mathematics of computation
volume 59,number 199
july
1992, pages
165-180
A TABLE OF ELLIPTIC INTEGRALS:
TWO QUADRATICFACTORS
B. C. CARLSON
Abstract.
Thirteen integrands that are rational except for the square root of
a quartic polynomial with two pairs of conjugate complex zeros are integrated
in terms of Ä-functions of real variables. In contrast with previous tables,
the formulas hold for all real intervals of integration for which the integrals
exist (possibly as Cauchy principal values). This is achieved by using Landen's
transformation and the duplication theorem. In an appendix, an elliptic integral
of the third kind with a restricted complex parameter is transformed to make
the parameter real. Also, a degenerate integral of the first kind is separated into
real and imaginary parts.
1. Introduction
This paper treats integrands that are rational except for the square root of a
artic polynomial with two pairs of conjugate complex zeros. Integrals of the
quartic
rm
form
(1.1)
rx 5
[p] = [Pi,..-,P5]=
/
Jia, + bit)»-'2dt,
Jy ¡=i
where p\, ■■■, Pu are odd integers and p$ is even, are treated in [4, 5] if all
quantities are real. Reference [8] deals with cases where p2 = pi and a%+ b$t
is the complex conjugate of a2 + b2t. Here we assume further that px = p¡, and
at, + bi,t is the complex conjugate of ax + bit. That is, we consider
(1.2)
[Pi,P2,Pi,Pi,
rx 2
Pi] = /
J(fi + git + hlt2)p'l2(a, + b5t)p>'2dt,
where all quantities are real, x > y , fi + gjt + hjt2>0 for all real t, px and p2
are odd integers, and p$ is even. We retain the redundant notation on the left
side of (1.2), omitting p5 if it is 0, for consistency with [5,8]. Section 2 contains
the 11 cases (apart from exchange of px and p2) with 2\px\ + 2\p2\ + \p5\ < 8
and 2px + 2p2 + p5 < 0, as well as [1,1,1,1,-2]
and [1,1,1,1].
The
formulas hold for all x and y for which the integral exists (possibly as a
Cauchy principal value if p$ = -2).
Received by the editor June 26, 1991.
1991 Mathematics Subject Classification. Primary 33E05; Secondary 33C75.
This work was supported by the Director of Energy Research, Office of Basic Energy Sciences.
The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University
under contract W-7405-ENG-82.
© 1992 American Mathematical Society
0025-5718/92 $1.00+ $.25 per page
165
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B. C. CARLSON
166
In Byrd and Friedman's table [1, §267] such integrals are listed only with a
lower limit that depends on the parameters in the integrand, and a restriction
on the upper limit is added in [9, 3.145(4)] so that <pm Legendre's F(<¡>,k)
is between 0 and n/2. Also, there is an ambiguity of sign; e.g., [1, 267.00]
with ax = a2 = \[2 is correct if b2 = -bx — 1/2 and y — 1 but incorrect if
bx = -bi = 1/2 and y = 2 unless gx is taken to be the negative square root
of gx2.
The integrals (1.2) are expressed in terms of four ^-functions:
1
(1.3)
RF(x,y,z)
= -J
/-oo
[(t + x)(t + y)(t + z)]-['2dt,
3 r°°
= lJ
[(t + x)(t + y)(t + z)]-x'2(t + w)-xdt,
(1.4) Rj(x,y,z,w)
and two special cases,
RD(x,y,
z) = Rj(x,y,
z, z)
and
(1.5)
Rdx,y)
= RFix,y,y)
= ^j
(t + x)~x'2(t+ y)-x dt.
The functions RF, Rd , and Rj respectively replace Legendre's elliptic integrals of the first, second, and third kinds, while Rc, which requires special
attention in this paper, includes the inverse circular (if 0 < x < y) and inverse
hyperbolic (if 0 < y < x) functions. Fortran codes for numerical computation
of all four functions are listed in the Supplements to [4, 5] and are available in
several major software libraries.
In [8] a Landen transformation was used to change the first two variables
of Rf, Rp, and Rj from complex to real numbers; the remaining variables,
including those of Rc , were never complex. In the present paper the complex
variables are the parameter (the fourth variable) of Rj and both variables
of Rc ■ However, a Landen transformation of Rf and Rd is used in §3 to
eliminate a restriction on the interval of integration that arose in [3] because
of a branch point. In §4 the complex parameter of Rj is made real, not by a
direct Landen transformation but by an inverse Landen transformation followed
by the duplication theorem (see Appendix A), a combination that also takes
care of the branch-point problem. The function Rc with complex variables
is separated into real and imaginary parts in Appendix B, and the imaginary
part cancels another Rc that comes from the inverse Landen transformation
of Rj.
The formulas of [5, 8] made it unnecessary to do any further work with
recurrence relations, although conversion to notation appropriate for this paper sometimes entailed tedious algebra. The integrals h, h, and I'3 used
previously are now complex, but the eventual cancellation of imaginary terms
provided a partial check. The 13 integral formulas in §2 were checked by numerical integration; some details of the checks are given in §5. The variables of
Rj and Rc are nonnegative, even when the three integrals with p$ — -2 have
their Cauchy principal values.
2. Table of integrals
We assume x > y and f + g¡t + h¡t2 > 0 for i = 1,2 and all real t.
