) formulas in previous tables are unified. The present table

includes only integrals of the first and second kinds having integrands with real singular

points. The 216 integrals of this type listed in Gradshteyn and Ryzhik's table constitute a

small fraction of the special cases of 13 integrals evaluated here. The interval of integration is

not required, as it is in previous tables, to begin or end at a singular point of the integrand.

Fortran codes for the standard /^-functions are included in a Supplement.

1. Introduction. Let

(1-1) [p} = [Pi,P2,---,Pn}=f(ai + b1tr/2---(an + bnt)''"/2dt,v

where px,...,pn are nonzero integers, the integrand is real, and the integral is

assumed to be well defined. Many integrals like

/ (1 - ;c2sin2<í,)"/2¿<í, and f (a + bz2)Pl/2(c + dz2)Pl/1 dz

can be put in the form (1.1) by letting t = sin2<f> or / = z2.

For purposes of classification we assume the b's are nonzero and no two of the

quantities a¡ + b,t are proportional. If at most two p's are odd, the integral (1.1) is

elementary. If exactly three p's are odd (the "cubic case"), the integral is elliptic of

the first or second kind if all the even p 's are positive, and otherwise it is third kind.

The only such integral of the first kind is [-1, -1, -1]. If exactly four p's are odd (the

"quartic case"), the integral is elliptic of the first or second kind if all the even p's

are positive and px + • ■ • +pn < -4; otherwise it is third kind. The only such

integral of the first kind is [-1,-1,-1,-1]. If more than four p's are odd, the

integral is hyperelliptic.

Integrals of the first kind are traditionally expressed in terms of Legendre's

F(<p, k) with 0 < k < 1 and 0 < <j> < it/2. Integrals of the second kind require

Received October 6, 1986.

1980 Mathematics Subject Classification (1985 Revision). Primary 33A25; Secondary 33A30.

*Part of this work was done at the University of Maryland, where the author was a visitor at the

Institute for Physical Science and Technology, with the support of AROD contract DAAG 29-80-C-0032.

The rest was done in the Ames Laboratory, which is operated for the U. S. Department of Energy by

Iowa State University under contract no. W-7405-ENG-82. The work was supported by the Director of

Energy Research, Office of Basic Energy Sciences.

©1987 American Mathematical Society

0025-5718/87 $1.00 + $.25 per page

595

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

596 B. C. CARLSON

E((p, k) and usually F also. We shall replace F by the symmetric integral

(1.2) RF(x, y, z) = \Ç [(t + x)(t + y)(, + z)Yl/2dt

and E by

(1.3) RD(x, y, z) = f jf° (t + Xyl/2(t + y)-1/2(t + z)"3/2dt.

These Ä-functions are homogeneous:

RF(Xx,Xy,Xz) = X-l/2RF(x,y,z),

RD(Xx,Xy,Xz) = X-3/2RD(x,y,z),

and they are normalized so that

(1.5) RF(x,x,x) = x~1/2, RD(x,x,x) = x"3/2.

Fortran codes [6] for computing RF and RD when x, y, z are real and nonnegative

are listed in the Supplements section of this issue and can be found also in most of

the major software libraries.

Customary integral tables [1], [7], [9] assume that the interval of integration begins

or ends at a branch point of the integrand, and many special cases are listed

according to the positions of the other branch points relative to the interval of

integration and to one another. If the integral at hand does not have either limit of

integration at a branch point, it must be split into two parts that do. In the present

paper these two parts are recombined by the addition theorem, and the need to

specify the relative positions of the branch points then disappears. The use of

/^-functions greatly facilitates the application of the addition theorem and leads to a

further unification that cannot be achieved with Legendre's integrals, because the

expressions for RF(x, y, z) and RD(x, y,z) in terms of Legendre's integrals with

0 < k < 1 and 0 < </> < 77/2 depend on the relative sizes of x, y, and z (see [5, (4.1),

(4.2)], (5.25), and (5.32)).

