A Table of Elliptic Integrals of the Third Kind...MATHEMATICS OF COMPUTATION VOLUME 51, NUMBER 183 JULY 1988, PAGES 267-280 A Table of Elliptic Integrals of the Third Kind By B

MATHEMATICS OF COMPUTATIONVOLUME 51, NUMBER 183JULY 1988, PAGES 267-280

A Table of Elliptic Integrals of the Third Kind*

By B. C. Carlson

Abstract. As many as 72 elliptic integrals of the third kind in previous tables are unified

by evaluation in terms of ñ-functions instead of Legendre's integrals. The present table

includes only integrals having integrands with real singular points. In addition to 31

integrals of the third kind, most of them unavailable elsewhere, 10 integrals of the first

and second kinds from an earlier table are listed again in new notation. In contrast

to conventional tables, the interval of integration is not required to begin or end at a

singular point of the integrand. Fortran codes for the standard ñ-functions Re and Rj,

revised to include their Cauchy principal values, are listed in a Supplement.

1. Introduction. A table [4] of elliptic integrals of the first and second kinds

is extended in this paper to integrals of the third kind. We choose a standard form,

/x nJ(at + bltY'/2dt,- i=i

where Pi,... ,pn are integers and the integrand is real. The integral is assumed to

be well defined; in particular, if p, is odd, a¿ + bit must be positive in the open

interval of integration. Many integrals like

1(1 + nsin2 <A)_1(1 - fc2 sin2 <P)~1/2 d<t>

and

/"(l+n^rMu-^Hi-fcV)]-1/2^

can be put in the form (1.1) by letting t = sin2 tp or t = z2.

If we assume the 6's are nonzero and no two of the quantities a¿ + bit are pro-

portional, then [pi,... ,p„] is an elliptic integral if the number of odd p's is exactly

three (the "cubic case") or four (the "quartic case"). It is elliptic of third kind if

at least one of the p's is even and negative or if pi +•■•■+ pn = -2,0,2,4,..., the

latter condition being impossible in the cubic case. Otherwise, it is first or second

kind, and the only integrals of first kind are [—1, —1, —1] and [—1, —1, —1, —1].

The conditions that the integral be of third kind are deduced by using [2,(8.1-2)]

to express [p] as an i?-function with "6-parameters" —pi/2,..., —p„/2, 2 + £)p,/2.

The integral is of third kind if at least one of these parameters is a positive integer

(see [2, §8.5, §9.2]).

Received October 14, 1987.

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.

©1988 American Mathematical Society0025-5718/88 $1.00 + $.25 per page

267

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

268 B. C. CARLSON

We shall replace Legendre's integrals of the first and third kinds by

1 f°°(1.2) RF(x,y,z) = -J \(t + x)(t + y)(t + z)\-ll2/dt,

(1.3) Rj(x,y,z,p) = -I [(í + x)(í + y)(í + 2)]-1/2(í + p)-1fií.

Besides being symmetric in x, y, z, these i?-functions are homogeneous:

RF(Xx, Xy, Xz) = X~^2RF(x, y, z),(1.4)

Rj(Xx, Xy, Xz, Xp) = X~2/2Rj(x, y, z, p),

and they are normalized so that

(1.5) RF(x,x,x) = x~1//2, Rj(x,x, x,x) = x-3/2.

Two special cases are denoted by

(1.6) Rc(x,y)=RF(x,y,y), RD(x,y,z) = Rj(x,y,z,z).

The function Rd replaces Legendre's integral of the second kind, and Re is an

elementary function** embracing the logarithmic, inverse circular, and inverse hy-

perbolic functions [3, (4.9)-(4.13)]. Use of Re allows unification of formulas for

circular and hyperbolic cases of integrals of the third kind.

Fortran codes [5] for Re and Rj are listed in the Supplements section of this

issue, and codes for RF and Rd are given in a Supplement to [4]. All four can be

found also in most of the major software libraries, but the codes for Re and Rj

in the Supplement have recently been modified to compute the Cauchy principal

value of the integral when the last variable is negative. As shown in Sections 5 and

6, Cauchy principal values are sometimes needed when using the formulas of this

paper.

As explained in [4], the present table differs from customary integral tables

[1], [7]***, [8] in two respects: We do not require the interval of integration to

begin or end at a branch point of the integrand, and we do not separate special

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

integration and to one another. The unification of these special cases is made

possible by application of the addition theorem and by the use of Ä-functions

instead of Legendre's integrals. The present table includes only integrands with

real branch points; conjugate complex branch points resulting from an irreducible

polynomial al + bit +• ctt2 are deferred to a later paper.

The table in Section 2 consists of quartic cases, since many cubic cases are

included in these by taking on — I and bi — 0 for various values of i. To select

integrals that are relatively simple, we arbitrarily require J2\Pi\ ^8 and J2pi <

0. Apart from permutation of subscripts in (1.1), there are just 37 quartic cases

satisfying these criteria. We omit nine with p5 = ±2, p^ = ±2 since they can

** Although not in connection with elliptic integrals, the use of a function equivalent to the

reciprocal of Re was proposed in 1897 by Fubini, while still a student at Pisa, in his first published

paper [6], I am obliged to Professor Luigi Gatteschi for this reference.

***Tables 3 and 4 in [7] use Legendre's IT as defined by [1, (110.04)] and are inconsistent with

[7, 8.111(4)], which differs in the sign of the parameter. The same is true of the 1965 edition.


A TABLE OF ELLIPTIC INTEGRALS OF THE THIRD KIND 269

easily be reduced to the others by (2.51)-(2.53), but we add three with Y^Pi >

0: [1,1,1,-1], [1,1,1,1], and [1,1,1,1,-2]. Thus we list 31 quartic cases, most

of which are not available in previous tables. The formulas for [1,-1,-1,-1],

[1,1,-1,-3], [1,1,-1,-1], and [1,1,1,-1] respectively unify 32, 72, 48, and 32

special cases in Byrd and Friedman's table [1].

All cubic cases of third kind with £) |p¿| < 7 are contained in the quartic cases of

Section 2 with three exceptions: [1,1,1, —2], [3,1, -1, —2], and [3,1,1, —2]. Cubic

cases of all three kinds will be discussed in a later paper.

Because some of the formulas are cumbersome when written out in the style of

[4], we have introduced abbreviations to save space. For uniformity of style, the

quartic cases of first and second kinds listed in [4] are listed again in Section 3 in

the new notation. Derivation of formulas by recurrence relations is discussed in

Section 4. The fundamental formula for [1, —1, —1, —1, —2] is proved in Section 5,

and Cauchy principal values are discussed in Section 6. All integral formulas have

been checked by numerical integration; some details of the checks are given in the

Supplement.

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

i = 1,..., 4. Assumptions about a^ + b^t will be stated where necessary. We define

(2.1) dij = axbj - ajbi, r^ = 7-^- = -1 - -¡±;bibj bi bj

(2.2) X% = (at + hxf'2, Yi = (at + ky)1'2;

(2.3) Uij = (XiXjYkYm + YiYjXkXm)l(x - y),

where i,j, k, m is any permutation of 1,2,3,4;

(2.4) W2 = U?2-di3di4d25/di5;

(2.5) Q2 = (X5Y5W/XiYi)2, P2 = Q2+d25d35d45/di5;

(2.6) A(pi,...,Pn)=XPi1---Xn">-Y?1---Y¿«.