Some relations useful for numerical checks are included among the following
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A TABLEOF ELLIPTIC INTEGRALS:TWO QUADRATICFACTORS
167
definitions:
(2.1)
(2.2)
(2.3)
Çi= ifi + gix + hix2)1/2,
^ = (gx+ 2hxx)/2ix,
B = ^i - r,[ri2,
n, = (f + g,y + hiy2)x'2,
r,[ = igx + 2hxy)/2nx,
£ = #£,& - "["Im,
(2.4)
dt = 2f + gi(x + y) + 2htxy = $ + n2 - h¡(x - y)2,
(2.5)
Ci = (%*, + di)1'2 = m + m? - hiix - y)2]1'2,
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
U = (c;ltl2+ rhÇ2)/(x-y),
M = CxCl/ix-y),
ô,j = (2f,hj + 2fjh, - gigj)x/2,
A = io*2- Ô2XÔ22)X'2
A±=r522±A,
L| = M2 + A±, L+L-=2MU,
G = 2AA+RD(M2, L2_, L2 )/3 + A/2U
+ (ô22ex-ô2ue2)i^xnxu,
RF = RF(M2,L2_,L2+),
Ï = G-A+RF
+ B.
For integrals with p$ ^ 0 we also define
(2.11)
(2.12)
(2.13)
a,5 = 2fibs - g¡a5,
ßi5 = g¡b5 - 2h¡a5,
y¡ = (aisbs - ßisa5)/2 = fb¡ - glasbs + hia\ > 0,
A = ö2xy2/yx,
Q2 = M2 + A,
W= (aXsß2s-a2Sßxs)ß
(2.14)
= (gxh2- g2hx)a2- 2(fh2 - f2hx)a5bs+ (fxg2 - f2gx)b2,
V2 = -à\xy22 + 2ô2x2yxy2- o%y\ = yxy2(A+ - A)(A - A_)/A,
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
Çs = a5 + bsx,
A(px ,p2,p2,px,
ns = as + b5y,
Ps)=ex^p22e55'2 - rftrftrft12,
X = [Í5(ai5 + ßisy)rl2/nx + r]s(axs + ßxsxfa/idßix
= ísifc[M(-l,
1,1, -l)/2 - ísiMU ,1,1,1,
- y)
-4)]/(x - y)2,
5 = (M2 + S¡2)/2 -U2 = (Çxrix02+ Ç2ri20i)/(x-y)2,
p = yxísr]s/íim,
T = pS + 2yxy2,
V2 = p2(S2 + AU2),
(2.20)
(2.21)
a = SÇi2/U + 2AU,
b2 = (S2/U2 + A)Q4,
a2 = b2 + AV¡yxy2 = b2 + A(A+- A)(A - A_),
(2.22)
H = S2Xy/[Rj(M2, L2_, L\, Q2)/3 + Rc(a2, Z>2)/2]/y2
-XRC(T2,V2).
We shall want some of the quantities above when a5 = 1 and ¿>5= 0. These
will be labeled by a subscript 0:
(2.23)
A0 = S2xh2/hx,
n2o= M2 + A0,
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168
(2 24)
B. C. CARLSON
Vo= g\h2-g2hx,
V2 = -62xxh\+ 2ô\2hxh2 - ô22h\ = hxh2(A+- Ao)(Ao - A_)/A0,
= [6xA(-l, 1, l,-l)/2-A(l,
1, 1, l)]/(x-y)2,
(2.26)
(2.27)
Po = hxl^xm,
T0 = p0S + 2hxh2, V2 = p2(S2 + A0U2),
flo = 5í2§/t/ + 2AoC/,
b¡ = (S2/U2 + An)Q¿,
(2.28)
a2 = b2 + A2ip2/hxh2 = b¡ + A0(A+ - A0)(A0 - A_),
H0 = ô2nWo[Rj(M2 ,L2_,L\,
Q2)/3 + Rc(a2o, b¡)/2]/h
-Xotfc^o2,^2)-
If the interval of integration is infinite, convergent integrals (with 2px + 2p2 +
Ps < -4) do not involve Hq . If x —>+oo and y is finite, we find (for i = 1,2)
that
(2.30)
Si~h\l¿x,
(2.31)
U = h\l2n2 + h\l2r]x,
di~(gi + 2hiy)x,
2
M2 = \[(2h)l2ni + g, + 2hiy).
i=\
If y —>-oc and x is finite, then (for i = 1, 2)
(2.32)
ifc~A}/2M,
0/~(ft + 2Ä/Jc)y,
2
(2.33)
Í7 = A|/26 + /^/2£, ,
M2 = l[(2hx% - g, - 2h.x).
If x = —y —>+00 , then (for i = 1, 2)
(2 34)
f/~*i~A//2*,
0i~-2hiX2,
d = Sii/h]/2,
l/U = M = Rc(a2, b2) = XRC(T2, V2) = 0.
In all three of the limiting cases an identity useful for (2.41) is
fi-M(l,
1,1,1,-2)
= («is + ßny)mßn\ns
- (ais + ^15^)6/2^^5.
Aside from interchange of px and /?2, there are 11 integrals
(2.35)
[px,p2,P2,Pi,
Ps] = /
Jifi + git + hit2)p''2(as+ bst)^2 dt
Jy i=i
with odd integers px, p2 and even integer p5 such that 2|/?i | + 2\p2\ + \ps\ < 8
and 2px+2p2+ps < 0. We shall include also [1, 1, 1, 1, -2] and [1, 1, 1, 1].
The integral of the first kind is
(2.36)
[-1,-1,
-1,-1]
= 4RF,
and the next two integrals are of the second kind:
(2.37)
[-3,1,1,-3]
= 4(-G + A+RF)/Sfx,
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A TABLEOF ELLIPTICINTEGRALS:TWO QUADRATICFACTORS
(2 38)
[-l,-U-l,-3]
169
= &hx[(Ao-ô22)G/A-(Ao-A+)RF]/S2lA
-4îM(-l,l,l,-l)/A2.