Integrals of the third kind and integrands with conjugate complex branch points,

resulting from an irreducible quadratic factor a¡ + b¡t + c¡t2, will be deferred to

later papers. (Integrals of the first kind with quadratic factors are treated in [3].) The

main table in Section 2 consists of quartic cases, since cubic cases can be obtained

from these by putting at = 1 and b¡ = 0 for various choices of i. To select integrals

that are relatively simple and occur most commonly in practice, we arbitrarily

require T.\p¡\ < 8. Apart from permutation of subscripts in (1.1), there are just nine

quartic cases of the first or second kind satisfying this criterion: [-1,-1,-1,-1],

[1,-1,-1,-3], [-1,-1,-1,-3,2], [-1,-1,-3,-3], [1,-1,-3,-3], [1,1,-3,-3],[-1,-1,-1,-5], [1,-1,-1,-5], and [1,1,-1,-5]. The integral [-1,-1,-1,-3] is a

special case of [-1, -1, -1, -3,2] with ai = 1 and b5 = 0.

Section 3 presents four cubic cases not contained in the nine quartic cases:

[3, -1, -3], [3, -1, -1], [-3, -3, -3], and [1,1,1].

The method of evaluating the integrals is discussed in Sections 4 and 5. The

fundamental integrals are [-1,-1,-1,-1] and [1,-1,-1,-3], and the rest are ob-

tained from these by recurrence relations. The single formula (2.7) for [1, -1, -1, -3]

replaces 72 cases occupying the nine pages of §3.168 in Gradshteyn and Ryzhik's


A TABLE OF ELLIPTIC INTEGRALS OF THE SECOND KIND 597

table [7], as well as 72 cubic cases: 18 cases of [-1,-1,-3] in §3.133, 18 cases of

[1, -1, -1] in §3.141, and 36 cases of [1, -1, -3] in §3.142.

By using [5, (4.1), (4.2)], (2.6) was checked against formulas 1,3,5,7 of [7, §3.147],

and (2.7) was checked against formulas 1,5,42,70 of [7, §3.168]. The nine integrals

in Section 2 and the four in Section 3 were checked numerically to 6S for y = 0.5,

x = 2.0, a, = 0.5 + /, b¡ = 2.5 - i by the SLATEC numerical quadrature routine

QNG and the routines for RF and RD in the Supplements section of this issue.

2. Table of Quartic Cases. We assume x > y and a¡ + b¡t > 0, y < t < x, for ail

i, and we define

(2.1) dIJ^aibJ-aJblt

(2.2) Xi=(a, + b,x)l/2, Yl = (al + biyf/2,

(2-3) UtJ = {XtXjYkYm + YtYjXkXm)/{x-y),

where i, j, k, m is any permutation of 1,2,3,4. These definitions imply

(2-4) U2 - U2m = d,jdkm,

and consequently the arguments of the Ä-functions appearing in the table differ by

quantities independent of x and y. If one limit of integration is infinite, (2.3)

simplifies to

Uu = (btbj)l/2YkYm + YtYj{bkbm)l/2, x=+œ,

Uu = XiXJ(bkbm)l/2+(bibj)1/2XkXm, y = -œ,

all square roots being nonnegative.

If one limit of integration is a branch point of the integrand, then X¡ or Y¡ will be

0 for some value of i (with pi > -1 since we assume that the integral exists), and one

of the two terms in every U¡j will vanish. If both limits of integration are branch

points, the elliptic integral is called complete, and one of the U¡¡ will be 0. It is not

assumed that bi ¥= 0 nor that d¡¡ + 0 unless one of these quantities occurs in a

denominator. The relation d¡¡ = 0 is equivalent to proportionality of a¡ + b¡t and

a + bjt. The nine quartic cases listed in Section 1 follow. Only the first two are

treated by Gradshteyn and Ryzhik [7, §3.147, §3.168].

(2-6)

(2.7)

¡\(ax + bxt)(a2 + b2t){a3 + b3t)(a4 + b4t)]~l/2 dtv

= 2RF{U22,U23,U24).

f (d + bxt)l/2[(a2 + b2t)(a, + b3t)Yl/2(a4 + b4t\V1 dty

= ldl2duRD{U22,Ux\,Ux24) +X4Y4UX4


598 B. C. CARLSON

The next equation remains valid even if a5 + b5t changes sign in the interval of

integration.

f [(ax + bxt)(a2 + b2t)(a3 + b3t)Yl/2(a4 + b4t)-y\a5 + b5t) dt

(2.8)2dX2dud54_RÁuaMaM^

3d14

IS ( 1 ~) 1 \ 54 11

+ ^7R^2'U»'U"> + diáXAYAU

I [(ax + bxt)(a2 + b2t)Yl/2[(a, + b3t)(a4 + b4t)\

14yl4-'4<-'14

3/2 dt

(2.9)3^4

{b23dX4d24 + b2dX3d23)RD{Ux\,U24,U22)