These definitions imply, if P is chosen positive,

(2-7) C/2 -U2k = d%md]k,

(2.8) P = (X~lX2X3X4Y2 + Y1-1Y2Y3Y4X2)/(x - y),

and hence, with the help of [4, (5.22)],

, . f/13 = Ui2 — di4d23, U i4 — U ï2 — di3d24,

W2 = uu - dijdikdl5/di5,

where i,j, k is any permutation of 2,3,4. When 05 = 1 and b5 = 0, the quantities

W,P,Q become Wi,Pi,Qi:

,21fJ) W2 = Uf2-b2di3di4/bi,

Q\ = (Wi/XiYi)2, P2 = Q\ + b2b3b4/h.

If one limit of integration is infinite, (2.3) simplifies to

Í2 111 Uij = (bibj)1/2YkYm + YiYj(bkbm)1/2, x = +oo,

l' ' UlJ=XlXJ(bkbm)1/2 + (blbjy/2XkXm, 2/=-00,


270 B. C. CARLSON

all square roots being nonnegative, while

(i)^ Q2 = (bM(Y5W/Yi)2, x = +oo,

1 ' Q2 = (bM(X5W/Xi)2, y = -oo.

All quartic cases will be expressed in terms of the quantities

(2.13) h=2RF(Ui2,U(3,U(i),

(2.14) h = 2-di2di3RD(U22,U23, U24) +™r¿™íó*û\î2,î3,î4) ' v y rj '

(2.15) I3=2d^dï3dliRj(U2i2,U23,U24,W2) + 2Rc(P2,Q2),

o"15

(2.16) /3 = -2dl2dl3dl4 Rj(U22, U23, U24,W2) + 2Re(P2,Q2).

áOi

It will be seen from the tables that

,91~ h = [-1,-1,-1,-1], h = [1,-1, -1,-3],

[ ' h = [1,-1, -1,-1, -2], I3 = [1,-1,-1,-1].

Thus, I3 reduces to I3 if 05 = 1 and 65 = 0, and I3 reduces to I2 if 05 = a4 and

65 =64.

If one limit of integration is a branch point of the integrand, then Xi or Yz is 0

for some value of i < 4, and one of the two terms on the right-hand side of (2.3)

vanishes. If X1F1 =0 then P,Q,Pi, and Qi are infinite, and the i?c-functions

in (2.15) and (2.16) vanish by (1.6) and (1.4). If both limits of integration are

branch points, the elliptic integral is called complete, and Ui2Ui3Ui4 = 0. It is not

assumed that 6¿ ^ 0 nor that d{j ^ 0, unless one of these quantities occurs in a

denominator. The relation dt] = 0 is equivalent to proportionality of o, + 6¿í and

üj + bjt.We are now ready to list 31 cases of

(2.18) [px,... ,p5] = / (ai + M)Pl/2 ■ • ■ (a5 + b5t)**'3 dt,Jy

15 with p5 = 0,2, or 4 and 16 with p5 = -2 or -4. The first 10 have p5 = 0 and

involve I'3 but not I3.

(2.19) [1,-1, -1,-1] =4

(2.20) [1,1, -1, -3] = (b2I'3 + d24I2)/b4.

(2.21) [3, -1, -1, -3] = (M3 + di4I2)/b4.

(2.22)[1,1 - 1, -1] = [b2(n3 + r24)I'3 + d24r34I2 - di2r13h}/2

+ A(l,l,l,-l)/63.

[3,-1,-1,-1] = (6i/2)

(2.23)

yruI3 + b4r24r34I2

1=2

-&iri2ri3/i+2A(l,l,l,-l)/¿>2¿>3



(2.24)

(2.25)

(2.26)

[1,1,1,-3] = (6263/264) y rl4I3 + 3b4r24r34I2 - biri2r13Ii

L i=i

(2.27)

+ A(l,l,l,-l)/64.

[3,1, -1, -3] = (6i6a/264)[(ri3 + 2r14 + r24)/3

+ 64r24(2r14 + r34)I2 - 6iri2ri3/i]

+ (6i/6364)A(l, 1,1,-1).

3

[1,1,1, -5] = (6263/62)/3 + (d24d34ßb\) yr-4lI2

i=l

- (di2di3/3b4di4)Ii - (2/364)A(l, 1,1, -3).

[1,1,1, -1] = (6263/8)(r24 - r22 - r23 + 2r24r34)/3

+ (¿^4/8j [b3d24r34I2 + 2A(1,1,1,-1)]

+ (63di2ri3/8)(r12 - r34)h + A(l, 1,1, l)/264.

(2.28)

[1,1,1,1] = (626364/16)(r12 - r34)(r24 - r223)I'3

-l(ri2-r34)2+2(r24+r23)]

■ [d24d34I2 + 264A(1,1,1 - l)]/48

+ (6364d12r13/48)(4?-24r34 - 3r24 + 3^3)/!

- í y ruin \ A(l, 1,1,1) + A(3,1,1, 1)/36l

The next five integrals, with p5 = 2 or 4, involve I3 but not I3. No restriction is

placed on 05 or 65.

[-1,-1,-1,-1,2] = (65/3 -dX5h)/bi.(2.29)

(2.30)

(2.31)

(2.32)

[1, -1, -1, -1,2] = (65/2)[(r12 - r35 - r45)/3 + 64^34/2

-61r12r13/1 + (2/6263)A(l, 1,1,-1)].

[1,-1, -1,-3,2] = (65/3 -d45I2)/b4.

[1,1, -1, -3,2] = (6265/264)[(r13 + r24 - 2r45)/3

+ 64r24(r34 - 2r45)/2 - biri2ri3h\

+ (65/6364)A(l, 1,1,-1).

[-1,-1,-1,-1,4]

(2.33)(62/26i - y rl5I'3 + b4r24r34I2 + 6i(2r25 - ri2ri3)/i

+ (2/6263)A(l, 1,1,-1)

L 1=1


272 B. C. CARLSON

The next 11 integrals have p5 = -2, and we assume a§ + 65Í either is positive on

the closed interval of integration or changes sign in the open interval of integration.

In the latter case the integral is interpreted as a Cauchy principal value; see Section

6. All 11 integrals involve I3, and the six with J^P¿ > _2 involve J3/65 also.

(2.34) [1,-1,-1,-1,-2] = 73.

(2.35) [-1, -1, -1, -1, -2] = (65/3 - bih)/di5.

(2.36) [1,1, -1, -1, -2] = (d25/3 + 62/3)/65.

(2.37) [1, -1, -1, -3, -2] = (65/3 - b4I2)/d45.

(2.38) [1,1, -1, -3, -2] = (d25/3 - d24I2)/d45.

(2.39)

(2.40)

(2.44)

[-1, -1, -1, -3, -2] = (b25/di5d45)I3 - (b24/di4d4b)I2

+ (b\/di4dib)Ii.

[1,1,1, -3, -2] = (d25d35/b5d45)I3 + (6263/6465)/3

- (¿24^34/64^45)^2-

(2-41) [3, -1, -1, -1, -2] = (d15/3 + 61 J3)/65.