Like the three preceding integrals, three integrals of the third kind with 2p x+
2p2 + Pi < -4 are not restricted to finite intervals of integration. They involve
H but not H0 :
(2.39)
(2.40)
(2 41)
[-1,-1,-1,-1,-2]
= -2(b5H + ßxsRF/yi),
[1,-1,-1,1,-4]
= [<pH+ G + (A-A+)RF]/y2
-[ßxsA(-l, I, l,-l) + 2yxA(-l, I, l,-l,-2)]/2bsy2,
[-1,-1,-1,-1,-4]
= bsißxs/yx + ß2s/y2)H + ßf5RF/y2
+ b25[l-bsAil, 1, 1, l,-2)]/yl72.
Seven integrals of the third kind have 2px + 2p2+ps > -2 and exist only for
finite intervals of integration. Four of them with ps > 0 involve Ho but not
H:
(2.42)
(2.43)
[-1, -1, -1,-1,2]
= 2bsH0- 2ßx5RF/hx,
[1,-1,-1,
l] = (y/oHo+ ï + A0RF)/hi,
[-1,-1,-1,-1,4]=-
(2.44)
[1,1,1,1]
b5(ßxs/hx + ßis/h2)H0
+ b2l/hxh2 + ß2sRF/h2,
= (o¡2/h¡ - ô2xxlh2)[tpoHo
+ (Ao - Ó22)RF]/S
- (3^02 - 4hxh2ô22)(l + o22RF)/24h2h¡
(2.45)
+ [A2RF-VoA(l,
1,1, l)]/l2hxh2 + E/3hx.
The final three integrals have Ps <0 and involve both H and Ho :
(2.46)
[i,_i,_i,
[1,1,1,1,-2]=-
l,-2]
= 2(-71//
+ /Jl//o)/è5,
2yxy2H/b¡ + [ihxy2+ h2yx)/b¡ - y/¡/'4hxh2b5]H0
(2.47)
+ ißX5/hx + ß25/h2)(l + A0RF)/4b¡
-ô2xy/oRF/2h2bs + A(l, 1,1,
(2 48) [í'1'1'
l)/2¿>5,
l'-4^\.-(y^2s
+ y2ßi5)H + (hxß2s + h2ßXs)H0]/bl
+ [2Z+(A + Ao)RF)]/b2-A(l,l,l,l,-2)/bs.
3. Integrals
of the first and second kinds
In this section we derive (2.36), (2.37), and (2.38). By [5, (2.13), (2.17)],
rx
/. = /
4
~\iai + bit)-1/2 dt = 2RF(U22, U23, Ux\),
3y i=i
(3.1)
(x -y)U¡j = XiXjYkYm+ YlY]XkXm,
Xi = (a, + biX)x/2,
Yi = (ai + b,y)xi2,
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170
B. C. CARLSON
where i, j, k, m is any permutation of 1, 2, 3, 4. Let
(3 2)
(ax + bxt)(a4 + b4t) = fx + gxt + hxt2 >0,
-co<i<oo,
(a2 + bit)(a3 + bjt) = f2 + git + hit2 > 0,
-co < t < co.
Then the biquadratic polynomial in the integrand of
(3.3)
/,=
[X[ifx+gxt + hxt2)if2 + g2t + h2t2)Tx'2dt
Jy
has two pairs of conjugate complex zeros. Equation (3.1) remains valid by the
permanence of functional relations if the Uy are in the open right half-plane.
We shall see that C/14> 0, and we may choose t/13 > 0. Although Ux2 is
real, it may be negative, and (3.1) is then invalid if RF is taken to have the
principal value represented by the integral (1.3). The reason is that RF , as a
function of any one of its variables, has a branch point at the origin [2, §8.3].
When Uxi describes a semicircle about 0 from the positive to the negative
real axis, U22 makes a complete circle and RF returns to a different branch.
Negative values of Ux2 may occur, as shown in [3, §4], when the quadrilateral
whose vertices are the complex zeros has diagonals intersecting at an interior
point of the interval of integration. The integral Ix can then be expressed in
terms of two standard integrals by breaking the interval of integration at the
intersection. In the present paper we shall eliminate this complication by using
Landen's transformation [7, (5.5)] to write
/, = 4RFiM2 , Ll,
(3.4)
L2+),
M=UX2 + UX3,
L± = [(C/,4 + c712)(c714+ Ux})]1'2 ± [(t/,4 - UX2)(UX4- Ux3)]x'2 ,
L+L- = 2MUxa ,
L\ - M2 = [(U24 - Ux22)xl2
± (U24 - Ux\)x'2]2 ,
where M, L_ , and L+ will be proved nonnegative for every interval of integration. Alternatively, the duplication theorem could be used for the same
purpose, but the resulting expressions are less simple.
In (3.2), since only f, g¡, and h¡ are given, we may choose bx = b4 —A, >
0, bi = bi — h2 > 0, Im(ai) > 0, and Im(û2) > 0. If we assume x and
y to be finite and take the principal branch of the square roots in (3.1), then
Xx, Yx, Xi, Yi lie in the open first quadrant of the complex plane, while their
respective complex conjugates X4, Y4, X$, Y3 lie in the open fourth quadrant.
It is clear that [/14 > 0 because both terms of (x - y)Ux4 are strictly positive
and we assume x > y. The same assumption, along with Im^) > 0, shows
that
lrrxiX2xY¡)= Im[(a, + bxx)(a4 + b4y)] = h\/2(y - x)lm(ax) < 0.
Thus, Xx7/4 is in the open fourth quadrant if x and y are finite, and a similar
argument shows that X^Y2 is in the open first quadrant. Hence the product
XxX^Y2Y4is in the open right half-plane, its real part is positive, and UXi> 0.