A-RF(U22,U23,U24) + ~

& dUUi34l-'12

>b2X4Y4 | b2X3Y3\

X3Y3 X4Y4 J

f (ax + bxt)l/2(a2 + b2t)-1/2[(a3 + b3t)(a4 + b4t)\ 3/2 dt

(2.10)

2-^(b3d24 + b4d23)RD(U23,U24,U22)id34

■ —(Mi4 + b4dx3)RF(U22,U23,U24)"34

2b3dX3X4Y4 | 2b4dX4X3Y3

d34UX2X3Y3 d34UX2X4Y4

f\(ax + bxt)(a2 + b2t)]1/2[(a3 + b3t)(a4 + b4t)\3/2 dt

(2.11) ^-RD(UX\,UX\,U24) + %*RF{Uà,Uà,Uà)3d34 "34

2 ¡d24XxYx dl3X2Y2

d34Ux41 X4Y4 X3Y3

f [(ax + bxt)(a2 + b2t)(a3 + b3t)Yl/2(a4 + b4t)' 5/2 dt

(2.12)

-4M 6, b2 b3\j , u 3XXYX^\i;4 + i;4 + i;4)\d^RÁ^2'u^u^ + -

+ 2¡2b¡ + _bxb^ + Jhb^__b2b3_\

3 1 d\4 dX4d24 dx4d34 d24d34

X4Y4UX4

RF{U22,U23,U24)

2 b2

3dX4d24d34(XxX2X3X4i- YxY2Y3Y4i).



Í (ax + bxt)l/2[{a2 + b2t)(a3 + b3t)YW\a4 + b4t)'5/2dt

.2| bx 2b2 2b3\l Â D ,tj2 jj2 ii2, , 3XXYX-r2\\dX2dX3RD{Ux22,Ux23,U24) +

(2.13)

d24 ^-«-»"«v"«'-«'""/ X4Y4UX4

■RF(U22,U23.U24)2b4dx2dx3

3dX4d24d34

? h

— — ( XxX2X3X4 - - YXY2Y3Y4 -,

C [(ax + bxt)(a2 + b2t)Y/2(a3 + b3tyl/2(a4 + b4tYyidtJy

-2 Í 3X Y-(d13d14 + d23dX4)[dX2dX3RD(Ux\,Ux23,Ux24) •

(2.14)

9dX4d34 v u i4 ~«-w^-"-»"«v-«'"«'-i«/ j^

-)"14"34

(A-^A^AV3- y,y2nn~3).

4

3^34

3. Cubic Cases. By putting a¡ = 1 and bi = 0 for various choices of /, 13 cubic

cases can be evaluated from the quartic cases in Section 2 and do not need to be

listed separately. Eight of these are given by Gradshteyn and Ryzhik [7,

§§3.131-3.135, 3.141, 3.142]: [-1,-1,-1], [-1,-1,-1,2], [-1,-1,-3], [-1.-1,-5],[-1.-3,-3], [1.-1,-1], [1,1.-1], and [1,-1.-3]. They do not give the other five:

[1.1, -3], [1.1, -5], [1, -1. -5], [1, -3, -3], and [-1. -1. -3,2].

In this section we list four cubic cases not contained in the quartic cases of Section

2: [3,-1,-3], [3,-1,-1], [-3,-3,-3], and [1,1.1]. Only [-3,-3.-3] is given by

Gradshteyn and Ryzhik [7, §3.136], and only two cases of this are listed, each with

an infinite limit of integration, because the integral diverges if it begins or ends at a

finite branch point with p¡ = -3. If the closed interval of integration lies in the open

interval between two finite branch points with pt = -3, there is no way to evaluate

the integral by using previous tables.

In place of (2.3) we define

(3.1) U, = (X,Y/Yk+ Y,XjXk)/(x -y),

where /. j, k is any permutation of 1,2, 3. Since this implies

(3.2) U2-U2 = bkd,r

the arguments of the /^-functions in the table differ by quantities independent of x

and y. If one limit of integration is infinite, (3.1) simplifies to

(3.3) U, = (6A)1/2y,. if x = +00, U, = {bjbk)l/2X¡ if y = -x.


