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Asymptotic formulas for elliptic integrals
by
John L. Gustafson
A Dissertation Submitted to theGraduate Faculty in Partial Fulfillment of theRequirements for the Degree ofDOCTOR OF PHILOSOPHY
Department: MathematicsMajor: Applied Mathematics
Approved:
!!!!!!!!!!!!!!!!!!!!!! !!!!!!In Charge of Major Work
!!!!!!!!!!!!!!!!!!!!!! !!!!!!For the Major Department
!!!!!!!!!!!!!!!!!!!!!! !!!!!!For the Graduate College
Iowa State UniversityAmes, Iowa1982
[Revised and re-typeset, February 2002]
TABLE OF CONTENTS
ABSTRACT............................................................................................................iii
INTRODUCTION....................................................................................................1
TABLES OF ASYMPTOTIC FORMULAS............................................................5
PRELIMINARY THEOREMS................................................................................ 9
PROOFS OF THE FORMULAS FOR LARGE ARGUMENTS.......................... 14
PROOFS OF THE FORMULAS FOR SMALL ARGUMENTS.......................... 30
ACKNOWLEDGMENTS......................................................................................39
LITERATURE CITED...........................................................................................40
APPENDIX A: NUMERICAL EXAMPLES........................................................ 41
APPENDIX B: PROGRAMS USED TO CREATE EXAMPLES........................ 51
ii
ABSTRACT1
Asymptotic formulas are derived for incomplete elliptic integrals of all three kinds when the arguments are real and tend to infinity or to zero. Practical error bounds are found for the asymptotic formulas. Several techniques are used, including a method recently discovered by R.!Wong for finding asymptotic expansions with remainder terms for integral transforms. Most of the asymptotic formulas and all of the error bounds appear to be new.
We use incomplete elliptic integrals that possess a high degree of permutation symmetry in the function arguments. The asymptotic formulas are applicable to complete elliptic integrals as a special case; some of the error bounds are treated separately in the complete case.
Numerical examples are given to demonstrate the typical accuracy that can be expected from the formulas, as well as the closeness of the error bounds.
DOE Report IS-T-1014. This work was performed under Contract No. W-7405-Eng-82 with the U.S. Department of Energy.
iii
INTRODUCTION
Elliptic integrals are classically defined as integrals of the form
†
r(x,y)dxÚ (1.1)
where r is a rational function of x and y, and
†
y 2 is a cubic or quartic polynomial in x. If
†
y 2 is linear or quadratic in x, then the integral may be evaluated using logarithms and rational functions of x and y, but if
†
y 2 is cubic or quartic then the integral is said to be elliptic and is not in general expressible in terms of elementary functions. Legendre [9] showed that only three nonelementary functions are needed to express (1.1) in the elliptic case.
We will choose as our three basis functions
†
RF ,
†
RD , and
†
RD, which have the integral definitions
†
RF (x, y,z) = 12 x + t( )- 1
2 y + t( )- 12 z + t( )- 1
2 dt0
•
Ú (1.2a)
†
RD (x,y,z) = 32 x + t( )- 1
2 y + t( )- 12 z + t( )- 3
2 dt0
•
Ú (1.2b)
†
RJ (x, y,z,) = 32 (x + t)- 1
2 (y + t)- 12 (z + t)- 3
2 (r + t)-1dt0
•
Ú (1.2c)
We assume x, y, z, and r are real and positive. The
†
RF function is the elliptic integral of the first kind and is symmetric in x, y, and z. The elliptic integral of the second kind,
†
RD , is symmetric in x and y only; it is related to
†
RF by
†
RD (x,y,z) = -6 ∂∂x
RF (x,y,z). (1.3)
The
†
RJ function is symmetric in x, y, and z only. The constant in front of each integral is chosen so that
†
RF (x, x,x) = x- 12
RD (x, x,x) = x- 32
RJ (x, x, x,x) = x- 32
(1.4)
1
Furthermore, the functions are homogeneous of degrees
†
- 12 ,
†
- 32 , and
†
- 32 ; that is,
†
RF (Cx,Cy,Cz) = C- 12 RF (x,y,z)
RD (Cx,Cy,Cz) = C- 32 RD(x, y,z)
RJ (Cx,Cy,Cz,Cr) = C- 32 RJ (x,y,z,r)
(1.5)
These basis functions are not the classical ones used by Legendre that became established as standard after his work was published in 1825. Legendre’s basis functions are denoted by
†
F(j,k),E(j,k), andP(j,k,n);
however, this notation hides the underlying permutation symmetry in the variables, introduces unnecessary linear transformations, and makes the quadratic Gauss and Landen transformations as well as other identities more cumbersome than with the present choice of basis functions. Carlson [3], [4] has shown elliptic integrals to be hypergeometric functions of several variables and has shown many classical results concerning properties of elliptic integrals to be special cases of the general properties of R functions. References [2], [3], [4], and [14] provide methods of converting between the classical basis functions and the ones used here. In particular,
†
F(j,k) = sinj( )RF (cos2 j,1- k 2 sin2 j,1), (1.6)
†
E(j,k) = F(j,k) - 13 k 2(sin3 j)RD(cos2 j,1- k 2 sin2 j,1), (1.7)
†
P(j,k,n) = F(j,k) - p3 sin3 j( )RJ (cos2 j,1- k 2 sin2 j,1,1+ nsin2 j). (1.8)
We will also require the following definitions of complete elliptic integrals:
†
RK (x,y) = 1p t- 1
2 (x + t)- 12 (y + t)- 1
2 dt0
•
Ú (1.9a)
†
RQ (x, y) = 2p t- 1
2 (x + t)- 12 (y + t)- 3
2 dt0
•
Ú (1.9b)
†
RM (x,y,r) = 2p t- 1
2 (x + t)- 12 (y + t)- 3
2 (r + t)-1dt0
•
Ú (1.9c)
2
where the constants in front of the integrals are chosen so that properties analogous to (1.4) are satisfied. These complete cases are related to the incomplete cases by
†
RK (x,y) = 2p RF (x,y,0) (1.10a)
†
RQ (x, y) = 43p RD (x,0,z) (1.10b)
†
RM (x,y,r) = 43p RJ (x,y,0,r). (1.10c)
We will find it convenient to state some of the formulas in terms of the elementary function
†
RC :
†
RC (x, y) = 12 x + t( )- 1
2 y + t( )-1dt0
•
Ú
=
x - y( )- 12 log x
12 + x - y( )
12
y12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ , 0 < y < x,
y - x( )- 12 arccos x
12
y12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ 0 £ x < y.
Ï
Ì
Ô Ô
Ó
Ô Ô
(1.11)
This is simply a special case of the
†
RF function, as can be seen from (1.2a):
†
RC (x, y) = RF (x,y,y). (1.12)
There is a similar relationship between
†
RD and
†
RJ , evident from (1.2b) and (1.2c):
†
RD (x,y,z) = RJ (x,y,z,z). (1.13)
Despite the intense activity that surrounded elliptic integrals and their inverses (elliptic functions) during the last century, much was left undiscovered concerning the behavior of elliptic integrals as the arguments tend to zero or to infinity. Kaplan [8, page 13] gives an asymptotic series for F(j,k) that implies the formula
†
RF (x, y,z) ~ z- 12 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ as
†
z Æ • . (1.14)
Kaplan also gives an asymptotic series for
†
E(j,k) that can be shown to imply the one-term asymptotic formula for
†
RD derived in the sections that follow. Both formulas also appear in reference [11, page 228]. No series is given in either work for the elliptic integral of the third kind, and no error bounds are given for truncations of the
†
F(j,k) and E(j,k) series.