Because X2Y^ is the complex conjugate of X^Y2 and so is in the open fourth
quadrant, XxXiY^Y4 is in the open lower half-plane, and UXi may be positive
or negative. However, XXY4+ YXX4and Xi 7/3+ YiX3 are both strictly positive,
whence
(3.5)
C/,2 + £7,3= (Xx74 + YXX4)(X2Y3+ Y2X3)/(x -y)
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> 0.
A TABLEOF ELLIPTIC INTEGRALS:TWO QUADRATICFACTORS
171
It follows from (3.1) that
(3.6)
Ufk- Ufm= dijdkm,
du = aibj - a¡bi.
In particular, we see that
Ux\-U^ = dxidu = \dxi\2>0,
(3-7)
U23-U22 = dX4d32>0,
Uh-Ux22 = dx3d42= |i/13|2>0.
Note that dx2 / 0 because we exclude the degenerate case where ax + bxt is
proportional to a2+b2t (whence fx+gxt+hxt2 is proportional to f2+g2t+h2t2,
and the integral (3.3) is elementary). The second inequality holds because dX4
is positive imaginary and d32 is negative imaginary. Finally, dX3^¿ 0 because
lmax > 0 and Im 03 < 0. We may now conclude, if x and y are finite, that
(3.8)
UX4> UX3> 0 and
- Ux3< Ux2< Ux3.
By (3.4) it follows that M > 0, L+ > L_ > 0, and L2_- M2 > 0, whence
(3.9)
L+>L->M>0.
Since L2±- M2 depends only on the quantities listed in (3.7), which are independent of x and y, both (3.9) and (3.8) are still valid if either x or y is
infinite, but not both. If the interval of integration is the whole real line, then
Ux4= £/i3 = -Ux2 = -(-co and M = 0, as we shall show later.
Again assuming x and y to be finite, we shall now express M and L± in
terms of f, g¡, and h¡. Let
t)x = YXY4= (fx + gly + hxy2)1'2,
t2 = X2X3 = (f2 + g2x + h2x2)x'2,
r,2 = Y2Yi = (f2 + g2y + h2y2)x'2,
Cx=XxY4 + YxX4= 2Re(XxY4),
Ç2= X2Y3+ Y2X3= 2Re(X2Y3),
0i = xf y2 + y2x¡ = c2- 2í, m,
e2 = X2Y2
Çx=XxX4 = (fx+gxx + hxx2)x'2,
+ Y2X2 = C22-2^2tl2.
Then &, r¡¡, and Ç, are positive, but 0, need not be. By (3.4) and (3.5) we
see that
(3.11)
M = dC2/(x-y),
C?= «/ + !fc)2-AiCx-y)2,
where the second equation follows from
(3 12)
C' = (Xl Y*+ y,X,)2 = {XxX*+ YxYa)2~ {Xx " Yx){X*" Y*]
= (Üi + nx)2-hx(x-y)2,
and similarly for Ç2.
If we define
(3.13)
ôu = (2fihj + 2fjhi-gigj)1/2,
then on > 0 because f + g¡t + h¡t2 > 0 for all real t. A stronger result than
¿12 > 0 will be given in (3.17). By (3.2) we have
axb4 + a4bx= gx,
(axb4)(a4bx) = fxhx.
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172
B.C.CARLSON
We solve these two equations for axb4 and use the assumption Im(fli) > 0 to
find
(3.14)
axb4 = (gx + iôxx)/2,
dx4 = axb4-a4bx
= iôxx.
A similar procedure yields
(3.15)
a2*3 = (g2 + ia22)¡2,
d23 = iô22.
Thus we find
|ú?22|= dx2d43 = (axb2 - a2bx)(a4b3 - a3b4)
(3.16)
=/i*2 + /2A1 - 2Re[(*, + iöxx)(g2 - iô22)/4]
= (S2X2-ÔXXÔ22)I2.
Since í/12 # 0, except in the excluded degenerate case (cf. (3.7)), we have
(3.17)
ô22>ôxxô22>0.
A similar calculation leads to
(3.18)
\d23\ = (ô22 + ôxxS22)/2.
We define
(3.19)
A=(Ô*X2-Ô2XÔ22)XI2,
A± = á22±A>0,
and use (3.7) and the last equation of (3.4) to get
(3 20)
V24-U22 = (ô22 + ôxxô22)l2,
U24-U23 = (ô2x2-ôxxô22)/2,
Ux23-U22 = ôxxô22,
L\-M2
= A±.
The last equation and (3.11) allow calculation of M2 and L\ .
If the interval of integration is infinite, we take the appropriate limit in (3.11)
to find (2.31), (2.33), or (2.34). In the first two cases, M2 is a product of two
factors such as
2hx/2ii - g, - 2h,x = [ig, + 2h,x)2 + a2]1'2 - igi + 2hiX) > 0.
Hence, M > 0, except when the interval of integration is the whole real line.
An integral of the second kind used in previous parts of this table [5, (2.14),
(2.17)] is
I2 = [1,-1,
(3.21)
" J
-'
-1,-3]
"
'Tl2
Tl2
Tl2
= 2dx2dx3RDiU22,
Ux\,
U24)I3
+ 2XXYX/X4Y4UX4.
(Since px ^ p4 , this integral is now complex.) Putting w = z in [7, (8.5), (5.5)]
to obtain the Landen transformation of Rp, and using the notation in (3.4),
we find
ARD(U22 , U23, U}4) = 8A+JRD(M2, L2_, L2 )
- 12RF(M2, L2_,L\) + 6/Ux4.