600 B. C. CARLSON

the square roots being nonnegative. The remarks in the paragraph preceding (2.6)

apply, after replacement of U¡¡ by U¡, also to the following integrals.

f (fll + bxt)y\a2 + b2ty1/2(a3 + b3t\3/2 dt

(3-4)ii / 1

= b~a\3\ 3d^(bid23 + b2dx3)RD{U22,U2,U2)

-äuRF(Ui2,u2,u2) + d-^} + 2-^.

f (ax + bxtf/2\(a2 + b2t)(a3 + b3t)YV2dt

(3.5) (b2dx3 + b3dx2)(^dX2dX3RD(U22,U32,U2) + ^3b2b3 y ., ^

iu-RF{U2,Ui,U2)+^j-(XxX2X3-YxY2Y3).3b2b3 3b2b3

f [(ax+ bxt)(a2 + b2t)(a3 + b3t)YV2dty

Ab,b-, I b,b-, b-,b^ b^b,\ , -, -, -,\

(3.6) +

3d

2bxb2

»12(^ + ^\Rf(U2,U2,U2) +

2b\

d\-hd23X3Y3U3

2 [ b\X2Y2 b\XxYx

d2x2V3\d^XxYx d23X2Y2

^^{h^^^'^'^^^Ywhere £ denotes summation over cyclic permutations of the subscripts 1,2,3. The

same notation is used in the next formula.

f \(ax + bxt)(a2 + b2t)(a3 + b3t)Y/2dtJy

-2(Lb2d¡3) (i , , , ,, X3Y3]kdX3d23RD{U2,U2,U2) + -^

I5bxb2bj I 3

(3-7) -^7T(M23 + Ml3)A,(í/l2^22,í/32)I5bxb2b¡

*i , X¡

+ Î5X^XAt + t + t "Ï5™Y¿ Y¿

2 b3

Y2ÍL + i2_ + ±3

i bx b-, b

- li> -d!'15 I bxb2

\dX3d23RD{U2,U2,U32) + ^

+ b-x\X¡X2X3-Y?Y2Y3)í



4. The Two Fundamental Integrals. In this section we shall prove (2.6) and (2.7) for

[-1,-1,-1,-1] and [1,-1,-1,-3], from which the remaining integrals can be ob-

tained by the recurrence relations of Section 5. In order that the first part of the

proof shall apply for future purposes to [1, -1, -1, -1, -2], which is an integral of the

third kind, we do not restrict the number n of factors in (1.1) to be 4. It will be

important that these three integrals have px> -2 and Hp¡= -4.

In (1.1) we assume x > y and a, + b¡t > 0, y < t < x, for all i. In the notation of

Section 2 this implies X2 > 0 and Y2 > 0 for all i. Temporarily we assume further

that -ax/bx > x and that ai + b¡t > 0, y ^ ; < -ax/bx, for i > 1. This assumption,

which will later be removed by analytic continuation, means that -ax/bx is the first

singularity encountered to the right of the interval of integration. The first part of

the assumption implies (ax + bxx)/bx < 0, whence X2 > 0, bx < 0, and Y2 > 0,

since Y2 = A'2 - bx(x - y). The second part of the assumption implies ai +

b¡(-ai/bi) > 0, whence dx, > 0, i > 1.

We can now split (1.1) into two parts, both well defined if px > -2:

-ai/bln(ai + bityi/2dt

-p/h,U(a, + b,tr2dt -/,-/,.Jx ;=1

It suffices to consider / because Ix is the same with y replaced by x. The interval of

integration is mapped onto the positive real line by a change of integration variable:

_ t - y _ y + aiY2u

"~ Y2(ax + bxt)' ~ 1 - bjfu'

(4'2) dt Yt Y2dx,u + Y2~T =-ï ' ai + bJ =-:—,

. du {l-bxY2u)2 l-bxY2u

where dxx = 0. If L />, = -4 the powers of 1 - bxYxu cancel, and we find

iv=Yrp>r u(Y2dXiu + Y2yy2du

-nW^i Yl(u+Y2/Y2dX/2du.y-2 J0 , = 2

The integral Ix is the same with Y2/YxdXi replaced by Xf/X2du, and the

difference,

X2/X2dXl - Y2/Y2dXl = (x-y)/X2Y2,

is positive and independent of i. Using the notation

(4.4) X = (x-y)/X2Y2, z, = Y2/Y2dx„ z, + X - X2/X2du,

we find from (4.1), (4.3), (1.2), and (1.3) that

[-1,-1,-1,-1] = 2(dududi4y1/2(4-5)

■[RF(z2,z3,z4) - RF(z2 + X,z3 + A,z4 + A)J,

[1,-1,-1,-3] = hduduy1/2(dj-3/2

(4.6) J■[RD(z2,z3,z4)~ RD(z2 + X,z3 + X,z4 + A)].