[3,1, -1, -1, -2] = (d15d25/62)/3 + (6x62/265)

(2-42) ■ [(r13 + 2r,5 + r24)I3 + b4r24r34I2 - 6^12^3/1]

+ (6i/6365)A(l, 1,1,-1).

[1,1,1, -1, -2] = (d25d35/b25)I3 + (6263/265)

(2-43) • [(n5 + r35 + r24)I3 + b4r24r34I2 - &iri2ri3/i]

+ A(l,l,l,-l)/65.

[1,1,1,1, -2] = (d25d35d45/b¡)I3 + (626364/865)^ - 2r)J3

+ (<7/865)[d24d34/2 + 264A(1,1,1, -1)]

+ (64d12di3/86i65)(ri2 - r35 - 3r45)Ii

+ A(l,l,l,l)/265,

where

4

(2.45) o- = yrt5, r = yr2

425-

t=l i=l

The final five integrals have p5 = —4. All five involve ^3, and the two with

Z)p¿ > —2 involve I'3/b\ also. We assume 05 +65Í is positive on the closed interval

of integration.

Í1 1 -1 -1 -41 = - f— + — - — ^ I +• ^24d34 j(2 dg) ^ V''53 ^54 ^51/ 2d35d45

2dj5d35 d35



>.-w,-i,-<i4(lri:è) 6sd24d34 ,

(2 47) 2d25d35d45

65di2di3 65-Ji - A(l, 1,1, -1, -2).

2di5d2öd35 d2$d35

[1, 1, 1, -1, -4] = (6263/265)[65(»"25 + <-25r35Al5 - <-24r35/<-45)/3

(2-48) + 2/3 + (64r24r34/r45)/2 - (61r12r13/ri5)/i]

-A(l,l,l,-l,-2)/65.

[-1,-1, -1,-1, -4] = ^-±±h + -^^ r.2di5 f-J r¿5 " 2^15^25^35^45

1=1

+£('-SS)'-A(l, 1,1,-1,-2).

diöd25d35

4

[1,1,1,1,-4] = (d25d35d45/264)£r-51/3

i=l

4

(2-5°) + (626364/262) 53 rt5/3 + (d24d34/b\)I2i=i

- (64di2di3/265di5)(r15 + r45)Ji

- A(l, 1,1,1, -2)/65 + (264/62)A(l, 1,1, -1).

The nine integrals with p5 = ±2, pe = ±2, J2\Pi\ ^ 8, and X)p¿ < 0 can be

reduced to those listed above by using the identities

(2.51) (a5 + b5t)-l(a6 + b6t)-1 = [b5(a5 + b^)'1 - 66(a6 + b6t)-l}/d65,

(2.52) (a5 + b^-âe + b6t) = [66 + d65(o5 + hty^/b*,,

(2.53) (a5 + 65i)(a6 + 66i) = [d65(a5 + b5t) + 66(a5 + 65i)2]/65.

3. Quartic Cases of First or Second Kind. In [4] the nine quartic integrals

of first or second kind with ^|p¿| < 8 were evaluated in terms of RF and Rd-

Since Rd is symmetric in only its first two variables, the third was chosen vari-

ously as U22,U23, or U24 to simplify the formulas. In (2.14) and hence throughout

Section 2, U24 is the choice. For uniformity of notation and for use in deriving the

formulas of Section 2, we list again the nine integrals in present notation and add

for convenience [-1,-1,-1, -3], which is a special case of [-1, -1, -1, —3,2].

(3.1) [-1,-1,-1,-1] = /!.

(3.2) [1,-1,-1,-3] = /2.

(3.3) [-1,-1, -1, -3] = (64/2 - bih)/di4.

(3.4) [-1, -1, -1, -3,2] = (d54I2 + d15/i)/d14.

(3.5) [1,1, -3, -3] = [2d24/2 + di2/i + 2A(1,1, -1, -l)]/d34.


274 B. C. CARLSON

(3.6)[1, -1, -3, -3] = [(63d24 + 64d23)/2 + 63di2/i

+ 263A(l,l,-l,-l)]/d23d34.

[-1, -1, -3, -3] = (l/d13)(r231 - itf + 2r3il)I2

(3-7) +(l/d34)(r2-31-rr41)/i

+ (262/d13d23d34)A(l, 1,-1,-1).

(3.8)[1,1, -1, -5] = [(di3d24 + di4d23)I2 + di2di3/i

+ 2d14A(l,l,l,-3)]/3d14d43.

[1, -1, -1, -5] = (l/364)(r1-41 - 2r2¡ - 2r341)/2

(3.9) - (64di2di3/3di4d24d34)/i

-(264/3d24d34)A(l,l,l,-3).

(3.10)

[-1,-1,-1,-5] = (2/3d41) y r~AlI2

i=2

+ (6i/di4)2(l - ri2ri3/3r24?"34)/'i

-(26|/3d14d24d34)A(l,l,l,-3).

4. Reduction by Recurrence Relations. The integrals in Sections 2 and 3

are expressed in terms of Ii,I2,I3, and I3 by using recurrence relations, most of

which were proved in [4]. Let e¿ denote an n-tuple with 1 in the ith place and 0's

elsewhere (for example, [p + 2ei] = [pi + 2,p2,... ,p„]). We denote [4, (5.4)] by

(A*):

(Ai) (Pi + • • • + Pn + 2)6,-[p] = y Pjdjtip - 2e3\ + 2A(p + 2e¿).

jî

For concise reference we write [4, (5.1)] in two forms:

(Bij) dij\p] = b3[p + 2e%\ - bi\p + 2ej],

(Cij) bj\p\ = bi\p - 2e% + 2e3\ + dtJ\p - 2e¿].

Equation (Cij) is (Bij) with p replaced by p — 2e¿. Finally, [4, (5.22)] implies

^2(ai + bit) djk = 0 and hence

(4.1) y\p + 2el}djk=0,

where Yl denotes summation over cyclic permutations of i,j,k. Replacing p by

p — 2ek, we see that

(Dijk) dij\p] = dk][p + 2et - 2ek] + dik[p + 2e., - 2ek\.

We discuss first the reduction of the integrals in Section 3, because they will be

used to reduce those in Section 2 and because the procedure differs at some points

from that used in [4]. Equations (3.1) and (3.2) were proved in [4, §4]. Equation

(3.3) obviously comes from (B14) and (3.4) from (D145). For (3.5) we first use

(D342) to get

(4.2) d34[l, 1, -3, -3] = d24I2 + d32[l, -1, -3, -1].



To reduce the last term, we put [p] = [1,-1,-1,-1] in (Al) and in [4, (5.3)] to

find, respectively,

0 = d12[l, -3, -1, -1] + d13[l, -1, -3, -1] + d14/24.3)

+ 2A(3,-1,-1,-1),

6,/j - 62[1, -3, -1, -1] - 63[1, -1, -3, -1] - 64/24.4

= 2A(1,-1,-1,-1).

Elimination of [1, —3, —1, —1] yields

(4.5) d32[l, -1, -3, -1] = d24I2 + dnIi + 2A(1,1, -1, -1),

which we have simplified by using (4.8) and [4, (5.22)].

Equation (3.6) follows from (B23) and (3.7) from (B13). Equation (3.8) comes

from putting [p] = [1,1, -1, -3] in (Al); (3.9) then follows from (B24) and (3.10)

from (B14).