3
Convergent series expansions for functions
†
F(j,k),
†
E(j,k) , and
†
P(j,k,n) , for 0!<!j!<!
†
p2 and
†
k 2 <1 can also be found in [1]. These are power series in
†
k 2 ,
†
1- k 2, and n that do not converge for arguments in the neighborhoods being considered here. Asymptotic series have been long established for the complete elliptic integrals K(k) and E(k), and these series are equivalent to series for
†
RK and
†
RQ ; see, for example, reference [7].
Tables 1 through 4 indicate the asymptotic approximations and error bounds for incomplete and complete elliptic integrals as the arguments tend to infinity or to zero. Where there is symmetry in the arguments, only the case for one argument is shown.
4
TABLE 1
Incomplete Elliptic IntegralsBehavior for Large Arguments
†
- 34 z- 3
2 log 1+2z
xy( )14 r
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ +
12
Ê
Ë
Á Á
ˆ
¯
˜ ˜
< r < 0
†
3z- 12RC (X 2,Y 2) + r,where
†
X = xy( )12 + r,
Y = xr( )12 + yr( )
12
†
z Æ •
†
RJ (x, y,z,r)
†
0 < r < 3 2r-2 x + y + z( )12
†
3r-1RF (x, y,z) - 3p2 r- 3
2 + r
†
r Æ •
†
RJ (x, y,z,r)†
- 34 y- 3
2 log 1+2y
x14 z
34
Ê
Ë Á
ˆ
¯ ˜ + 1
2
Ê
Ë Á
ˆ
¯ ˜ < r < 0
†
y- 12
3x
12 z
12 + z
Ê
Ë Á
ˆ
¯ ˜ + r
†
y Æ •
†
RD (x,y,z)†
0 < r < 98 z- 5
2 x + y( ) log 1+4z
x + yÊ
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
3z- 32 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ -1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r
†
z Æ •
†
RD (x,y,z)†
0 < r < 18 z- 3
2 x + y( ) log 1+4z
x + yÊ
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
z- 12 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r
†
z Æ •
†
RF (x, y,z)
Error BoundAsymptotic FormulaArgumentFunction
5
TABLE 2
Incomplete Elliptic IntegralsBehavior for Small Arguments
†
- 3p2 2 z
12 x- 1
2 y- 12 r-1 < r < 0
†
3p4 RM (x, y,r) + r
†
z Æ 0+
†
RJ (x, y,z,r)
†
- 65 rl- 5
2 < r < 0†
3RC (a,b) + 2RJ (x + l,y + l,z + l,l) + rwhere
†
l = xy( )12 + xz( )
12 + yz( )
12 ,
†
a = r x12 + y
12 + z
12( ) + xyz( )
12( )
2,
and
†
b = r r + l( )2
†
r Æ 0+
†
RJ (x, y,z,r)
†
- 3p2 2 y
12 x- 1
2 z- 32 < r < 0
†
3p4 RQ (x,z) + r
†
y Æ 0+
†
RD (x,y,z)†
-6 x + y( )12
xy< r < 0
†
3 xyz( )- 12 + r
†
z Æ 0+
†
RD (x,y,z)†
- p2 2 z
12 x- 1
2 y- 12 < r < 0
†
p2 RK (x, y) + r
†
z Æ 0+
†
RF (x, y,z)
Error BoundAsymptotic FormulaArgumentFunction
6
TABLE 3
Complete Elliptic IntegralsBehavior for Large Arguments
†
- 1p y- 3
2 log 1+2y
x13r
23
Ê
Ë Á
ˆ
¯ ˜ + 2
3
Ê
Ë Á Á
ˆ
¯ ˜ ˜ < r < 0
†
4p y- 1
2 r- 12 RC (r,x) + r,
†
y Æ •
†
RM (x,y,r)
†
0 < r < 4 2p r-2(x + y)
12
†
2r-1RK (x,y) - 2r- 32 + r
†
r Æ •
†
RM (x,y,r)†
- 1p x- 3
2 log 1+2xy
Ê
Ë Á
ˆ
¯ ˜ + 2
3
Ê
Ë Á
ˆ
¯ ˜ < r < 0
†
4p x- 1
21y
+ r
†
x Æ •
†
RQ (x, y)†
0 < r < 32p y- 5
2 x log 1+4yx
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
2p y- 3
2 log 16yx
Ê
Ë Á
ˆ
¯ ˜ - 2
Ê
Ë Á
ˆ
¯ ˜ + r
†
y Æ •
†
RQ (x, y) †
0 < r < 14 p y- 3
2 x log 1+4yx
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
1p y- 1
2 log 16yx
Ê
Ë Á
ˆ
¯ ˜ + r
†
y Æ •
†
RK (x,y)
Error BoundAsymptotic FormulaArgumentFunction
7
TABLE 4
Complete Elliptic IntegralsBehavior for Small Arguments
†
0 < r <
†
12p y r + 2x
x32 r2
log 1+4xr
y 2x + r( )
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á Á
ˆ
¯ ˜ ˜
†
2p r-1 x- 1
2 log 16xy
Ê
Ë Á
ˆ
¯ ˜ - RC (x,r)
Ê
Ë Á
ˆ
¯ ˜ + r
†
y Æ 0+
†
RM (x,y,r)
†
- 85p r xy( )- 5
4 < r < 0†
4p RC (a,b) + 8
3 RD (x + l, y + l,l) + rwhere
†
l = (xy)12 ,
†
a = r x12 + y
12( )( )
2,
and
†
b = r r + l( )2
†
r Æ 0+
†
RM (x,y,r)
†
0 < r < 32p y- 5
2 x log 1+4yx
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
2p y- 3
2 log 16yx
Ê
Ë Á
ˆ
¯ ˜ - 2
Ê
Ë Á
ˆ
¯ ˜ + r
†
x Æ 0+
†
RQ (x, y)†
- 1p x- 3
2 log 1+2xy
Ê
Ë Á
ˆ
¯ ˜ + 2
3
Ê
Ë Á
ˆ
¯ ˜ < r < 0
†
4p x- 1
2 y-1 + r
†
y Æ 0+
†
RQ (x, y) †
0 < r < 14 p x- 3
2 y log 1+4xy
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜
†
1p x- 1
2 log 16xy
Ê
Ë Á
ˆ
¯ ˜ + r
†
y Æ 0+
†
RK (x,y)
Error BoundAsymptotic FormulaArgumentFunction
8
PRELIMINARY THEOREMS
Before proving the formulas summarized in Tables 1 through 4, we require several preliminary results.