602 B C. CARLSON

The addition theorem [4. (9). (13)] for RF is

RF(z2, z3, z4) = RF{z2 + A, z3 + A, z4 + A)

(4.7) +ÄF(z2-r-u,z3 +u,z4 +u).

z,. + u = A-2{[(z,. + A)z,z,]1/2 + [z,.(z, + A)(z, + A)]1/:

where ?'. /. k is any permutation of 2. 3,4. Thus (4.5) becomes

(4.8) [-l.-l.-l.-l] = 2(dX2dl3dl4yl/2RF(z2 + u, z3 + u, z4 + u).

(4.9) z, + n =a-^,.7/, + y^A,.^) i/,2

Vi/i^-^)1 ^12^3^14'

By the homogeneity property (1.4) we find

(4.10) [-1.-1.-1.-1] = 2RF(UX\.U23,UX24).

which is the same as (2.6).

This removal of the d 's from the arguments of R F is the critical step. As shown

by (1.2). an argument of RF must not be negative, and so the functions on the

right-hand side of (4.5) require the branch points to be ordered so that dX2. dx3. and

dx4 are positive. To show that (4.10) holds without the assumption that -ax/bx is the

first singularity to the right of the interval of integration, we use analytic continua-

tion in bx or more conveniently in w, where

Yfw = X2 = ax + bxx. bx =

(4.11)y

xY{ - yw -ax w(x - v)-'— • ~T~ = x +-i-:—

x - y bx y2 - w

We fix x. y. Y¡ > 0, 1 ^ / < n. and X, > 0. 2 *s / sS n. Then ax and bx are

functions of w\ and we can make -ax/bx be the first singularity to the right of the

interval of integration by choosing w positive and sufficiently small. For such values

of n- we have proved that (4.10) is true. We shall show that both sides of (4.10) are

analytic in w on the complex plane cut along the nonpositive real axis. It follows by

the permanence of functional relations that (4.10) holds in the cut plane and in

particular for all positive values of w. Therefore it holds for any real value of -ax/bx

outside the closed interval of integration. The last statement is immediately evident

from the graph of ax + bxt as a function of t. since ax + bxy has been fixed and

w = ax + bxx.

To prove analyticity, we recall that an ^-function is analytic when each of its

arguments lies in the plane cut along the nonpositive real axis [2, (6.8-6). Theorem

(6.8-1)]. Since (2.3) shows that U¡. = a,- w1/2 + j8, , where a(/ and ßu are positive.

U2 lies in the cut plane when w does, and so the right-hand side of (4.10) is analytic

in the cut u-plane. The left side is defined by (1.1), which can be rewritten, when

"L p, = -4. as

(4.12) [p] = (x-y)\T\YiP'\R -1

1

' -Pi -Pi, *{

2 . 2 ' y2



by taking s = (x - t)/(t - y) as a new variable of integration and using [2,

(6.8-6)]. Since Y2 is positive and X2 = w, the right side of (4.12) and the left side of

(4.10) are analytic in the cut w-plane, and the proof of (2.6) is complete.

A different proof of (2.6) was given in [3], but the present proof is adaptable to

(2.7) with only minor changes. The addition theorem for RD, obtained by putting

p = z in [11, (8.11)], is

RD(z2,z3,z4) = RD(z2 + X,z3 + X,z4 + X)

(4.13) +RD(z2 + n,z3 + ¡i,z4 + u)

+ 3[z4(z4 + A)(z4 + u)]-1/2,

where ¡i is the same as in (4.7). Thus (4.6) becomes

[1,-1,-1,-3] = \(dX2dX3yl/2(dx4y'/2

(4.14) ■{RD(z2 + ii,z3 + li,z4 + ii)

+ 3[z4(z4 + A)(z4 + M)]-I/2}.