Equation (2.19) is a special case of (2.34). Equations (2.20) and (2.21) come

from (C24) and (C14), respectively. To get (2.22) we use (A3) and evaluate

[-1,1,-1,-1] by (C21). Equations (2.23), (2.24), (2.25), and (2.26) come from

(C12), (C34), (C14), and (C34), respectively. For (2.27) we use (A4) and evalu-

ate [-1,1,1, -1] by (C31) and [1, -1,1, -1] by (C32). For (2.28) we use (Al) and

evaluate [1, -1,1,1] by (C42) and [1,1, -1,1] by (C43). Equations (2.29) to (2.33)

follow in order from (C51), (C52), (C54), (C54), and (C51).

We shall prove (2.34) in Section 5. Equations (2.35) to (2.44) follow in order

from (B15), (C25), (B15), (B45), (D452), (C35), (C15), (C15), (C35), and (C45).

To get (2.46) we put [p] = [1,1, -1, -1, -2] in (A3) and evaluate [-1,1, -1, -1, -2]

by (C21). Equations (2.47) to (2.50) then follow in order from (B25), (C35), (B15),

and (C45).

The formulas resulting from this procedure have often been simplified with the

help of various identities. Besides [4, (5.22)] we note

(4-6) rlk + rkj =ri:j = -r3i,

(4-7) r2k + r% - r2 = 2rikrjk,

(4.8) dtJA(p) = bjA(p + 2ex) - b%A(p + 2e.,),

(4.9) yA(p + 2ei)djk=0,

where ^ denotes summation over cyclic permutations oîi,j,k. Equation (4.6) is

obvious from (2.1), and (4.7) comes from squaring (4.6). Equation (4.8) follows

from (2.6) and [4, (5.7)], and the proof of (4.9) is like that of (4.1).

5. The Fundamental Integral. Equations (2.13) and (2.14), with left sides

identified in (2.17), were proved in [4, Section 4], initially with a restriction on the

o's and 6's that was later removed by using the analyticity of the R -function when

each of its variables lies in the complex plane cut along the nonpositive real axis.

The proof of (2.15) is similar except for a complication: The variables W2 and Q2

are sometimes real and negative.


276 B. C. CARLSON

In this section we assume x > y and at + bit > 0, y < t < x, for 1 < i < 5. From

[4, (4.1), (4.3)] and (1.3) we find the companion of [4, (4.5), (4.6)]:

. I3 = [1, -1, -1, -1, -2] = (2/3d15)(d12d13d14)-1/2(5.1)

• [Rj(z2, ..., 25) - Rj(z2 + X,...,z5 + X)],

where, for i = 2,..., 5,

(5.2) zl = Y2/Y2du, X = (x-y)/X¡Y2, zt + X = X2/X2du.

As in [4], this result depends on the temporary assumption that —ai/bi is the first

singularity of the integrand to the right of the interval of integration, which implies

du > 0, i > 1, and thus guarantees that all variables of Rj are positive. We now

apply the addition theorem [9, (8.11)],

Rj(z2,...,z5) = Rj(z2 + X,...,z5 + X) + Rj(z2 + p,...,z5 + p)(5.3)

+ 3Rc(l + è,l),

(5.4) * +p = X'2{\(zt + X)z3zk)1'2 + \zx(z3 + X)(zk + A)]1/2}2,

(5.5) 7 = 25(25 + X)(z5 + p), ¿ = (22 -25X23 - zb)(z4 - z5).

In (5.4), t,j, k is any permutation of 2, 3, 4. Equation (5.1) becomes

(5.6) I3 = (2/3d15)(d12d13d14)-1/2[i?j(22 + p,..., 25 + p) + 3Rc(l + 6,7)].

From (5.4) and (5.2) we find, for i = 2,3,4,

zx + p = U2i/di2di3di4,

(5.7) 25 + p = (21 + p) + (25 -2¿),

25 - 2t = (di,Y52 - diö^/disd!^2 = -di5/di5dii,

where [4, (5.23)] is used in the last step. It follows that

(5.8) 25 + p = W7di2di3di4,

(5.9) l = Q2/di2di3di4d215, l + 6 = P2/di2di3di4d2i5,

where W2, P2, and Q2 are defined by (2.4) and (2.5). Homogeneity of Rj and Re

[see (1.4) and (1.6)] now yields (2.15) from (5.6).

Since z2,z3,z4,X are positive by (5.2), expansion of the right-hand side of (5.4)

shows that p > 0. Since 25 > 0 by (5.2), it follows by (5.8) that W2 > 0 and by

(2.5) that Q2 > 0. Equation (2.8) shows that P2 > 0. Thus the ñ-functions in

(5.6) and (2.15) are well defined when the temporary assumption about -ai/61 is

satisfied. When it is not, the functions in (2.15) remain well defined if W2 and Q2,

which have the same signs by (2.5), are positive.

To examine the sign of W2, we let w — X2 — ai + bxx and allow w to vary while

fixing x and y and Xi > 0, 2 < i < 5, and Y¿ > 0, 1 < i < 5. If we define

(5.10) wt = XfY2/Y2,

then w2,..., w5 are fixed positive quantities, w = wi is a positive variable, and du

is a linear function of w because

(5.11) (x - y)du = X2Y2 - Y2X2 = Y2(wx - w).



We shall want also, for i = 2,3,4,

(5.12) (x - y)dx5 = X¡Y2 - Y2X¡ = (YXY5/Yi)2(w5 - wx).

Since Q2 and W2 are positive by (2.5) if d25d35d45 = 0, we may assume that w5 is

not equal to w2, w3, or w4.

The graph of ai + bit has the fixed positive ordinate Y2 at t = y, the variable

positive ordinate w at t = x, and the intercept —ai/bi on the i-axis. As w runs

through all positive real values, this intercept takes all values outside the fixed

interval [y, x] of integration. When w — w, for some i > 1, the intercept coincides

with the fixed intercept -a¿/6¿ of the graph of a¿ + bit. The temporary assumption

about —ai/bi means that 0 < w < wt, 2 < i < 5.

We find from (2.4) and (2.3) that

(513) W2 = [Y2Y3Y4/Yi(x-y)}2f(w),

f(w) = \(ww2)1/2 + (w3w4)1/2\2 - (viz - w)(w4 - w)(w5 - w2)/(w5 - w).

This is positive when w equals w2, wz, or w4, but it changes sign as w goes through

the singularity at w5. Writing ç for the positive square root w1^2 = X\, we find

(w - w5)f(w) = w5ç4 + 2(w2w3w4)1/2c3

(5.14) + [w2w3 + w2w4 + w3w4 - w*,(w2 + w3 + w4)]ç2

- 2wb(w2w3w4)ll2<; - w2w3w4.

By Descartes' rule of signs the polynomial in c has exactly one positive zero, say

ç0 = Wq . Thus f(w) changes sign exactly twice, at wq and W5, as w goes through

all positive values. Since / is positive at 0 and 00, it must be negative in the

open interval with endpoints Wq and w5, either of which may be the greater. This

interval cannot contain w2, w3, or w4 since / is positive at those points.