Substituting in (3.21) and using the identities
dx2dX3d24d34= \dx2dx3\2 = A2/4,
4d24d34X2Y2 = ô226x-ô2xd2-i2SxxCxrlxUX4A(-l,
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1,1,-1),
A TABLEOF ELLIPTICINTEGRALS:TWO QUADRATICFACTORS
173
we get
(3.24)
I2 = 4dx2dx3[2G- 2ARF - iôuA(-l,
1, 1,-1)]/A2,
(3.25) G = 2AA+RD(M2,L2_, L2 )/3 + A/217 + (S¡29x- Sfx62)/4ÇxtlxU,
where we denote RF(M2, L2_, L2) by RF and Í/14 by U for brevity.
The integrals [-3, 1,1, -3] and [-3, -1, -1, -3] can now be obtained
from [8, (2.24), (2.23)]. In the second case, we use the identity
(3 26)
ldndn = [M22 - Mn + '(£1*2 - gih\)ôu\lhx
= o¡2-A0 + ii//oox\/hx.
4. Integrals
of the third
kind
We shall encounter Rj(U22, U23, U24, W2), where the first three variables
are real but W2 is complex. The function can be changed by Landen transformation [8, (4.14)] into Rj(M2, L2_,L2+, W2), but W2 also is complex.
Instead, an inverse Landen transformation followed by the duplication theorem
leads to Rj(M2, L2_, L2 , Q2) with real Q2 . This combination of two transformations (see Appendix A) is equivalent to a direct Landen transformation
for integrals of the first and second kinds but not the third kind.
In (A.8) we identify (z_ , z+, a) with (Î/12, Ul3, UX4)and find from (A.9)
and (3.4) that (x2 + X, y2 + X, z2 + A) = (L2_, L\ , M2). Because of [5, (2.15),
(2.9)], we put
(4.1)
w2+= W2 = U24 - dx2dx3d4s/dX5.
Since dXs and d4s are complex conjugates, it follows that la2-«;2!2 = \dx2dx3\2.
By (3.7) the condition (A.2) is satisfied, and so w2_ is the complex conjugate
of w\ :
(4.2)
w2_ = Ux24- d43d42dx5/d4s.
We define
(4.3)
n2 = w2 + A,
w = w+W-/Ux4,
and find from (A.3), (3.7), and [4, (5.22)] that
Q2 - M2 = w2 - z2 = w2+wl-z2+-
zl
= (Ux24- U23) + (U24 - U22) - dx2dx3d45/dxs - d43d42dX5/d45
= idX2d45- d42dxs)id43dl5 - dX3d45)/dXsd4s
= I(¿12^45 - i/42Ö?15)/^15|2 = \dX4d25/dX5\2 .
Defining
(4.4)
y, = |i/;5|2 = fib] - gla5b5 + htal,
i =1,2,
we note that 7, > 0 because Irrxda / 0. By (3.14) we have
(4.5)
Q2 = M2 + A,
w2 = z2 + A,
A = ô2xy2/yx> 0.
Since Q2 > 0, we may choose Cl > 0. Then it follows from (3.9), (A.5), (4.3),
and (3.20) that
(4.6)
L+>Q>
L_ >M>0,
A+>A>A_>0,
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174
B.C.CARLSON
where M = 0 by (2.34) only if the interval of integration is the whole real line.
It will be useful to define also (for i = 1, 2)
(4.7)
q,5 = dy¡/dbs = 2fjb5 - gta5,
ßi5 = -dyi/da5
= gib5 - 2hiü5,
whence
(4.8)
yi = (a,5b5-ßi5a5)/2.
(The definitions (4.7) are equivalent to aXs = axd45 + a4dXs and ßxs = bxd45+
b4d\s, and similar relations for a25 and ß2$.)
From (4.1) and (4.2) we can obtain a coefficient in (A.8):
w2 - w2_ = -dl2dx3d45/dxs + d43d42dl5/d45
= -(2i/yx)lm(dX2dx3d¡5).
It is straightforward to show by (3.6) and (3.2) that
¿i2¿45 = a5(hxa2 - axb4b2) + b5(fb2 - a4bxa2).
Replacing the subscript 2 by 3, multiplying dl2d4s by dx3d4s, and taking the
imaginary part with the help of (3.14), we find
(4.9)
w2-w2=-iôxxy//yx,
where
¥ = (ai5#>5 -a25ßl5)/2
= a](gih2 - g2hx)- 2a5b5ifxh2- f2hx) + b¡(fxg2 - f2gx).
Incidentally, with the help of (A.3) and (3.19) we see that
(w\ - w2_)2 = (x2 - w2)(y2 - w2) = (L2_ - Q2)(L2 - Q2)
^22'
= (A_ - A)(A+ - A) = A2 - 28\2A + 62xxô2
Substituting
A from (4.5) and comparing with (4.9), we get
(4.11)
0>(io2
-w2_)2 = -Aif/2/yxy2,
where
¥2 = -à2xxy\ + 2ô22yxy2 - 3¡2y2 > 0,
a result that is tedious to obtain by squaring y/ . Although it provides a useful
numerical check, the last equation does not determine the sign of \p, which
may be positive or negative.
Since b = w(w2 + X) = wÙ2 by (A.6), we can now write (A.8) as
2(^2-C/24)JR7(ÍJ122,c/123,c/124,^2)
(4.12)
= -i(ôxxWlyx)[2Rj(M2, L2_, L\ , Q2) + 3Rc(a2, b2)]
+ 6RF(M2, L2_, L\) - 3Rc(z2, w2),
where W2 is given in (4.1 ) and
(4 13)
z = Ui2Ux3/Ui4,
b = wQ2,
w2-z2
= n2-M2
a2 - b2 = AV/yr/2
= A = S2xy2/yx,
= A(A+ - A)(A - A_).