Substituting (4.4) and (4.9) and using the homogeneity property (1.4), we find (2.7).

The temporary assumption about -ax/bx can again be removed by the permanence

of functional relations. In the first term on the right-hand side of (2.7), dX2 and dX3

are linear functions of w = A'2 by (2.1) and (4.11), and RD is analytic in the cut

w-plane by the same reasoning that applied earlier to RF. The second term also is

analytic because Xx/Ux4 = wl/2/(aX4wl/2 + ßX4), where aX4 and ßX4 are positive.

Since the left side of (2.7) is a special case of (4.12), the proof is complete.

5. Recurrence Relations. Let e¡ denote an n-tuple with 1 in the ith place and 0's

elsewhere (for example, [p + 2ex] = [px + 2,p2,..., /?„]). We shall first list some

relations between different integrals, then give their proofs, and finally show how

they can be used to obtain all the integrals in the table from the two fundamental

integrals (2.6) and (2.7). The most useful relation is

(5.1) dIJ[p} = b,[p + 2el]-bl[p + 2eJ}.

Two others, involving the quantity

(5.2) A(p)=Ylxr-flYp',i=\ ,=\are

(5.3) tp,b,[p-2e,] = 2A(p)i~i

and

(5.4) (Px+ ■■■+p„ + 2)b,[p]= tpJdJi[p-2eJ\+2A{p + 2ei).7 = 1

The latter, which can be used to raise the value of £/?,-, contains n integrals since

di: = 0.


604 B. C. CARLSON

Recurrence relations for a single />, depend on the value of n. For n = 3 and

i, j, k any permutation of 1,2,3, we have

(Pi+p2+p3 + 4)bjbk[p + 2ei]

(5.5) +{(/>, + Pj + 2)bjdki+(Pi + Pk + 2)bkd;,}[p]

+P,dJ,dk,[p -2e,] = 2b,A(p + 2eJ + 2ek).

The analogous relation for n = 4 and /, j, k,m a permutation of 1,2,3,4 is

(Px + p2 + p3 + p4 + 6)bjbkbm[p + 4e,]

+ E U + Pj + Pk + 4)bjbkdmi[p + 2e,]

+ L(Pi+Pj + 2)bjdkidmi[p] +pidJ,dk,dm,[p - 2e,]

= 2bfA(p + 2ej + 2ek + 2em),

where £ denotes summation over cyclic permutations of j, k, m. This relation is

especially useful if £ p, = -6, because the first term vanishes.

Equation (5.1) follows at once from the definition of [p] and the identity

(5-7) diJ = bJ(ai + bit)-bi(aJ + bjt).

To prove (5.3) we integrate both sides of

(5.8) 2^ n (-,+ b,ty/2 = E P,b,{a, + b,tyl U (-,■ + ty)• >i , = i y-i

with respect to t over the interval [y, x].

If p is replaced by p + 2e,, (5.3) becomes

(5.9) (/,,. + 2)¿,.[^] + ¿ PA^ + 2e,. - 2e\ =2A(p + 2e,),7-17'*'

and if /7 is replaced by p — 2e,, (5.1) becomes

(5.10) ô,.[/7 + 2e, - 2ey] = ¿,.[p] - d¡\p - 2e;].

Substitution of (5.10) in (5.9) yields (5.4). To prove (5.5) we use (5.7) twice to

write b2(aj + bjt)(ak + bkt) as a quadratic polynomial in a, + b,t, multiply by

ri(ar + brt)Pr/2, and integrate to get

b2[p + 2ej + 2ek\ = bjbk[p + 4e,] +{bjdk, + bkdß)[p + 2e,](5 11)1 j +dßdki[p].

Next we replace p by p + 2e¡ + 2ek in (5.3) with « = 3 and find

/7,A[/> - 2e, + 2ey + 2ek] +(Pj + 2)bJ[p + 2ek](5 12)

+ (P* + 2)bk[p + 2ej] = 2>l(/> + 2ej + 2ek).

In the first term we substitute (5.11) with p replaced by p - 2e¡; in the second and

third terms we use (5.1) with or without replacement of j by k. The result is (5.5),

and (5.6) has a similar proof starting from 6,3(a, + bft)(ak + bkt)(am + bmt) as a

cubic polynomial in a, + b¡t.