Consider functions that can be defined by
†
I(l) = f (t)h(lt)dt0
•
Ú (2.1)
where f(t) and h(t) are locally integrable functions on (0,∞) that have asymptotic expansions
†
f (t) ~ aktk +u-1
k= 0
n-1
 , as
†
t Æ 0+, (2.2)
where 0 < u ≤ 1, and
†
h(t) ~ bkt-k-v
k= 0
n-1
 , as
†
t Æ +•, (2.3)
where 0 < v ≤ 1. Wong [13] has found a method for finding the asymptotic expansion for I(l) that also provides an explicit error term. Define the generalized Mellin transform of h(t) by
†
M[h;z] = tz-1
0
1
Ú h(t)dt + tz-1
1
•
Ú h(t)dt . (2.4)
Near the origin, we assume that
†
h(t) = O(tb ) , u +b > 0. (2.5)
The first integral in (2.4) is analytic for Re z > –b, and the second integral is analytic for Re!z!<!v. By analytic continuation of the second integral, the Mellin transform M[h;z] can be extended to a meromorphic function in the half-plane Re z > –b. For
†
n =1,2,º define
†
hn (t) = h(t) - bkt-k-v
k= 0
n-1
 , (2.6)
9
and
†
jn (t) =h(t), 0 < t <1,hn (t), 1£ t < •,
Ï Ì Ó
(2.7)
With these definitions, the following result is easily established:
Lemma 1. Let h(t) be a locally integrable function on (0,∞) satisfying (2.3) and (2.5). Then for –b < Re z < n + v,
†
M[h;z] = M[jn;z] -bk
z - k - vk= 0
n-1
 , (2.8)
Analogous results are needed for f(t). Define
†
M[ f ;1- z] = t-z
0
1
Ú f (t)dt + t-z
1
•
Ú f (t)dt , (2.9)
As
†
t Æ +• , we assume that
†
f (t) = O(t-a ), a + v > 1. (2.10)
The first integral in (2.9) is analytic for Re z < u, and the second is analytic for Re z > 1–a. By analytic continuation of the first integral, M[f;1–z] can be extended to a meromorphic function in the half plane Re z > 1–a. For
†
n =1,2,º define
†
fn (t) = f (t) - aktk +u-1
k= 0
n-1
 (2.11)
and
†
yn (t) =f (t), 0 < t <1,fn (t), 1£ t < •,
Ï Ì Ó
(2.12)
Lemma 2. Let f(t) be a locally integrable function on (0,∞) satisfying (2.2) and (2.10). Then for 1 – a < Re z < n + u,
10
†
M[ f ;1- z] = M[yn;1- z] -ak
k - u - zk= 0
n-1
 . (2.13)
We can now give the asymptotic expansion of I(l) and the remainder term.
Theorem 1 (Wong). Let f(t) satisfy (2.2) and (2.10). Let h(t) satisfy (2.3) and (2.5). Then for any n ≥ 1,
†
I(l) = ak M[h;k + u]l-k-u +k= 0
n-1
 bk M[ f ;1- k - v]l-k-v
k= 0
n-1
 + dn (l) (2.14)
for the case
†
u ≠ v , and
†
I(l) = ck (v)l-k-v + loglk= 0
n-1
 akbkl-k-v
k= 0
n-1
 + dn (l) (2.15)
for the case
†
u = v , where
†
ck = ak M[jn;k + v] + bk M[yn ;1- k - v] -akbj - a jbk
k - jk= 0
n-1
 . (2.16)
In both cases, the remainder term is given by
†
dm (l) = fn (t)hn (lt)dt0
•
Ú . (2.17)
Proof. We write
†
I(l) = I1(l) + I2(l) , (2.18)
where I1 and I2 correspond to the intervals (0,1) and (1,∞), respectively. We break up the remainder term similarly:
†
dn (l) = dn,1(l) + dn,2(l). (2.19)
By applying (2.6) and (2.11) to
†
I1 and
†
I2 ,
11
†
I1(l) = ak tk +u-1h(lt)dt +0
1
Úk= 0
n-1
 bkl-k-v t-k-v fn (t)dt +
0
1
Úk= 0
n-1
 dn,1(k)
and
†
I2(l) = ak tk +u-1hn (lt)dt +1
•
Úk= 0
n-1
 bkl-k-v tk +u-1 f (t)dt +
1
•
Úk= 0
n-1
 dn,2(k) .
Assume for the moment that
†
u ≠ v . Since
†
yn (t) = fn (t) for 0 < t < 1 and
†
jn (t) = f (t) for
†
1 < t < •, adding the last two identities together gives
†
I(l) = ak tk +u-1h(lt)0
1
Ú dt + tk +u-1hn (lt)1
•
Ú dtÈ
Î Í
˘
˚ ˙
k= 0
n-1
Â
+ bk M[yn;1- k - v]l-k-v + dn (l).k= 0
n-1
Â(2.20)
The presence of l in the quantity in square brackets prevents us from similarly combining h and
†
hn into a Mellin transform of
†
jn ; however, we can rewrite that quantity as a Mellin transform of
†
jn plus other terms if we substitute
†
s = lt :
†
[º] = l-k-u sk +u-1h(s)ds0
l
Ú + l-k-u sk +u-1hn (s)dsl
•
Ú
= l-k-uM[jn ;k + u]+ l-k-u t k +u-1h(t)dt1
l
Ú + t k +u-1hn (t)dt1
l
ÚÊ
Ë Á
ˆ
¯ ˜
= l-k-uM[jn ;k + u]+ l-k-u t k +u-1 bjt- j-v
j= 0
n-1
 dt1
l
Ú
= l-k-uM[jn ;k + u]+l- j-vb j
k + u - j - vj= 0
n-1
 - l-k-u b j
k + u - j - vj= 0
n-1
Â
(2.21)
By Lemma 1, the sum of the first term and the third term in (2.21) is equal to
†
l-k-uM[h;k + u]. So (2.20) now becomes
†
I(l) = ak M[h;k + u]l-k-u
k= 0
n-1
 +l- j-vakb j
k + u - j - vj= 0
n-1
Âk= 0
n-1
 + bk M[yn;1- k - v]l-k-v + dn (l)k= 0
n-1
 , (2.22)
By applying Lemma 2 to (2.22), the double sum is completely canceled, leaving
12
†
I(l) = ak M[h;k + u]l-k-u
k= 0
n-1
 + bk M[ f ;1- k - v]l-k-v + dn (l)k= 0
n-1
 , (2.23)
which proves the case
†
u ≠ v . For the case
†
u = v , we apply a limiting process. Since
†
limuÆv
l-k-u - l-k-v
u - vÊ
Ë Á
ˆ
¯ ˜ = l-k-v -logl( ), (2.24)
this limit applies to the terms of the sums in (2.21) for which
†
k = j :
†
limuÆv
tk +u-1h(lt)dt +0
1
Ú tk +u-1hn (lt)dt1
•
ÚÊ
Ë Á
ˆ
¯ ˜
= l-k-v M[jn ;k + v]+ bk logl( )l-k-v +bj l- j-v - l-k-v( )
k - jj= 0j≠k
n-1
 .(2.25)
Therefore, instead of (2.22) and (2.23), we obtain the second case of the theorem:
†
I(l) = ck (v)l-k-v + loglk= 0
n-1
 akbkl-k-v
k= 0
n-1
 + dn (l),
where
†
ck = ak M[jn;k + v] + bk M[yn ;1- k - v] -akbj - a jbk
k - jk= 0
n-1
 . n
Both cases will be of considerable value to us in deriving asymptotic formulas and error bounds for elliptic integrals.
13
PROOFS OF THE FORMULAS FOR LARGE ARGUMENTS
We now prove the asymptotic formulas and error bounds summarized in Tables 1 and 3, and give more detailed results for certain cases. In attempting to bound the error term, we seek the following features in a bound:
1. The bound should be simple, i.e. an elementary function of the argument in question. It should not involve limits or integrals.
2. There should be little “waste” in the bound. The bound should not exceed the actual error by more than, say, a factor of three. Ideally, the bound should asymptotically agree with the actual error.
These features tend to compete directly with one another, since the actual error always involves an integral that is not expressible using elementary functions, and all simplifications of that integral introduce a difference between the actual error and the bound. Hence, the bounds presented here are the result of compromise. All satisfy the first condition in that they involve only logarithms and rational powers of the argument; the second condition is generally satisfied also, based on numerical tests (see Appendices).