We shall now establish (2.15) for all w > 0 by analytic continuation. We cut

the w-plane along the nonpositive real axis and make a second cut between w0

and W5 on the positive real axis. The corresponding region of the ç-plane is the1/2 1/2

open right half-plane with a cut between w0' and w5' on the positive real axis.

The coefficient of Rj in (2.15) and the variables of both Rj and Re are analytic

functions of w in the cut w-plane by (2.3), (2.4), (2.5), (5.11), and (5.12). We recall

that an 72-function is analytic when each of its variables lies in the complex plane

cut along the nonpositive real axis [2, (6.8-6), Theorem 6.8-1]. Therefore, if we can

show that none of the variables can be real and nonpositive when w is in the cut

ui-plane, it will follow that Rj, Re, and the right-hand side of (2.15) are analytic.

Since Ui2, Uiz, and <7i4 are linear functions of ç with positive coefficients by

(2.3), they have positive real parts when c does, and their squares cannot be real

and nonpositive. A similar remark, with ç replaced by 1/ç, applies to P2 by (2.8).

It remains to consider W2 and Q2.

Now W2 is real and nonpositive if and only if f(w) + r = 0 for some r > 0. The

quantity

(5.15) (w - w5)[f(w) +r] = (w - w5)f(w) + rç2 - rw5

is a quartic polynomial in ç whose coefficients, like those of (5.14), have the signs

(5.16) +, +, ±, -, -.


278 B. C. CARLSON

By Descartes' rule of signs there is exactly one positive zero, which must lie on the

cut between w0' and w5' where f(w) takes every nonpositive real value. Either

the other three zeros are real and negative or else one is negative and the other two

are conjugate complex. In the latter case the polynomial must have the form

, , w5(ç - 0(? + V){i - « - *7?)(í - a + iß)

= Mí2 + ÍV- 0Í - «2 - 2ac + a2 + ß2},

where £ and n are positive and a and ß are real. The coefficients of c3 and c have

the signs required by (5.16) if and only if

(5.18) r)-Ç-2a>0, (a2 + ß2)(n - Ç) + lafr < 0,

which imply 2q(q2 + ß2 + tin) < 0 and hence a < 0. Thus the conjugate complex

roots have negative real part. We conclude that a quartic polynomial whose coef-

ficients have the signs (5.16) has one real positive zero and three zeros in the open

left half-plane. Thus no zero of (5.15) lies in the cut right-half f-plane, and W2

cannot be real and nonpositive at any point in the cut w-plane.

By (2.5) and (5.13), Q2 is a positive multiple of f(w)/w. Since w ^ 0 in the cut

plane, Q2 is real and nonpositive if and only if f(w) + rw = 0 for some r > 0. The

quantity

(5.19) (w - w5)[f(w) + rw] = (w - w5)f(w) + rç4 - w5rc2

is a quartic polynomial in ç whose coefficients, like those of (5.14), have the signs

(5.16). Thus Q2, like W2, cannot be real and nonpositive at any point of the cut

plane. This completes the proof that the right-hand side of (2.15) is analytic in w

on the cut wi-plane.

The left side of (2.15) is defined by (1.3), which can be rewritten in the form of

[4, (4.12)]:

I3 = [1, -1, -1, -1, -2] = (x - y)(Yi/Y2Y3Y4Y2)

(5.20) /-I l 1 1 w_ XI X¡\

1 V 2 '2'2'2' 'Y^Yf'Yg)'This is analytic in the w-plane cut along the nonpositive real axis, even without the

second cut from u>o to w§. Since (2.15) is known to hold if 0 < w < Wi, 2 < i < 5, it

holds throughout the twice-cut tu-plane by the permanence of functional relations.

Each of the two terms on the right-hand side of (2.15) is discontinuous across

the second cut, but the discontinuities must cancel (as can be verified explicitly)

because the left side is continuous. Hence the second cut is unnecessary and can

be removed after defining Rj and Re on the cut so as to make the right-hand side

continuous. A suitable definition for each is its Cauchy principal value, which is

the arithmetic average of its values on the upper and lower edges of the cut. With

this definition, (2.15) holds on the w-plane cut only along the nonpositive real axis.

In particular it holds for all positive values of w, which was to be proved.

At each endpoint of the second cut the Cauchy principal value is to be interpreted

as a limit of values off the cut or of principal values on the cut. The limit is

easy to evaluate at the end where w = w5, since this implies by (5.11) that dis

vanishes, a5 + 65£ is proportional to ai + bit, and [1,-1,-1,-1,-2] reduces to

(a,/a5)[-l, -1,-1,-1].



6. Cauchy Principal Values. According to Section 5, the evaluation of I3

may involve Cauchy principal values of both Rj and Re, even though a5 +• 65 í is

positive on the closed interval of integration. We shall prove now that formulas

(2.34) to (2.44) are still valid if 05 +65Í changes sign in the open interval of integra-

tion. If the left side of each formula is taken to be a Cauchy principal value, then

Iz as defined by (2.15) involves the Cauchy principal value of either Rj or Re but

not both, since W2 and Q2 have opposite signs by (2.5) when X2Y2 < 0.

It suffices to consider (2.34), since the proofs of (2.35) to (2.44) by recurrence

relations (Bij), (Cij), and (Dijk) do not depend on the sign of a$ + 65Í. We shall

assume a5 +65x < 0 < (25+651/, but the proof with x and y interchanged is similar.

The method is much like that in Section 5.

We fix x and y and Xi > 0, 1 < i < 4 and Fj > 0, 1 < i < 5, while allowing

X2 = 05 +65X to vary in the complex plane. By (5.10), u>i,..., w4 are fixed positive

quantities while w5 = X2Y2/Y2 is variable. Since w is an abbreviation for wi, we

see from (5.11) that the coefficient of Rj in (2.15) is analytic except for a pole at

u>i. The first three arguments of Rj are fixed and positive, but it follows from

(5.13) that the fourth argument has the form

where A and B are real and independent of W5. A little algebra shows that A > 0

if B = 0, and hence W2 cannot be real and nonpositive unless v>$ is real. We cut

the ?¿>5-plane along the nonpositive real axis and also along every interval of the

positive real axis where W2 < 0. Unless B = 0, one of these intervals will have the

pole at Wi as one endpoint, and so the cut plane will in all cases include an open

interval of the positive real axis, where X2 > 0 and (2.15) is valid by Section 5. In

the cut plane, W2 is analytic in w§ and cannot be real and nonpositive. Hence, the

first term on the right-hand side of (2.15) is analytic in w§ on the cut plane.

We rewrite the second term, using the homogeneity of Re, as

(6.2) 2w;1/2Rc((Pw;1/2)2,Q2/w5).

Equation (2.8) shows that

(6.3) Pws1/2 = Mw~1/2 + Nwl12,

where M and A'' are positive and independent of w§. If w$ is in the cut plane,

both terms on the right-hand side of (6.3) lie in the open right half-plane, and

hence (Pw^1' )2 cannot be real and nonpositive. Neither can Q2/w5, since it is a

positive multiple of W2 by (2.5). Thus, the second term on the right-hand side of

(2.15) also is analytic in W5 on the cut plane.