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A TABLEOF ELLIPTICINTEGRALS:TWO QUADRATICFACTORS
175
An integral of the third kind used in previous parts of this table [5, (2.15),
(2.17)] is
/3 = [1,-1,
(4 14)
-1,-1,
-2]
= 2dx2dx3dx*Rj(U¡2, í/,23, «724,W2)/3dx5 + 2RC(P2, Q2).
Substituting (4.12) and using (4.1), (3.14), and (4.4), we find
h/2dl5=
-ô2xip[Rj(M2,L2_,L2+,a2)/3
+ Rc(a2,b2)/2]/y2
- iöxxRF/yx + iaxxRc(z2, w2)/2yx + RC(P2, Q2)/dx5,
where RF = RF(M2,L2_,L\).
We shall separate the real and imaginary parts of the last term,
(4.16)
Rc(P2,Q2)/dx5 = Rc((dxsP)2,(di5Q)2),
and find that the imaginary part cancels the next to last term. Putting dx$P =
X + iY and referring to (B.l) in Appendix B, we shall need X, Y, X2 +
Y2, \dX5Q\\ and
(4.17)
c = d25(Q2 - P2) = -d25d35dX5d45= -yxy2,
where we have used [5, (2.5)] and (4.4). It follows from [5, (2.8)] and (3.10)
that
(4.18) (x-y)(X
+ iY) = (x - y)dX5P= (m^/^d^X2
+ OW/iWis^2,
where
Í5 = X] = a5 + b5x,
r]5 = Y¡ = a5 + b5y.
Using (3.14) and (4.7), we find
(4.19)
2dl5X¡ = 2(axb5 - a5bx)(a4 + b4x) = aX5+ ßx$x + iôxxÇ5,
and similarly for dx$Y42.Substitution in (4.18) yields
(4 20)
2{-X~ y^X = ^a'5
+ ßX5X^2^x + ^a'5
2Y = oixí5m^2^i + m/m)/(x-y)
+ AsíO^/fi
>
= Sxi^tj5ui^t¡x.
Instead of squaring X and Y, it is easier to get X2 + Y2 by calculating
\dX5P\2= yx\P\2 from [5, (2.8)]:
(4 21)
{X ~y)2\P\2
= {Í5r,2)2 + (^2)2
+ ^l5^ri2(X,Yx/XxY4
+ XXY4/X4YX)
= (&m)2 + (r¡¿2)2 + ^t]iex^2n2l^xnx.
From 72 = ^25^35 and (x - y)di5 = ¿Í5Y/2- n5X2 it follows by (3.10) that
(4.22)
(x - y)2y2 = (&m)2 + (U2)1 - fcfcfc ,
and hence
(4.23)
(x - y)2\P\2 = ^5(02
+ 0,í2^2/íiffi)
+ (JC- y)2y2 ■
Defining
(4.24)
S=Uz
= Ux2Ui3,
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176
B.C.CARLSON
we find
(x - y)2S = iXxX2Y3Y4+ YxY2X3X4)iXxX3Y2Y4+ YxY3X2X4)
(4-25)
= ZxnxiX2Y2 + Y2X2) + £2rç2(X2742+ Y2X2)
Çxnxe2+ Ç2n26
and thus
\P\2 = Í5rl5S/txrix+y2,
(4.26)
x2 + Y2 = y&tlsS/Çim+7ir2
= pS + yxy2= T- y,72,
P = yi&m/Zili,
T = pS + 2yxy2.
Finally, [5, (2.5)], (4.1), and (4.3) imply
|<05Ô|4 = (pw+w.)2
= (pUw)2 = p2U2(z2 + A) = V2,
V2 = p2(S2 + U2A),
while
(4.28)
;r2 + y2 + c = /jS +7172-7172
=/¿t/z.
From (B.l) we now have
(4.29)
Rc(P2,Q2)/dxi
= XRC(T2 , V2) - i(ôxxpU/2yx)Rc((pUz)2 , (pUw)2))
= XRC(T2 , V2) - i(ôxx/2yx)Rc(z2 , w2).
The last term cancels a term in (4.15) to yield
(4.30)
h = -2dX5(H+iôxXRF/yx),
where
H = S2X>p[Rj(M2 ,L2_,L\,
Q2)/3 + Rc(a2 , ¿>2)/2]/72 - XRC(T2 , V2).
Using a subscript 0 to label quantities in which we have put as = 1 and bs = 0,
we obtain from [5, (2.17)] also
(4.31)
T3= [1, -1, -1, -1] = 2bx(H0 + iôxxRF/hx),
where
Ho = Ô2X
tpo[Rj(M2 ,L2_,L\,
Q2)/3 + Rc(a\,
b¡)/2]/h2
- X0RC(T¡, V02).
Since Ix, I2,13, and /j have now been reduced to Ä-functions of real variables, the ten integrals of the third kind in §2 can be derived by substitution
in the formulas of [5, 8] (the latter if the odd p's are not in decreasing order). Converting coefficients to the notation of this paper is straightforward
but sometimes tedious. In addition to the recurrence relation [5, (4.8)], the
following identities are useful:
(4.32) 2d24d34X2Yx2
= (ô226x- ó¡xd2)/2 - iôxxc:xnxUA(-l, 1,1,-1),
(4.33)
(4.34)
h\l2A(l, 1, I,-I)
= B + iôxxAi-l, 1, 1, —1)/2,
A,1/2^(3, 1,1, I) = E + iôxxAil, 1,1, l)/2,
where B and E are defined in (2.3).
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A TABLE OF ELLIPTIC INTEGRALS: TWO QUADRATIC FACTORS
5. Numerical
177
checks
The 13 integrals in §2 were checked numerically when x = 2, y = —3,
C/i, g\, Ai) = (2.7, -1.8, 0.9), (fi,gi,hi)
= (2.0, 2.4, 0.8), and (a5,b5) =
(1.1, -0.4).