Theorem 2. If x, y ≥ 0 and x+y, z > 0, then
†
RF (x, y,z) = z- 12 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r, (3.1)
where the error term, r, is bounded by
†
0 < r < 18 z- 3
2 x + y( ) log 1+4z
x + yÊ
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜ . (3.2)
Proof. If we lets
†
s =1 t in the integral representation (1.2a) of
†
RF (x, y,z),
14
†
RF (x, y,z) = 12 x + t( ) y + t( ) z + t( )( )- 1
2 dt0
•
Ú
= 12 x +
1s
Ê
Ë Á
ˆ
¯ ˜ y +
1s
Ê
Ë Á
ˆ
¯ ˜ z +
1s
Ê
Ë Á
ˆ
¯ ˜
Ê
Ë Á
ˆ
¯ ˜
- 12 -1
s2
Ê
Ë Á
ˆ
¯ ˜ ds
0
•
Ú
= 12 s 1+ xs( ) 1+ ys( ) 1+ zs( )( )- 1
2 ds0
•
Ú
Hence,
†
RF has the alternative representation
†
RF (x, y,z) = 12 t 1+ xt( ) 1+ yt( )( )- 1
2 1+ zt( )- 12 dt,
0
•
Ú (3.3)
which is in the form (2.1) for which Theorem 1 applies, with
†
z = l . Let
†
f (t) = t- 12 1+ xt( ) 1+ yt( )( )- 1
2
and
†
h(zt) = 1+ zt( )- 12 = zt( )- 1
2 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 12
.
The functions f and h have asymptotic formulas
†
f (t) ~ t- 12 1-
x + y2
Ê
Ë Á
ˆ
¯ ˜ t
Ê
Ë Á
ˆ
¯ ˜ as t Æ 0+,
h(t) ~ t- 12 1- 1
2 t-1( ) as t Æ +•,(3.4)
which means that
†
u = v = - 12 , and the second case of Theorem 1 applies. Here,
†
a0 =1,
†
b0 =1,
†
a1 = - 12 x + y( ) , and
†
b1 = - 12 . Observe that
†
f (t) = O(t- 32 ) as
†
t Æ • , and
†
h(t) = O(1) as
†
t Æ 0, so conditions (2.5) and (2.10) are satisfied. Using (2.15) with
†
n =1, we obtain the first term of the asymptotic expansion:
†
RF (x, y,z) = 12 c0(1
2)z- 12 + log z
12( )z
12 + 1
2 d1(z) , (3.5)
where
†
c0 and
†
d1(z) will be derived as follows: By (2.7) and (2.12),
15
†
j1(t) =h(t), 0 < t <1h(t) - t-1/2, t ≥1,
Ï Ì Ó
(3.6)
and
†
y1(t) =f (t) - t-1/2, 0 < t <1f (t), t ≥1.
Ï Ì Ó
(3.7)
The fact that
†
a0 = b0 permits considerable simplification of the expression (2.16) for
†
c0( 12) :
†
c0( 12) = M[j1; 1
2]+ M[y1; 12] = M[j1 +y1; 1
2]
= h(t) + f (t) - t- 12( )t- 1
2 dt0
•
Ú .
This integral can be evaluated using elementary functions. By writing
†
1+ xt( ) 1+ yt( ) =1+ x + y( )t + xyt2 and expressing the improper integral as a limit, we find
†
c0( 12) = t- 1
2 1+ t( )- 12 + t-1 1+ x + y( )t + xyt2( )
- 12 - t-1Ê
Ë Á ˆ
¯ ˜ dt
0
•
Ú
= limb Æ•e Æ0
log t12 (1+ t)
12( ) + t + 1
2( ) - log1+ x + y( )t + xyt 2( )
12
t+
1t
+x + y
2
Ê
Ë
Á Á
ˆ
¯
˜ ˜
- log tÏ
Ì Ô
Ó Ô
¸
˝ Ô
˛ Ô e
b
= 2log 4x
12+ + y
12
Ê
Ë Á
ˆ
¯ ˜
(3.8)
(See [6], entries 2.261 and 2.266). Combining (3.8) with (3.5) provides the first term of the asymptotic expansion:
†
RF (x, y,z) = z- 12 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ +
12 d1(z).
As has already been mentioned, this first term is not new to the literature; however, we now have an exact expression for the error:
†
= 12 z- 1
21t
1- 1+ x + y( )t + xyt 2( )- 1
2Ê Ë Á ˆ
¯ ˜ 1- 1+
1zt
Ê
Ë Á
ˆ
¯ ˜
- 12Ê
Ë Á Á
ˆ
¯ ˜ ˜
0
•
Ú dt . (3.9)
16
This integral is of course not elementary, but the integrand may be bounded rather closely by a function that has an elementary integral. Consider the factors in square brackets in (3.9); letting
†
w =1 zt( ) ,
†
1- 1+ w( )- 12 =
1+ w( )12 -1
1+ w( )12
=w
1+ w( )12 1+ 1+ w( )
12( )
(3.10)
where we have multiplied top and bottom by
†
1+ 1+ w( )12 to rationalize the numerator. The
denominator of (3.10) may be rewritten and bounded from below by
†
1+ w( ) + 1+ w( )12 > 1+ w( ) +1= 2 + w .
Here we use the fact that
†
w =1 zt is positive for t in the range of integration, (0, ∞). For the same reason, the quantity in (3.10) is positive. Hence,
†
1- 1+ w( )- 12 < w 2 + w( ) , or
†
0 <1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 12
<1
1+ 2zt. (3.11)
It will turn out to be convenient to specialize further this inequality according to whether t is small or large. Since
†
1 1+ 2zt( ) <1 2zt( ) ,
†
1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 12
<1
1+ 2zt"t > 0, close for small t,
1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 12
<1
2zt"t > 0, close for large t.
(3.12)
The term “close” here means that the ratio of the terms being compared approaches unity as t becomes small or large, appropriately.
The first term in square brackets in (3.9) is bounded differently; we use
†
1- 1+ x + y( )t + xyt 2( )- 12 <1 "t > 0, close for large t,
1- 1+ x + y( )t + xyt 2( )- 1
2 < 12 x + y( )t "t > 0, close for small t.
(3.13)
17
The first inequality is obvious; the second follows from the inequality of arithmetic and geometric means:
†
xy( )12 £
x + y2
for x, y ≥ 0. (3.14)
As a result,
†
1- 1+ x + y( )t + xyz2( )- 1
2 £1- 1+ x + y( )t +x + y( )2 t 2
4
Ê
Ë Á Á
ˆ
¯ ˜ ˜
- 12
=1- 1+x + y( )t
2Ê
Ë Á
ˆ
¯ ˜
-1
<1- 1-(x + y)t
2Ê
Ë Á
ˆ
¯ ˜ =
x + y( )t2
.
By partitioning the interval of integration into (0,s) and (s,∞) for any s > 0, we may bound the remainder term r defined by (3.9):
†
r < 12 z- 1
2x + y
2Ê
Ë Á
ˆ
¯ ˜
11+ 2zt
Ê
Ë Á
ˆ
¯ ˜
0
s
Ú dt + 12 z
12
12zt2
Ê
Ë Á
ˆ
¯ ˜
s
•
Ú dt
= 14 z- 3
2x + y
2Ê
Ë Á
ˆ
¯ ˜ log 1+ 2zs( ) +
1s
Ê
Ë Á
ˆ
¯ ˜ .