The following special cases of (5.1) show how to obtain (2.8), (2.9), (2.10), and

(2.11) from (2.6) and (2.7):

(5.13) d14[-l,-1,-1,-3] = 64[1,-1,-1,-3] -bi[-l,-1,-1,-1],



(5.14) 64[-l, -1,-1, -3,2] = ¿541-1, -1,-1, -3] + b5[-l, -1,-1,-1],

(5.15) dM[-l,-1,-3, -3] = ft4[-l,-1, -1,-3] - b3[-\, -1, -3, -1],

(5.16) 63[l,-l,-3,-3] = d13[-l,-l,-3,-3] + bx[-l,-l,-l,-3],

(5.17) ¿>3[l,l,-3,-3] = £.23[l,-l,-3,-3] + b2[l,-l,-l,-3].

We have omitted ps = 0 in the two integrals on the right-hand side of (5.14). In

(5.15), [-1, -1, -3, -1] is found by interchanging the subscripts 3 and 4 in formula

(2.8) specialized to [-1, -1, -1, -3]. Letting [p] = [-1, -1, -1, -3] and i = 4 in (5.6),

we get [-1, -1, -1, -5] from [-1, -1, -1, -3] and [-1, -1, -1, -1], since the first term

of (5.6) is 0. Equations (2.12) and (2.13) then follow from two more special cases of

(5.1):

(5.18) b4[l, -1,-1, -5] = ¿14[-1,-1, -1,-5] + ^[-1,-1,-1, -3],

(5.19) 64[l,l,-l,-5] = ¿24[l,-l,-l,-5] + 62[l,-l,-l,-3].

The formulas resulting from this procedure can sometimes be simplified with the

help of various identities:

(5.20) ¿.A,2 - bjX2 = b,Y2 - b¡Y2 = dy„

(5.21) X2Y2 - Y2X2 = (x - y)dJt,

(5-22) £ a,d]k = £ b,djk = E d,md]k = 0,

(5-23) E^* = Lr,X* = 0,

(5.24) ZdtjUijXJ.-O,

where £ denotes summation over cyclic permutations of i, j, k. These identities are

obtained from definitions (2.1) to (2.3). Equation (5.22) is used to prove (5.23) and

(5.23) to prove (5.24). Since RD, unlike RF, is symmetric in only its first two

arguments, another useful relation is

dijd^RoiUilUÛ2,) = d^d^ÛÛÛ2)

(5.25) , - , , v 3C/,,+ 3RF{UX22,UX23,U24)-

UijUu'

where i, j, k is any permutation of 2,3,4. This can be proved by using [5, (4.14)] to

express both sides in terms of the symmetric functions RG and RF and simplifying

with the help of (2.4).

The four cubic cases in Section 3 can be obtained from (5.1) and (5.4) as follows:

(5.26) b3[3, -1,-3] = ¿13[1, -1, -3] +bx[l, -1,-1],

(5.27) f>2[3,-l,-l] = d12[l,-1,-1] -r-èj[1,1,-1],

(5.28) dx2Y3, -3, -3] = b2[-l, -3, -3] - feJ-3, -1,-3],

(5.29) 5*i[l,l,l] =</21[l,-l,l] +d31[l,l,-l] + 2/1(3,1,1).

Aside from permutation of indices, each integral on the right-hand side of these

equations is among the 13 cubic cases listed in the first paragraph of Section 3.

Equation (5.24) is replaced by two identities,

(5.30) Y,duUkXkYk = 0,

(5.31) bfíXjY.-bjUjXfi-djM,


606 B. C. CARLSON

and (5.25) is replaced by

bjdkiRD(u2,U2,U2) = b^R^U2,!/2,!!2)

(532) I 2 2 2^ 3«+ 3*f(t/12,t/22,[/32)

In these three equations i,j,k is any permutation of 1,2,3, and £ denotes

summation over cyclic permutations of /', j, k. Equation (5.23) is used to prove

(5.30), and (5.20) to prove (5.31). Equation (5.32) is proved in the same way as (5.25)

except that (3.2) is used in place of (2.4).

Department of Mathematics

Iowa State University

Ames, Iowa 50011

1. P. F. Byrd & M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed..

Springer-Verlag, New York, 1971.