(The zeros of the quadratic polynomials are 1 ± iy/2 and
(-3 ± 0/2. Because S < 0, the validity condition [3, (33)] is violated (cf.
(4.25)), and the integral of the first kind would have to be split in two parts
before using [3, (34)].) In each of the formulas (2.36) to (2.48) the integral
on the left side, defined by (2.35), was integrated numerically by the SLATEC
code QNG. On the right side the quantities RF, G, H, Hq were calculated by
using the codes for 7?-functions in the Supplements to [4, 5], and the remaining
calculations were done with a hand calculator. For each of the 13 cases the
values obtained for the two sides agreed to better than one part in a million
(better than the claimed accuracy of QNG).
Some intermediate values are
M2 = 0.36362947,
L2_= 0.53423014,
RF(M2 ,L2_,L2+) = 0.54784092,
L\ = 24.673029,
Q2 = 21.199185,
Rj(M2,L2_,L\,
il2o= 6.1236295,
a2= 11237.193,
b2 = 9741.4746,
a2 = 844.71933,
b2Q
= 247.52253,
T2 = 10.288757,
V2 = 1.1362990,
T02= 1.1328716,
V02= 1.1077327,
AT= -1.1571677,
X0 = -0.093427949,
RD(M2, L2_, L2+)= 0.042910488,
il2) = 0.048599080,
Rj(M2,L2_,L\,
Q2,)= 0.12739513,
Rc(a2,b2) = 0.0098889795,
Rc(a2,,bQ1)= 0.050085175,
RC(T2, V2) = 0.58372845,
Rc(Tç2,V02)= 0.94657139,
C7= 10.495586,
77 = 0.049905556,
770=-1.8557835,
Ai-l, 1, 1, -1)= 1.5731367,
¿(1,1,1, 1) =-0.49594737,
Ai-l, 1,1,-1, -2) = 6.2622360,
¿(1,1,1,1,-2)
= 14.845682.
As a test of Cauchy principal values, the three integrals with p$ = -2, viz.
(2.39), (2.46), and (2.47), were checked numerically with the same values of
x, y, f, gi, and h¡ as before but with a5 + bst = t, so that each integrand
has a simple pole in the open interval of integration. In each case the Cauchy
principal value of the left side was computed by the SLATEC code QAWC,
and the right side was calculated as before. Cauchy principal values are not
required for either 7?7 or Rc, as one can see from (4.6) and (2.19), since
V2 > 0 whether £5n¡ is positive or negative. For each of the three cases the
values obtained for the two sides agreed to better than one part in a million,
even though the SLATEC code issued a warning about impairment of accuracy
by roundoff error in the case of (2.47).
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178
B. C. CARLSON
Appendix A. Rj with a restricted
complex parameter
When the fourth variable of Rj is complex but has a special relation (see
(A.2)) to the first three variables, which are real, transformation (A.8) leads to
Rj with four real variables. To derive this, we start from the inverse Landen
transformation [7, (8.5), (5.7), (7.2)],
2(w2+ - *2)Rj(z2-
,z2+,a2,
w2) = (w2 - w2_)Rj(x2 ,y2,z2,
w2)
+ 3RF(x2, y2, z2) - 3Rc(z2, w2),
y + x = 2a,
y - x = (2/a)[(a2
w = w+W-/a,
- z2+)(a2 - z2_)]x'2 ,
(a2 - w2)(a2
z = z+z_/a,
- w2:) = (a2 - z2)(a2
- z2_).
We are concerned with the case in which w2 is not real, a > z+ > 0, and
-z+ < z_ < z+. (We exclude the degenerate case a = z+, in which Rj
is elementary, w2: = a2, and w2 = w2 , whence w2 is complex.) The last
equation in (A.l) defines w2. and shows, since the right side is positive, that
a2 - w\ and a2 - w2. have equal and opposite complex phases. If they have
also the same absolute value, i.e., if
(A.2)
\a2-w2+\2
= (a2-z2+)(a2-z2_),
then w2_ is the complex conjugate of w\ , and hence w2 > 0. Since w cannot
vanish, we may choose w > 0, whence w- is the complex conjugate of w+ .
From (A.l) and (A.2) we see that y > x > 0 and
xy + z2 = z2 + z2_ ,
xy + w2 = w\ + w2_ ,
(x + y)z = 2z+z-,
(x + y)w = 2w+w-,
(x±z)(y±z)
1
?
= (z+ ±z_)2,
?
7
(x±w)(y±w)
7
= (w+±W-)2,
7
w —z ■=■
w + + w± —z+ —z_.
We find also that
( ' j
x2_z2
= [(a2_z2_)l/2_(a2_z2)./2]2j
y2-z2
= [(a2-z2.)1/2
+ (a2-zi)'/2]2,
x2 - w2 = [(a2 - w2_)'/2 - (a2 - w2+)xl2]2 ,
y2 - w2 = [(a2 - wl)x/2 + (a2 - wl)1'2]2 .
Since (x - w)(y - w) = (w+ - W-)2 < 0 and x2 - z2 > 0, we have
(A.5)
y > w > x > 0 and
- x < z < x.
If z_ < 0, then z < 0, and the 7?-functions in (A. 1) do not take the principal
values represented by (1.3) to (1.5). A remedy is provided by the duplication
theorem [7, (6.1)(8.7)]:
RF(x2,y2,
(A.6)
z2) = 2RF(x2+À,y2
Rj (x2 ,y2,z2,w2)
+ À, z2 + A),
= 2Rj(x2 + À,y2+À,z2+À,w2
+ 3Rc(a2,b2),
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+ X)
A TABLE OF ELLIPTIC INTEGRALS:TWO QUADRATIC FACTORS
179
with
X = xy + xz +yz,
b = w(w2 + X),
a = w2(x + y + z)+xyz,
b ± a = (w ±x)(w ±y)(w ± z).