(3.15)
As s tends to infinity, the log term dominates and the bound in (3.15) tends to infinity; as s tends to zero, the
†
1 s term dominates and the bound tends to infinity. In either case, the bound is not useful. The minimum occurs for a particular value of s that is a function of x, y, and z. Minimizing the quantity in square brackets in (3.15) by differentiation,
†
x + y( )z1+ 2zs
-1s2 = 0,
which implies
†
x + y =1
zs2 +2s
. (3.16)
Since we are mainly interested in large z, drop the
†
1 zs2 term to obtain
†
smin ~ 2 x + y( ) . By
18
using this value of s in (3.15), the stated bound (3.2) is obtained. n
Since
†
RF is
†
O z- 12 log z( ) and the error bound is
†
O x + y( )z- 32 log z( ) , the relative error is
†
O x + y( ) z( ) . Therefore, one expects to be able to approximate
†
RF to about three significant figures if
†
z x + y( ) is on the order of 1000. Numerical tests bear this out; see Appendix A. Since there is complete symmetry in the arguments x, y, and z, one can always permute the arguments so that z is the largest argument.
As a corollary to Theorem 2, we can find an asymptotic formula and error bound for the complete case
†
RK (y,z) as
†
z Æ • . Using (1.10a) gives
†
RK (y,z) = 1p z- 1
2 log 16zy
Ê
Ë Á
ˆ
¯ ˜ + r, (3.17)
where
†
0 < r < 14 p z- 3
2 y log 1+4zy
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜ . (3.18)
The second-order asymptotic formula (
†
n = 2 in Theorem 1) is obtainable by methods similar to those above, and is stated here without proof:
†
RF (x, y,z) = z- 12 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ 1+
x + y4z
Ê
Ë Á
ˆ
¯ ˜ + 1
4 z- 32 xy( )
12 - x - y( ) + r, (3.19)
where r is
†
O(z- 52 log z). A simple bound for the second-order error is less easy to find because of
the complexity of the integrand in the remainder term. The complete asymptotic series for
†
RF appears to be very complicated, and the complexity lies in the evaluation of
†
ck ( 12) in (2.16) for
k!>!0. The logarithmic term in (2.16), however, is tractable as n approaches infinity. By using [4, (6.1–1), (6.1–4)], it can be shown to converge to
†
z- 12 log z
12( )RK (1-
xz
,1-yz
) , for 0 ≤ x,y < z.
We now prove the asymptotic formulas given for
†
RD . The proofs are simpler than that for
†
RF since they build on the results for
†
RF .
19
Theorem 3. If x, y ≥ 0, and x+y, z > 0, then
†
RD (x,y,z) = 3z- 32 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ -1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r , (3.20)
where the error term, r, is bounded by
†
0 < r < 98 z- 5
2 x + y( ) log 1+4z
x + yÊ
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜ . (3.21)
Proof. By (1.3),
†
RD can be obtained through differentiation of
†
RF with respect to z. Hence, we obtain the first-order term and remainder by applying
†
-6∂ ∂z to the expressions for
†
RF and r, (3.1) and (3.9):
†
RD (x,y,z) = 3z- 32 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ -1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ - 3z- 3
2 - 3 t- 12 - f (t)( ) ∂
∂zzt( )- 1
2 - h(zt)( )dt0
•
Ú . (3.22)
To justify the differentiation of (3.9) under the integral sign, we need to majorize the integrand of (3.22) for z in any closed interval [K,L], 0 < K < L, by a function that is integrable and does not depend on z; see [4, Appendix B.3]. This will be done below in the process of bounding the integral.
To bound the error term, we need to bound
†
-3 ∂∂z
zt( )- 12 - 1+ zt( )- 1
2( ) = 32 t- 1
2 z- 32 1- 1+
1zt
Ê
Ë Á
ˆ
¯ ˜
- 32Ï
Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô . (3.23)
We rewrite the quantity in curly brackets above by substituting
†
w =1 1+ zt( ) :
†
1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 32
=1-1+ zt
1+ zt( ) -1Ê
Ë Á
ˆ
¯ ˜
- 32
†
=1- 1- w( )32 , where 0 < w < 1.
This function has the same value and derivative as
†
32 w at
†
w = 0, but is concave down since its
20
second derivative is negative for 0 < w < 1. Hence,
†
=1- 1- w( )32 < 3
2 w , or
†
=1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 32
<3
2
1+ zt.
The above inequality can be used to bound (3.23) for 0 < K ≤ z ≤ L by
†
32 t- 1
2K- 32 3
2 1+ Kt( )( ) . This provides an integrable majorizing function for the integrand of (3.22) that is independent of z, as required to justify differentiation under the integral sign. The inequality is specialized for large and small values of
†
zt ;
†
1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 32
<3
2
1+ zt"t > 0, close for small t,
1- 1+1zt
Ê
Ë Á
ˆ
¯ ˜
- 32
<3
2
zt"t > 0, close for large t.
(3.24)
By combining inequalities (3.13) from the previous proof and (3.24),
†
0 < r < 94
x + y2
Ê
Ë Á
ˆ
¯ ˜
z- 32
1+ zt
Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt +
0
s
Ú 94 t-2z- 5
2dts
•
Ú
= 94 z- 5
2x + y
2Ê
Ë Á
ˆ
¯ ˜ log 1+ zs( ) +
1s
Ê
Ë Á
ˆ
¯ ˜ .
(3.25)
The quantity in square brackets in (3.25) is the same as that in (3.15), so for large z we approximate
†
smin by
†
2 x + y( ) as before. Using this value of s in (3.25) gives the error bound stated in Theorem 3. n
The relative error here is
†
O(1 z) . By applying (1.10b), the complete case of Theorem 3 is seen to be
†
RQ(y,z) = 2p z- 3
2 log 16zy
Ê
Ë Á
ˆ
¯ ˜ - 2
Ê
Ë Á
ˆ
¯ ˜ + r, (3.26)
where
†
0 < r < 32p z- 5
2 y log 1+ 4zy
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜ (3.27)
21
By applying the
†
-6∂ ∂z operator to (3.20), we can obtain an approximation for
†
RD (x,y,z) with an error that is
†
O(z- 72 log z):
†
RD (x,y,z) ~ 3z- 32 log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ -1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + 3z- 5
2 34 x + y( )log 4z
12
x12 + y
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ - x + y( ) + 3
4 xy( )12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ (3.28)
By making use of the permutation symmetry of the arguments of
†
RF (x, y,z), the asymptotic behavior of
†
RD (x,y,z) as
†
y Æ • can be derived by a method similar to that used to obtain Theorem 3.
Theorem 4. If x, y, and z are positive, then
†
RD (x,y,z) = 3y- 12
1xz( )
12 + z
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r , (3.29)
where
†
0 < -r < 34 y- 3
2 log 1+2y
x14 z
34
Ê
Ë Á
ˆ
¯ ˜ + 1
2
Ê
Ë Á
ˆ
¯ ˜ . (3.30)
Proof. Since
†
RF is symmetric in x, y, and z, we can permute the arguments in Theorem 2 and write
†
RF (x, y,z) = y- 12 log 4 y
12
x12 + z
12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + r, (3.31)
where
†
r = 12 y- 1
21t
1- 1+ xt( )- 12 1+ zt( )- 1
2( ) 1- 1+1yt
Ê
Ë Á
ˆ
¯ ˜
- 12Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt.
0
•
Ú (3.32)
The first term, (3.29), is obtained by applying
†
-6∂ ∂z to equation (3.31). Applying it to the error term as well yields
22
†
-r = 32 y- 1
2 1+ xt( )- 12 1+ zt( )- 3
2 1- 1+1yt
Ê
Ë Á
ˆ
¯ ˜
- 12Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt,
0
•
Ú (3.33)
where r now represents the
†
RD error. To justify the differentiation under the integral sign, observe that for z in any closed interval [K, L], 0 < K < L,
†
1+ zt( )- 32 can be majorized by
†
1+ Kt( )- 32 in (3.33) to yield an integrand that is integrable and independent of z; see
[4,!Appendix B.3]. The factor
†
1- 1+1 yt( )( )- 12 is bounded as it was in the proof of Theorem 2:
†
1- 1+1yt
Ê
Ë Á
ˆ
¯ ˜
- 12
<1
1+ 2yt"t > 0, close for small t,
1- 1+1yt
Ê
Ë Á
ˆ
¯ ˜
- 12
<1
2yt"t > 0, close for large t.