2. B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.

3. B. C. Carlson, "Elliptic integrals of the first kind," SIAMJ. Math. Anal., v. 8,1977, pp. 231-242.

4. B. C. Carlson, "Short proofs of three theorems on elliptic integrals," SIAM J. Math. Anal., v. 9,

1978, pp. 524-528.5. B. C. Carlson, "Computing elliptic integrals by duplication," Numer. Math., v. 33,1979, pp. 1-16.

6. B. C. Carlson & Elaine M. Notis, "ALGORITHM 577, Algorithms for incomplete elliptic

integrals," ACM Trans. Math. Software, v. 7, 1981, pp. 398-403.

7. I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New

York, 1980.

8. D. H. Lehmer, "The lemniscate constant," Math. Comp., v. 3,1948/49, pp. 550-551.

9. A. P. Prudnikov, Yu. A. Brychkov & O. I. Marichev, Integrals and Series, vol. 1, Gordon and

Breach, New York, 1986.

10. John Todd, "The lemniscate constants," Comm. ACM, v. 18,1975, pp. 14-19, 462.

11. D. G. Zill & B. C. Carlson, "Symmetric elliptic integrals of the third kind," Math. Comp., v. 24.

1970, pp. 199-214.


mathematics of computationvolume 49, number 180october 1987, pages s13-s17

Supplement to

A Table of Elliptic Integralsof the Second Kind

By B. C. Carlson

This supplement contains Fortran routines for the standard func-

tions Rp(x,y,z) and RD(x,y,z), followed by two examples of their use

in computing elliptic integrals.

CDOUBLE PRECISION FUNCTION RF( X,Y,Z,ERRTOL,IERR)

CC THIS FUNCTION SUBROUTINE COMPUTES THE INCOMPLETE ELLIPTICC INTEGRAL OF THE FIRST KIND,C RF(X,Y,Z) - INTEGRAL FROM ZERO TO INFINITY OFC

C -1/2 -1/2 -1/2C (1/2MT+X) (T+Y) (T+Z) DT,CC WHERE X, Y, AND Z ARE NONNEGATIVE AND AT MOST ONE OF THEMC IS ZERO. IF ONE OF THEM IS ZERO, THE INTEGRAL IS COMPLETE.C THE DUPLICATION THEOREM IS ITERATED UNTIL THE VARIABLES AREC NEARLY EQUAL, AND THE FUNCTION IS THEN EXPANDED IN TAYLORC SERIES TO FIFTH ORDER.C REFERENCES: B. C. CARLSON AND E. M. NOTIS, ALGORITHMS FORC INCOMPLETE ELLIPTIC INTEGRALS, ACM TRANSACTIONS ON MATHEMA-C TICAL SOFTWARE, 7 (1981), 398-403; B. C. CARLSON, COMPUTINGC ELLIPTIC INTEGRALS BY DUPLICATION, NUMER. MATH., 33 (1979),C 1-16.C AUTHORS: B. C. CARLSON AND ELAINE M. NOTIS, AMES LABORATORY-

C DOE, IOWA STATE UNIVERSITY, AMES, IA 50011, AND R. L. PEXTON,C LAWRENCE LIVERMORE NATIONAL LABORATORY, LIVERMORE, CA 94550.C AUG. 1, 1979, REVISED JAN. 15, 1987.CC CHECK VALUE: RF(0,1,2) - 1.31102 87771 46059 90523 24198C CHECK BY ADDITION THEOREM: RF(X,X+Z,X+W) + RF ( Y,Y+Z,Y+W )C - RF(0,Z,W), WHERE X,Y,Z,W ARE POSITIVE AND X*Y - Z*W.C

INTEGER IERR,PRINTR

DOUBLE PRECISION Cl , C2 , C3,E2,E3,EPSLON,ERRTOL,LAMDADOUBLE PRECISION LOLIM,MU,S,UPLIM,X,XN,XNDEV,XNROOTDOUBLE PRECISION Y,YN,YNDEV,YNROOT,Z,ZN,ZNDEV,ZNROOT

C INTRINSIC FUNCTIONS USED: DABS,DMAX1,DMIN1,DSQRTCC PRINTR IS THE UNIT NUMBER OF THE PRINTER.

DATA PRINTR/6/

©1987 American Mathematical Society

0025-5718/87 $1.00 + $.25 per page

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A Table of Elliptic Integrals of the Second Kind* Table of Elliptic Integrals of the Second Kind*...

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