It follows from (A. 5) that b-a<0.
Since z2 + X = (z + x)(z + y) is a product
of nonnegative factors, and w2 + X > z2 + X by (A.5), we have
(A.7)
a>b>0.
Finally, we combine (A.6) and (A.l):
Theorem. If (A.2) holds, let W- be the complex conjugate of w+ . Then
2(w2+-a2)Rj(z2_,z2+,a2,w2)
(A.8)
= (w2+- w2_)[2Rj(x2 + X, y2 + X, z2 + X, w2 + X) + 3Rc(a2, b2)]
+ 6RF(x2 +X,y2 + X,z2 + X)- 3Rc(z2, w2),
where
a>z+>0,
-z+<Z-<z+,
z = z+Z-/a,
Im(it;2)/0,
w = w+W-/a,
z2 + X = (z+ + z_)2 ,
x2 + X = (z+ + z_)2 + [(a2 - zl)x'2 - (a2 - z2+)x'2]2,
y2 + X = (z+ + z_)2 + [(a2 - z2_)x'2 + (a2 - z2+)x'2]2 ,
(A.9)
w2 + X = wl + w2: + 2z+z_ ,
a = [(u;2 + w2)z+z-
+ 2w\w2_\ja,
b = w+W-(wl + w2_+ 2z+z_)/a,
b ± a = (w+ ± W-)2(w+W- ± z+z-)/a,
b2 -a2 = (wl - w2_)2(w2 - z2) = (wl - w2_)2(w2++ w2_- z2+- z2_).
Note that z2 + X is the square of a nonnegative quantity, even if z < 0.
The first term on the right side of (A.8) is pure imaginary while the second
and third terms are real. If z < 0, the third term is not represented by (1.5)
until it is rewritten by the duplication theorem as -67?c((^ + ^)2, 2w(z + w)) ;
alternatively, it can be expressed in terms of an arctangent taken in the second
quadrant rather than the fourth. Neither procedure is needed in this paper.
Appendix B. Real and imaginary
parts of 7?c
In §4 we need to separate the real and imaginary parts of Rc when its two
variables are complex but differ by a real number. (If the difference is not real,
it can be made real by using the homogeneity of 7?c .)
Lemma. Let x, y, c be real, z = x + iy, r2 = x2 + y2 > 0, and z2 + c ^ 0.
If \c\ < r2, then
Rc(z2,z2
+ c) = xRc((r2 -c)2,\z2
-iyRc((r2
+ c\2)
+ c)2,\z2 + c\2),
where Rc(z2, z2 + c) denotes the branch that is continuous in c and takes the
value 1/z when c = 0.
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B. C. CARLSON
180
Proof. Let |c| = a2 < r2. Then (B.l) reduces by [2, (6.9-15), (6.9-16)] to the
correct equations
, z+a
log-=
z —a
a
1
log
z+a
z- a
2ax
¿.ay
..
i arctan -z-?
t,
2
if c = -a
rl — a1
r2 + 2ay + a2
arctan - = - arctan -z-T
- - log -=—r-=■
z
2
r2 - a2
4
r2 - 2ay + a2
,
.,
,
if c = a .
On each right-hand side, the logarithm is taken real and the arctangent is taken
in the first or fourth quadrant to get the principal value of the left side.
In the excluded case when |c| > r2, the arctangent must be taken in the third
or second quadrant, and the corresponding Rc in (B. 1) has the square of a
negative number as its first argument. This can be replaced by the square of
a positive number by using the duplication theorem [7, (3.7)], and c can then
have any real value provided z2 + c ^ 0. The result is given here although it
is not needed in the present paper; it provides a way of computing Rc with
complex arguments.
Theorem. Let x, y, c be real, z = x + iy, r2 = x2-\-y2 > 0, and s = |z2 + c| >
0. Then
Reiz2, z2 + c) = xRc(a2., sa-) - iyRc(ol,
(B.2)
sa+),
s2 = |z2 + cf = (r2 - c)2 + 4cx2 = (r2 + c)2 - 4cy2,
o± = (r2±c + s)/2>0.
In the first equation, 7?c(z2, z2 + c) denotes the branch that is continuous in
c and takes the value 1/z when c = 0. In the exceptional cases where z2 is real
and c/z2 < -1, it denotes the Cauchy principal value of (l/z)7?c(l,
1+c/z2).
On the right side of the first equation, each Rc denotes the principal branch
represented by (1.5).
Bibliography
1. P. F. Byrd and 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. _,
Elliptic integralsof thefirst kind, SIAMJ. Math. Anal. 8 (1977), 231-242.
4. _,
A table of elliptic integrals of the second kind, Math. Comp. 49 (1987), 595-606.
(Supplement, ibid., S13-S17.)
5. _,
A tableof ellipticintegralsof the third kind, Math. Comp.51 (1988),267-280. (Sup-
plement, ibid., S1-S5.)
6. _,
A table of elliptic integrals: cubic cases, Math. Comp. 53 (1989), 327-333.
7. _,
Landen transformations of integrals, Asymptotic and Computational
Analysis (R.
Wong, ed.), Marcel Dekker, New York, 1990, pp. 75-94.
8. _,
A table of elliptic integrals: one quadratic factor, Math. Comp. 56 (1991), 267-280.
9. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press,
New York, 1980.
Ames Laboratory
Iowa 50011
and Department
of Mathematics,
Iowa State
E-mail address: [email protected]
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University,
Ames,