(3.34)
In bounding the other term, we rely on the fact that x > 0, which means that the following bound is not useful for the complete case
†
RQ :
†
1+ xt( )- 12 1+ zt( )- 3
2 <1 "t > 0, close for small t,1+ xt( )- 1
2 1+ zt( )- 32 < x- 1
2 z- 32 t-2 "t > 0, close for large t.
(3.35)
Combining these inequalities in (3.33) gives
†
0 < -r < 32 y- 1
21
1+ 2ytdt + 3
4 y- 32 x- 1
2 z- 32 t-3dt
s
•
Ú0
s
Ú
= 34 y- 3
2 log 1+ 2ys( ) + 12 x- 1
2z- 32s-2( ).
(3.36)
The quantity in square brackets in (3.36) can be minimized by differentiation with respect to s:
†
2y1+ 2ys
- x- 12 z- 3
2 s-3 = 0. (3.37)
As
†
y Æ •,
†
smin ~ x- 14 z- 3
4 , which when applied to (3.36) yields the bound (3.30). n
As
†
x Æ 0, the bound in (3.30) approaches infinity, and hence a slightly different approach is needed for the complete case.
23
Theorem 5. If y and z are positive, then
†
RQ(y,z) = 4p y- 1
2 z-1 + r, (3.38)
where
†
0 < -r < 1p y- 3
2 log 1+2yz
Ê
Ë Á
ˆ
¯ ˜ + 2
3
Ê
Ë Á
ˆ
¯ ˜ . (3.39)
Proof. Equation (3.38) is easily obtained from (1.10b) and (3.29). To obtain a bound, apply (1.10b) and (3.33) with
†
x = 0:
†
r = - 2p y- 1
2 1+ zt( )- 32 1- 1+
1yt
Ê
Ë Á
ˆ
¯ ˜
- 12Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt
0
•
Ú . (3.40)
Here we use a slightly different bound on the first factor:
†
(1+ zt)- 32 < z- 3
2 t- 32 "t > 0, close for large t,
(1+ zt)- 32 <1 "t > 0, close for small t.
(3.41)
So the bound on r here takes the form
†
-r < 2p y- 1
21
1+ 2ytdt +
12yz
32t
52
dts
•
Ú0
s
ÚÊ
Ë Á
ˆ
¯ ˜
= 1p y- 3
2 log 1+ 2ys( ) + 23 z- 3
2 s- 32( ).
. (3.42)
As
†
y Æ •,
†
smin ~ 1 z , which gives the bound stated in (3.39). n
The relative error here is
†
O(y- 12 log y), so the approximation is less accurate for the complete
case than for the incomplete case.
We now turn our attention to elliptic integrals of the third kind,
†
RJ . Since
†
RJ is symmetric in x, y, and z, we need only treat the cases
†
r Æ • and
†
z Æ • .
Theorem 6. If r > 0 and x, y, z ≥ 0 with at most one of x, y, z equal to 0, then
24
†
RJ (x, y,z,r) = 3r-1RF (x,y,z) - 3p2 r- 3
2 + r, (3.43)
where
†
0 < r < 3 2r-2 x + y + z( )12 . (3.44)
Proof. By using the substitution
†
s =1 t in (1.2c),
†
RJ (x, y,z,r) = 32 x + t( )- 1
2 y + t( )- 12 z + t( )- 1
2 r + t( )-1dt0
•
Ú
= 32 r- 1
2 1+ xs( ) 1+ ys( ) 1+ zs( )( )- 12 rs( )
12
1+ rsds
0
•
Ú .(3.45)
This is of the form
†
32 r- 1
2 f (t)h(pt)dt0
•
Ú , where
†
f (t) = 1+ xt( ) 1+ yt( ) 1+ zt( )( )- 12 ~ 1 as t Æ 0+ (3.46)
and
†
h(t) =t
12
1+ t~ t- 1
2 as t Æ +•. (3.47)
So the first case of Theorem 1 applies, with
†
u =1 and
†
v = 12 . Since
†
f (t) = O(t- 32 ) as
†
t Æ +• and
†
h(t) = O(t12 ) as
†
t Æ 0+ , conditions (2.5) and (2.10) are satisfied. By (2.14),
†
RJ (x, y,z,r) = 32 r- 1
2 M h;1[ ]r-1 + M f ; 12[ ]r- 1
2{ } + r, (3.48)
where by (3.3),
†
M f ; 12[ ] = t- 1
2
0
•
Ú 1+ xt( ) 1+ yt( ) 1+ zt( )( )- 12 dt = 2RF (x, y,z).
Using the beta function,
†
M h;1[ ] = 1+ t( )-1t12 dt
0
•
Ú = B( 32 ,- 1
2) = -p ,
25
we arrive at the first asymptotic term, equation (3.43). The error term, r, is by (2.17) equal to
†
32 1- 1+ xt( ) 1+ yt( ) 1+ zt( )( )- 1
2{ }t12
1rt
-1
1+ rtÏ Ì Ó
¸ ˝ ˛ 0
•
Ú dt, (3.49)
where we bound each factor in curly brackets as follows:
†
1rt
-1
1+ rt=
1rt 1+ rt( )
<1
r2t 2 "t > 0, (3.50)
and
†
1- 1+ xt( ) 1+ yt( ) 1+ zt( )( )- 12 < 1
2 x + y + z( )t "t > 0, close for small t, (3.51a)
1- 1+ xt( ) 1+ yt( ) 1+ zt( )( )- 1
2 <1 "t > 0, close for large t. (3.51b)
It might at first appear that we should specialize the inequality in (3.50) by bounding
†
1 rt 1+ rt( ) by
†
1 rt for small t; however, this turns out to yield an error bound that is only
†
O(r- 74 ) . Inequality (3.51a) follows from the following general inequality for b,
†
xi > 0:
†
1+ xi( )-b
i=1
n
’ = e-b log 1+xi( )
i=1
n
Â>1- b log 1+ xi( )
i=1
n
 >1- b xii=1
n
 . (3.52)
Applying these bounds to (3.49) gives
†
0 < r < 34 x + y + z( )r-2 t- 1
2dt0
s
Ú + 32 r-2 t- 3
2dts
•
Ú
= 32 r-2 x + y + z( )s
12 + 2s- 1
2( ).(3.53)
The minimum value of the quantity in square brackets occurs exactly when
†
s = 2 x + y + z( ) , resulting in the bound stated in (3.44). n
The complete case,
†
RM (x,y,r) , may be easily obtained as a corollary to Theorem 6 by letting
†
z = 0 and multiplying the formula and the error bound by
†
43p (see (1.10c)).
The first term of Theorem 6 is in a sense not a convenient asymptotic approximation because it involves the nonelementary function
†
RF . However, the variable r is involved only as a
26
polynomial in
†
r- 12 . In using Theorem 1 to find higher-order approximations to
†
RJ , one finds that the coefficients of that polynomial are nonelementary R functions. Of the five formulas for large arguments,
†
RJ as
†
r Æ • has the most complicated approximation but the simplest error bound. The approximation for
†
RJ as
†
z Æ • involves only elementary functions.