Finally, the left side of (2.15) is analytic in w$ on the cut plane because the first

four arguments of R-i in (5.20) are fixed and positive while the last argument is a

positive multiple of W5. Since the cut plane contains an open interval of the positive

real axis and (2.15) holds on that interval, it holds everywhere in the cut plane by

the permanence of functional relations. The equality clearly persists if each term

in the equation is replaced by the arithmetic average of its values on the upper and

lower edges of the cut along the negative real axis. On the left side this average is

the Cauchy principal value of I3. On the right-hand side, either Rj or Re takes


280 B. C. CARLSON

its Cauchy principal value, but since Q2 and W2 have opposite signs when wb is

negative, one of the two functions will be continuous across the cut.

A numerical example in which .13 and Re take their principal values is given at

the end of the Supplement.

Ames Laboratory and 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, "Computing elliptic integrals by duplication," Numer. Math., v. 33, 1979,

pp. 1-16.4. B. C. CARLSON, "A table of elliptic integrals of the second kind," Math. Comp., v. 49, 1987,

pp. 595-606. (Supplement, ibid., S13-S17.)5. B. C. CARLSON & ELAINE M. NOTIS, "ALGORITHM 577, Algorithms for incomplete

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

6. G. FUBINI, "Nuovo método per lo studio e per il calcólo delle funzioni trascendenti elemen-

tan," Period. Mat, v. 12, 1897, pp. 169-178.7. I. S. GRADSHTEYN & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press,

New York, 1980.

8. A. P. PRUDNIKOV, YU. A. BRYCHKOV & O. I. MARICHEV, Integrals and Series, Vol. 1,

Gordon and Breach, New York, 1986.

9. D. G. ZILL & B. C. CARLSON, "Symmetric elliptic integrals of the third kind," Math. Comp.,

v. 24, 1970, pp. 199-214.


MATHEMATICS OF COMPUTATIONVOLUME 51, NUMBER 183JULY 1988, PAGES S1-S5

Supplement to

A Table of Elliptic Integrals of the Third Kind

By B. C. Carlson

This supplement contains Fortran codes for the functions

Rc(x,y) and Rj{x,y,z,p). If y or p is negative, the Cauchy princi-

pal value is computed. The codes are followed by some numerical

values that were used to check the formulas in Sections 2 and 3.

cDOUBLE PRECISION FUNCTION RC(X,Y,ERRTOL,IERR)

CC THIS FUNCTION SUBROUTINE COMPUTES THE ELEMENTARY INTEGRALC RC(X,Y) - INTEGRAL FROM ZERO TO INFINITY OFC

C -1/2 -1C (1/2MT+X) (T+Y) DT,cC WHERE X IS NONNEGATIVE AND Y IS NONZERO. IF Y IS NEGATIVE,C THE CAUCHY PRINCIPAL VALUE IS COMPUTED BY USING A PRELIMI-

C NARY TRANSFORMATION TO MAKE Y POSITIVE; SEE EQUATION (2.12)C OF THE SECOND REFERENCE BELOW. WHEN Y IS POSITIVE, THEC DUPLICATION THEOREM IS ITERATED UNTIL THE VARIABLES AREC NEARLY EQUAL, AND THE FUNCTION IS THEN EXPANDED IN TAYLOR

C SERIES TO FIFTH ORDER. LOGARITHMIC, INVERSE CIRCULAR, ANDC INVERSE HYPERBOLIC FUNCTIONS ARE EXPRESSED IN TERMS OF RCC BY EQUATIONS (4.9)-(4 . 13 ) OF THE SECOND REFERENCE BELOW.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 SEPT. 1, 1987.CC CHECK VALUES: RC(0,l/4) - RC(l/16,l/8) - PI,

C RC(9/4,2) • LN(2),C RC(l/4,-2) = LN(2)/3.C CHECK BY ADDITION THEOREM: RC(X,X+Z) + RC(Y,Y+Z) - RC(0,Z ) ,C WHERE X, Y, AND Z ARE POSITIVE AND X * Y ■ Z * Z.

c

SI

©1988 American Mathematical Society

0025-5718/88 $1.00 + $.25 per page


S2 SUPPLEMENT

4 2U 4Z ED E-«»r> CO

CO

u u> J r- -a- •-{M HHHCOin

H K I I 1 I I4 O Q Q Q O QiJ OS O O O O oId PS.(S W N (S (S OÍ (S

U E.. j Ktú CU UQ £ Eho oUUJ

4AJEO 4 PSOS £ OOS OS 2

2 42OS OS OSD O OE- t. tx.

rH PS PSOS OS

CO u u

co tou DO CQZ O< os

2 IdPS Eu4 X

J \Q OS— — o orH fN - .PS * o

G £ LO -iN* • Cuo J ¡x DÛ 04o D • E-< ̂vo * OS O cwEO • --MM » —■

• J —<N O O ÎHi j a —■ —o co

m — ■ mJ • O 4

• Eh ■ Q>h J E-i +

■ J «IX' —

OS T OS -ïïEh -Eh -2 o 2m E m EPS .-h PS r

\ I PS H O O

Cu —— En — E- H 2U < U < KEh E Eh E PS Dih OS m OS OS Eh fa-PS O PS O U U m

O Eh 11 4 |lO 2 Z U PS Z Z

Il -X^SwEhX^SDu

UOUUUUEJCJUUUUUUCJUCJUUU

O JH CuOS DQS -u H- ta

Dj (H »H O- W W t-i

«HHÜWPS U CJ < 2Ulllu] JH

m PC PS Du PSG. CL, Eh

PS J 2bl fa] U 4 i-io .j j u6J CQ CQ mEh D D O2 O O Om Q Q J

PS OSEh Eh

Z 2

Eh U in inm EE H >h >H

« -mmJ E 3

UOÛQB¡h JUUU E Du m D0* S OS J ma. to eu >D OS > m w

O O H QU Lu JE OC D EH >> O E D

EE Q S E mm Z O D XJ 4 ^ £ 4CU Lu i-i ED X PS Z

Q b- Q E Z2 O 2 h.4 DUE

CO 2 UEh [i] Eh m <m D D X EE J O Un < X < LdJ > Eh E E

os u s uW J E 25 CQ Q Eh <OhU EJ to U 2 Eh

CO CJ 4W m O E «E E PS Eh WEh Q CU Eh

4 "4CO JIOUb] u. j u asz o m j e>

EU Eh EH

PS U Z O OU Z O Z ZEh < M

M PS Eh in toa 4|hh

U EHE E D E EHE.D.HH•J E .-J vJO (fa O O CM-] O CJ J D

.h r- *rm <n o r-j r~ r-r-- m m +mm

E + + + Q + +»h û a Q o Q QJ O O O sr o oeu.DHHHirmm

m a uco u zH U MU 2 EU < UOS E 4a, o £

2 ÛHJUCO D to

O DO E 4Eh CO U

r œ cdîn r—CO Cn O

r- (N m iEl 1 I Q a QM Q Q Q O O OjooominiiiO.JWHHISHH

COLUi ta

co u os uU M U MD OS CO PSiJ w u4 co o to

u r- r- o PCJ ci \«H Mm \o .H co< o oEh *0 O CJ rH UCu m \o 4 H hU > ¡HU E U w 4 X Eo m q 2 os 4 m4 M CJ D CJ > M

w

PS OS f*i •U U + 2> CO u oZ O MO o • toU O .H 1-1

M uiah n uM PS

U £ Cu£ 4 H4 > J u05 n Cu EO 2 D Eho aos a iLU LU Z H

e 4 mt0 Eh Sh r-E PS m UH O I O

U U Z[fa o <m CO ■ E

U .-h U.. dO J H <Z < £>-< > £ UZ M PSOS U i-] Eh< E O XS Eh .J U

Q PS toU U UH CO UDCU PS OS£ O OO J OSu >■< ce

< uU Eh

M ï»

Où -I2 Q - Jm 4 OS 4S O DO 2 OS to

OS du

to

XEh < >

OS Eh <D 4UUU3O Z U< D U

PS DU Eh ÛE O

U J 22 < mM OSE O OSPS U UU Eh QEh 2 «

U EO Œ EhHhu.