Theorem 7. If x, y, z, and r are positive, then
†
RJ (x, y,z,r) = 3z- 12 RC (X 2,Y 2) + r, (3.54)
where
†
X = xy( )12 + r ,
†
Y = xr( )12 + yr( )
12 ,
and
†
- 34 z- 3
2 log 1+ 2z xy( )- 14 r- 1
2( ) + 12
Ê Ë Á ˆ
¯ ˜ < r < 0. (3.55)
Proof. As was done in the proof of Theorem 6, we may substitute
†
s =1 t in (1.2c) to obtain
†
RJ (x, y,z,r) = 32 t
12 1+ xt( ) 1+ yt( )( )- 1
2 1+ rt( )-1{ } 1+ zt( )- 12 dt.
0
•
Ú (3.56)
Rather than use Theorem 1, rewrite (3.56) using
†
t- 12 1+ xt( ) 1+ yt( )( )- 1
2 1+ rt( )-1< t
12 xy( )- 1
2 r-1,
†
"t > 0 :
†
RJ (x, y,z,r) = 32 z- 1
2 1+ xt( ) 1+ yt( )( )- 12 1+ rt( )-1dt
0
•
Ú
+ 32 t
12 1+ xt( ) 1+ yt( )( )- 1
2 1+ rt( )-1 1+ zt( )- 12 - zt( )- 1
2( )dt.0
•
Ú(3.57)
The first integral is elementary, being expressible as
†
3z–1/2RC (X 2,Y 2) , where
†
RC is defined by equation (1.11) and X, Y are defined above. A derivation of this may be found in [4, pp. 290, 314]. The second integral is the error term, for which we seek a bound. Bounding each factor, we find
27
†
t- 12 1+ xt( ) 1+ yt( )( )- 1
2 1+ rt( )-1< t
12 "t > 0, close for small t,
t- 12 1+ xt( ) 1+ yt( )( )
- 12 1+ rt( )-1
< t12 xy( )- 1
2 r-1 "t > 0, close for large t,(3.58)
and
†
zt( )- 12 - 1+ zt( )- 1
2 <zt( )- 1
2
1+ 2zt"t > 0, close for small t,
zt( )- 12 - 1+ zt( )- 1
2 < 12 zt( )- 3
2 "t > 0, close for large t.(3.59)
The latter inequality follows from multiplying (3.12) by
†
zt( )- 12 . Applying these inequalities to
the error term, for any s > 0, gives
†
0 < -r < 32 z- 1
21
1+ 2ztdt
0
s
Ú + 34 z- 3
2 xy( )- 12 r-1 t-3dt
s
•
Ú ,
= 34 z- 3
2 log 1+ 2zs( ) + 12 xy( )- 1
2 r-1s-2( ).(3.60)
We can minimize the quantity in square brackets by differentiation:
†
2z1+ 2zs
- xy( )- 12 r-1s-3 = 0, (3.61)
which implies that as
†
z Æ • ,
†
smin Æ xy( )- 14 r- 1
2 . Using this value in (3.60) gives the inequality (3.61). n
The asymptotic formula and bound given for
†
RD (x,y,z) as
†
y Æ • are special cases of the preceding theorem, in accordance with (1.13). Hence, this gives an independent proof of Theorem 4.
In the complete case,
†
y = 0, the error bound of Theorem 7 is not useful. An error bound for
†
RM (x,z,r) may be found by a slight modification of inequality (3.58).
Theorem 8. If x, z ≥ 0 and x+z, r > 0, then
†
RM (x,y,r) = 4p z- 1
2 r- 12 RC (r,x) + r , (3.62)
where
28
†
0 < -r < 1p z- 3
2 log 1+ 2zx- 13r- 2
3( ) + 23( ). (3.63)
Proof. The first term on the right side of (3.62 is obtained by setting
†
y = 0 in (3.54) and then using (1.10c) and homogeneity to bring the
†
r- 12 outside the
†
RC function. To bound the error term, we use
†
t12 1+ xt( )- 1
2 1+ rt( )-1< t
12 "t > 0, close for small t,
t12 1+ xt( )- 1
2 1+ rt( )-1< t-1x- 1
2 r-1 "t > 0, close for large t.(3.64)
Inequality (3.59) is still useful in this case; by combining it with (3.64), we obtain a variation on (3.60):
†
0 < -r < 2p z- 1
21
1+ 2ztdt
0
s
Ú + 1p z- 3
2 x- 12r-1 t- 5
2 dt,s
•
Ú
= 1p z- 3
2 log 1+ 2zs( ) + 23 z- 1
2 r-1s- 32( ).
(3.65)
We can minimize the quantity in square brackets by differentiation:
†
2z1+ 2zs
- x- 12r-1s- 5
2 = 0 , (3.66)
which implies that as
†
z Æ • ,
†
smin Æ x- 13 r- 2
3 . Using this value in (3.65) gives the bound on the error, (3.63). n
This completes the proofs of formulas for large arguments.
29
PROOFS OF THE FORMULAS FOR SMALL ARGUMENTS
The formulas for arguments tending to zero are of a different character from those for arguments tending to infinity. In four of the five incomplete cases, setting an argument equal to zero gives a finite value that can be used as an approximation. In three of the cases, that first term is simply the complete version of the elliptic integral. In addition to Theorem 1, a variety of approaches is used to obtain approximations and error bounds.
Theorem 9. If x, y >0 and z ≥ 0, then
†
RF (x, y,z) = p2 RK (x,y) + r , (4.1)
where
†
0 < -r < p2 2 z
12 xy( ) - 1
2 . (4.2)
Proof. By (1.10a), the first term is the complete case,
†
RF (x, y,0) . By using the alternative representation (3.3), the error term can be expressed as
†
-r = 12 t- 1
2 1+ xt( ) 1+ yt( )( )- 12 1- 1+ zt( )- 1
2( )dt.0
•
Ú (4.3)
We bound
†
t- 12 1+ xt( ) 1+ yt( )( )- 1
2 by
†
xy( )- 12 t- 3
2 for all t > 0. Using inequality (3.11), we bound
†
1- 1+ zt( )- 12 by
†
zt 2 + zt( ) for all t > 0. Combining these yields an elementary integral as a bound:
†
0 < -r < 12 z xy( )- 1
2 t- 12 2 + zt( )-1dt
0
•
Ú
†
= xy( )- 12 RC (0,2 z)
†
= p2 2 z
12 xy( )- 1
2 . n
The asymptotic formula for the complete case
†
RK (x,y) as
†
y Æ 0 is the same as the formulas for
†
RK (x,y) as
†
x Æ •, (3.17) and (3.18). This follows from homogeneity, since
†
RK (x,y) = y- 12 RK (x y ,1). Hence, by letting
†
z = x in (3.17) and (3.18), we obtain a formula and
30
an error bound that are
†
O(log y) and
†
O(y log y), respectively:
†
RK (x,y) = 1p x- 1
2 log 16xy
Ê
Ë Á
ˆ
¯ ˜ + r , (4.4)
where
†
0 < r < 14 p x- 3
2 y log 1+4xy
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á
ˆ
¯ ˜ . (4.5)
For the same reason, the asymptotic formula and bound for
†
RQ(x, y) as
†
y Æ 0 are the same as those for
†
x Æ •, derived in Theorem 5. The formula is
†
O(y- 12 ) and the bound is
†
O(log y). The behavior of
†
RQ(x, y) as
†
x Æ 0 is described by the formulas for
†
y Æ •, (3.26) and (3.27); the formula is
†
O(log x), and the bound is
†
O(x log x). The first three cases shown in Table 4, therefore, are equivalent to the first three cases in Table 3.
In the formula for
†
RD (x,y,z) as
†
z Æ 0, we require that z be the smallest of the three arguments; this allows considerable simplification of the error bound.