2 [faU E O toE D Q UEh J Z CJ

O D OSU U O DO OS O

Jt- JOO J CQO U M

ESEQ Eh O2 U OSO Z PS U

U iEh E

co < PSm £ U

M Eh

J XO O OSEh PS UPS Ou EhOS CU b-U 4 4

i u

U Eh C¿ KE OS O UH OS OS

U OS JZ u <i-i [fa Eh

O 2 OZ O Eh

O D Eh UŒ J 4 ECO 4 U Eh

> 2Eh DUZ E OS CJD O H mO 4 HE U U O4 E 4

PS Eh PSu o eue u oEh Eh Z

U MZ J Z4 CQ O EhE 4 m DEh Eh H CQ

4 J 4 <CJ O E OS2 Eh M fa]D PS CJ H05 OC U mEh U] O

UU « U OSE OS OEh <N O E

• vo J blU U ZJ* hOCQ * >h4 uEh J O O

O Û H fa] •E PS Du CO £Eh PS O 2 U

U fa] OS2 PS CU Oi-t * O X U

Eh U EZ VÛ U EhU m 4 fa]> U E 2m Z Eh OO 4 4 m

E Eh EhEh h i- 4 4Z DO (JD to i« MOWJUJ£ u o 4 eu*i Jri«S

OS D Ofa] to PS oE m DJ u UEh 4 E

PS O EhZ O 2 u4 PS m O fa-E CC CO OEh Ei] 4 CO

U Eh tOCO 2 OS i-c 2CO O U O Ou > LO v

J Eh q a Eh

UUU UUUUUUUUUUUUUUUUUUUU UUUUCJtJCJtJCJCJCJCJUCJOUCJCJ


SUPPLEMENT

X PSH Eh

S Z

J E PSu u

; m >

) u s 4: o >i CU J

- Eu U: w e e4 »tO Ehj :* Q- -2 Du1 X D O

) [fa Eh Eh: O D O; o o

CO E PSi to Ehi D m U: j s «

• U M

U E HPS Eh D

OPS co PSO U COu E D

J>ÛUH

: u u uI JUI: E O H> M OS1 10 eu z

co 4l m j E: E J E-

PS 2 mCO 4D E QCO Eh Ei]

COU PS D

: u Eh h; o 4 o) 2 Eh Z

j ce o, co :J £ h t-

u O »: E CJ £ cH Eh W ►

] U J) Efa E OlOHJ

: uEh Eh E

) 4 Mliauj] m PS eu1 O D

Qfa] Eh (faCO O OD 2

E + + + Û + +w Q û Q o Q ûJ O O O i-H o o

£ I I I Q O Qm û Û D O O OjooominmO.JNiniílMMM

CO i-Ld

CO

.-i utu cu EO D Eh

LO U PS UU M Ei] MD PS to PS■J U u4 co o to

o o O Mfa] r- r» o osJ m \.h UCQ \o .H to400Eh \0 O CJ HUCU Cl VD 4 HPiU > >iO £ O m 4 X £U E Q 2 PS 4 CQ4 m CJ D U > m

Z Eh

Om O tu-to W 2M O M

UZEu 4 cjPS E 4Cu O £

O H2 Ûm J fa]CO D to

O DO E 4Eh CO O

UÛ to EU uHhhE PS -Hu u + ;> to u c

CO M II [M f

O £ C

4 > J Ips m eu :O 2 D EO Dos û :CU U 2 E

E 4 ^CO Eh ¡im ciE E m IHO I l

tu b] ;

O J II .Z 4 :M > £ t

2 1-1 (OS U J E4 E o ;5 H J t

CO 1U i-tM CO O Q

Û PS CO 2 Ûfa] U U M 4Eh 10 J SD 0 2eu os ps j m£ O O -JOJEOCJ ?h PS U ■

4 U ZU Eh U EE U E DH 4 > Eh J

XZU2WCJ O E £ PS4 m DmE Eh 4 J tuD 4 OU U co CJ UUZW E4 D U Q Eh

OS D ZU Eh Q O 2E O CJ MEh OS U

• Eh CO J

U J Z oZ 4 m UJ E-.m OS E E£ O E Eh OSE U U UU H Q ZH Z E m Lu

fa] E S UO E H O DH Eh Eu E J

m CO 4Z O Cu >U E-» EhtO Du Z EO 2 O D Oœ o 04U M to E U

M £ 4tO 4 E Em £ U U O

M Eh E DuJ X EhO O E UEh E U Z JE Cu EH 4 mE Ou (fa X 4U 4 4 M H

- J 4E 4 OO D ZE LO DE D E

<N J I\4 <m E E

Ld[Q

[fa M fa]U ŒO CO EHQ UZ U UDEL)O D ZE O m

CO COfa]m e

EHj O uj m jm CQS E 4

O EHU EE fa- UU EE « HEh O

E 2E E mO UE Z

* fa] MQ

— UU Cd etO E OH O EE £E OU X SI M H

M COw «

CO O\Q

J U\D U Z

co Z4 OCU H

J DJ Em EH

Eh El] Z

4 E DO Eh O

Z £D U 4E UH m fa]

EH EU O EHŒ 4Eh E Z

E 4O EEH Z EH

O EH tOm D Ueh m j

U E EO X

« EH U

EtO ><10 J UU O 4J EH E

E DCO E CJ1-1 U O

4E OO Z DuE m OE COU 4 CO

U HZ E mO U ÜM fa] MEH Q Q

• M tO1 E E ùC: u os 4] E U £

OUDOE Z Eh •EH 4 n

tu EZOEO E Em J Q OH O Z Efa4 Eh 4ou zM — u 4J E ECU CO EhDUEÛ U O tO

Z tu COfal 4 UX E to JEh U fa]

J Z EuU O m EO EH Eh

D QCO E O ZZ O E 40 E mm E D O(H DO CO E

UOOOOOOOOOUOOOOOOOOOOOOO OOUUOOOOOOOOOCJOOOOOOO

<N — O(N D

* o\

ZmO LO —

H QU ZE Ld

LO U Jm H 4

Z E

X U ZH X m

H E(fa E

O -O N

fal E EZ U OO N D

4Eh tO CJ E

O -EO Û O

Z EU m tuZ SiM kl

EH Q 4DEEO m OE E UE Eh Eh

D 2[O U i-t

2 J ND 4 -[fa K >h

O -tO U Xm EH ~E Z -")

fa, fajU E

2 E OO Um Q EEH 2 EH4 OE O 2E U OO tO m ,.ffa EH *to id < :z x o c4 H m E

U E EX U OEH û U

4 E XEh EH :

.Ihh(

4 2 -■ £ m ~

U EH OiE D r-EH e 014 Em£ O —

U

) 2 CO E1 O J Eh; m e <

M 4 E-U O

-s* >U M> E Ehm 0 4

U E UE U m4 N

J UU 2 >E O mE m EH

Eh m4 4 to

D OO O E2 UM COCO U t-iD U

CO E

m -zu u

Û > Eu 1-1 3H EHD mE tO ■E O 3O E O

4 PS M :1 E U O «: 2 to 2 l• E

U E •E O £ :4 -A <

. CO CC

1 to œ 4¡dijiu: 4ro m

E D 2 C4 U O t> D to E

2 J ¡U 4 E yX E 4EH X O t

I 2 ■>h O oE EH inO X mEH LO -g*4 E en

u Q OE 2 E4 4 E

EH©OO -Z m >*

E■ 4 O ■

E m eh r-4 œ

u -E tr>2 co OMm U m4 £ 4 -J 4 U MU

- J •Û >h 4 Eh

2 H Z E4 m O U

U

M Z OM falZ X •D Eh m

û O E U Eh

■ Eh r- JE 4M -EJ U OJEUU 4 M

3 ZU E-- MEh Du

U O U

: E DO m U E

CO Lf)m mLO CO

CO i CO

koj:w E 4 ►EUE«3 m o :

> Eh £h E: m u ui 2 us to d u1 M Pu E

EJEO 4 mO O J

2 E £hO u 4 ûLO > 2 UJ M COE Z U M4 D E >O OU

U E E■ M E

O 4 U -EH > en

■ to m r^CQ JOl

to o O -E m Z M

Uvo E - E •-i Eh fa] 3 OI D O 4 D

m 4 Q J 4

— >1<CH >4 +E E OO * w E

in m,-H Ol —r- fM E

+\o -o- Xl£> O -

ai © 3r- ci +

NQ ->Z X U

ci 4 + E •E Eh —

- Ld - m

— 1-1*04

<^ —-m E £ O

CQ CO DE; -o II w-ta E *1 » mu■ 4 U X ci: — E EH- rs < . to: e — m- E 3 •) * - + — fa]; 3 n 3 D

— - + I M• E N >h E CO

: 1 - + —1 4 s* X Eh! ^, w^ * [fa

U N

E <N M"H

•• n MLO E Q

— E EEU E E+ X * — EH:* 3

E « Z3 - O+ — II E>h E tO- -4 « E

N 3 X+ - 4 U><r3 - Eh

- -3 l>* O X— — * E Hr> n EE E M Û D

Z O+ 11 n 4 fa*

cw zJ E X HtO U -OE 3 Z OU O X E

-E -Zci -X N

: J Q M. E Z E

4 oH EU JU O- EH

4 U

EH 4 UZm z Z Z ZE O O O O

-co CO CO 10[tí M M M M

E U UU U U1-1 E E

E E

- -N Z5 Cl -M: to H E- -O Û; in o -h tO E «M1 -Z Xï m >h 41 to -E- -> aC 4 u -3 £ Q tO: o z coC M >H 41 LO -Û- -z

o ;h ..e - aE >i u

- >• cofa, -DE >■

- - toX EH ZOOOE O M

-EHO Z CJE X Z

U U UO J JU CQ EHDDZOOm Q Q

LOU O 2W fa] ME E EE E H

ZU U M•J Jm mD DO OC G

U U UOOUOOUOUUOUOOOOUOOOOOOOOOOOOUOUOOU


S4 SUPPLEMENT

o xo« +z

* CJo u *D >O U * Dm Q E

• 2 > —m E U

Q \1 + 2

E —fN Cl Cl

u zC¿ PSJ DO HH LdU E

WWW E

N ——.,z z :

+ X >H Io

O Q 2 11û o ><o • d d :

■ m + e E :

I H H H: j e e; • oo: to toi 2 a o

x n *~> >

N U UQ Q

D 2 2(O Cb E X ¡n

< E£ UO 3

> > OLd Ld JQ Q to2 2 ENEU

COE H Hfal O O— O O

E Efa- 2 2m X ><

— EE- ZO NO

H ZO XoE +Z>h H

--O 4Z O fa-N * E J~ 2 4H H XE O — *OtOtO E * 4Q Z Du

X 2 JI E 4

O 4O Û 4 4 'E E U Du E2 4 J J lN J < 4 c

D -3*£ E4 uj 3 u o

o o © oC D D o

o o o © oQ m m m m

2 O <N rs (N <NPC m ■ . - ■Drsoooo

— 4 4 4 4■ o -a- a a a a

E £ £ £ £; • u 4 4 4 4: u 3 j j J j) 2 Oi • E + + + +

o o o >Û O Q Û Uo a © © û

U O UM

• u > *■ Û u> 2 Û t

E 2 EE U

+ 3

w w*o © o o >û Q D Q UO O O O Q

O CD CM CD û U uO O

> >U uQ QZ Z 'X E l

> >o1 * U Q> Q O• E Z •i DO E ci

Z ©E QDO

Eh Q

U zE Ld

E 2— UtO XE H

4 ZO 4Z ED HEH LO

tOU U> JtOl/lN»H HHHOMC

H E I I I I I4 O Q D D Û QJ E © o o o ofal E.E U î* ci t ro ^r

Cl fl es) (N M

Eh mU £

••JEU E falQ £ HO OU O J

4EJEO 4 EE £ OE E Zu o m

Z 4zE E ED O OH Efa [u

H E EE E

tO fal U

LO tOfal DO taZ o4 E

cp uZ COM ZZ ME E4 X3 u

— EN'

; >< ci; »oíi x a -u x a —

JE©«©• O m X m ;

— ■ -<n - ;

XOEmEmEm— J E — E — X

Z H u 4 u 4m J H E H £ X K D

T E H fcuU U MM E

H —— 14 >« Z

; Û"-Nr3>i>i2* N1 Z>h>---->i \: 4--XX22— -- 2* XXXXXX2NZ

N— — — — — — \iHffl E*-4 *-<•-< r-1 r4 r4 *

) >«ZXXZXZoi + *1 -m44m4mQ z> XEEEEEE022XE■ XXNE QÛÛÛQD -NX

H Q H H O D02222UE2X25HZ2 2X4

• X>>NEC0OX>iN><>HX4mEEH

E JE OU H

OOOOUOOOOUUOOUUOUUOU


SUPPLEMENT S5

M -P -P C

■rt > -H

O 4J +J 4J

fNt-H. CMM fNi-H (NH

fNrH ÍNM CNM INH

T3 *t-l -H

rvjM r-NjM

(MM (NH (NH OJ>-


Documents

A Table of Elliptic Integrals of the Third Kind*...MATHEMATICS OF COMPUTATION VOLUME 51, NUMBER 183 JULY 1988, PAGES 267-280 A Table of Elliptic Integrals of the Third Kind* By B

A Table of Elliptic Integrals of the Third Kind...MATHEMATICS OF COMPUTATION VOLUME 51, NUMBER 183 JULY 1988, PAGES 267-280 A Table of Elliptic Integrals of the Third Kind By B