Theorem 10. If x, y, and z are positive, and
†
z = min(x, y,z), then
†
RD (x,y,z) = 3 xyz( )- 12 + r, (4.6)
where
†
0 < -r < 6 x + y( )12
xy. (4.7)
Proof. Let
†
g(z) = z12RD (x,y,z) . By homogeneity (1.5),
†
g(z) =RD ( x
z , yz ,1)
z= 3
2 x + zt( )- 12 y + zt( )- 1
2 1+ t( )- 32 dt
0
•
Ú . (4.8)
Hence,
†
g(0) = 3 xy( )- 12 , and this gives the term
†
3 xyz( )- 12 in (4.6). The error term can be bounded
by using
31
†
z12 r = g(z) - g(0) = 3
2 xy( )- 12 1+
ztx
Ê
Ë Á
ˆ
¯ ˜ 1+
zty
Ê
Ë Á
ˆ
¯ ˜
Ê
Ë Á
ˆ
¯ ˜
- 12
-1Ï Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô 1+ t( )- 3
2 dt.0
•
Ú (4.9)
We can bound the quantity in curly brackets in (4.9) by the method used in Theorem 2:
†
1- 1+ztx
Ê
Ë Á
ˆ
¯ ˜ 1+
zty
Ê
Ë Á
ˆ
¯ ˜
Ê
Ë Á
ˆ
¯ ˜
- 12
<1 "t > 0, close for large t,
1- 1+ztx
Ê
Ë Á
ˆ
¯ ˜ 1+
zty
Ê
Ë Á
ˆ
¯ ˜
Ê
Ë Á
ˆ
¯ ˜
- 12
< 12 Ct "t > 0, close for small t,
(4.10)
where
†
C = z x + z y . So for any positive s,
†
0 < -z12r < 3
2 xy( )- 12 C 1
20
s
Ú t 1+ t( )- 32 dt + 1+ t( )- 3
2 dts
•
ÚÊ
Ë Á
ˆ
¯ ˜
= 32 xy( )- 1
2 C 1+ s( )12 + 1+ s( )- 1
2( ) + 2 1+ s( )- 12Ê
Ë Á ˆ
¯ ˜ .
(4.11)
If we let
†
v = 1+ s( )- 12 and minimize (4.11) with respect to v, we obtain
†
v 2 = C 2 + C( ). Substituting this value of v in the quantity in square brackets in (4.11) gives
†
º[ ] = 2C12 2 + C( )
12 < 4C
12 ,
where the latter inequality follows from the assumption that
†
z = min(x, y,z), which implies
†
C < 2.The bound on r, therefore, is
†
0 < -r < 32 xy( )- 1
2 ⋅ 4 1x
+1y
Ê
Ë Á
ˆ
¯ ˜
12
†
=6 x + y( )
12
xy. n
Hence, the bound in this case is actually independent of the argument tending to zero. But since the approximation is
†
O(z- 12 ), the relative error is
†
O(z12 ). A similar relative error is obtained for
the asymptotic approximation of
†
RD (x,y,z) as y tends to zero; that approximation is found below as a special case of
†
RJ (x, y,z,r) as z tends to zero.
32
The behavior of
†
RJ (x, y,z,r) as r tends to zero requires a different approach. If x, y, and z are all nonzero, then
†
RJ (x, y,z,r) has a logarithmic singularity at
†
r = 0; specifically, it is shown below that
†
RJ (x, y,z,r) ~ - 32 xyz( )- 1
2 logr as
†
r Æ 0. However, the integral that defines the error of this approximation is very difficult to bound. Instead we make use of the Duplication Theorem, trading a more complicated approximation for a simpler error bound.
Theorem 11. If r > 0 and x, y, z, ≥ 0 with at most one of x, y, z equal to 0, then
†
RJ (x, y,z,r) = 3RC (a,b) + 2RJ (x + l,y + l,z + l,l) + r, (4.12)
where
†
l = xy( )12 + xz( )
12 + yz( )
12 ,
a = r x12 + y
12 + z
12( ) + xyz( )
12( )
2,
b = r r + l( )2,
and the error term, r, is bounded by
†
0 < -r < 65 rl- 5
2 . (4.13)
Proof. References [13] and [14] give proofs of the Duplication Theorem in the notation used here; it says that
†
RJ (x, y,z,r) = 3RC (a,b) + 2RJ (x + l,y + l,z + l,r + l), (4.14)
where l, a, and b are defined above. If we let
†
r = 0 in the last term in (4.14), we obtain the first term of the approximation in (4.12). Observe that this meets our requirement that the approximation be an elementary function of r, since l does not depend on r, and
†
RC is an elementary function. The error term, then, is
†
r = 2RJ (x + l,y + l,z + l,r + l) - 2RJ (x + l,y + l,z + l,l)
= 3 x + l + t( ) y + l + t( ) z + l + t( )( )- 12 1
r + l + t-
1l + t
Ê
Ë Á
ˆ
¯ ˜ dt
0
•
Ú
= -3rx + l + t( ) y + l + t( ) z + l + t( )( )- 1
2
r + l + t( ) l + t( )dt
0
•
Ú .
(4.15)
33
The last integrand in (4.15) is positive and is easily bounded above by dropping x, y, z, and r:
†
0 < -r < 3r l + t( )- 72 dt = 6
50
•
Ú rl- 52 ,
This is the error bound claimed in (4.13). n
If
†
xyz is nonzero, then as r tends to zero, a tends to
†
xyz , and b tends to
†
rl2 ; it is easy to show from (1.11) that the
†
RC term thus agrees with the
†
- 32 xyz( )- 1
2 logr term mentioned earlier, and that the difference is O(1). Since the
†
RJ function on the right hand side of (4.12) is also O(1) as
†
r Æ 0,
†
RJ (x, y,z,r) = 32 xyz( )- 1
2 logr + O(1), (4.16)
which may be more convenient to use than Theorem 11 in situations where an error bound is not required.
If one of x, y, z is zero, then we have the complete case,
†
RM . Theorem 11 yields a result for the complete case
†
RM if we let
†
z = 0 and use (1.10c):
†
RM (x,y,r) = 4p RC (a,b) + 8
3p RD (x + l,y + l,l) + r (4.17)
where
†
l = xy( )12 ,
a = r x12 + y
12( )( )
2,
b = r r + l( )2,
and the error term, r, is bounded by
†
0 < -r < 85p rl- 5
2 . (4.18)
If
†
r = min(x, y,r), then the
†
RC function in (4.17) is an arc cosine (see (1.11)) and can be written as
34
†
RC (a,b) = r x - r( ) y - r( )( )- 12 arccos rx( )
12 + ry( )
12
r + xy( )12
Ê
Ë Á Á
ˆ
¯ ˜ ˜ . (4.19)
Therefore, the complete case of (4.16) is
†
RM (x,y,r) = 2 rxy( )- 12 + O(1),
†
r Æ 0. (4.20)
We now derive the behavior of
†
RJ (x, y,z,r) as
†
z Æ 0.
Theorem 12. If x, y, r > 0 and z ≥ 0, then
†
RJ (x, y,z,r) = 3p4 RM (x,y,r) + r, (4.21)
where
†
0 < -r < 3p2 2 z
12 xy( )- 1
2 r-1. (4.22)
Proof. The
†
RM term follows from (1.10c), with corresponding error
†
-r = RJ (x,y,0,r) - RJ (x,y,z,r)
= 32 t
12 1+ xt( ) 1+ yt( )( )- 1
2 1+ rt( )-1 1- 1+ zt( )- 12( )dt.
0
•
Ú(4.23)
The first factor in brackets can be bounded by
†
xy( )- 12 r-1t- 3
2 , and the second factor can be bounded by
†
zt 2 + zt( ) , as was done in the proof of Theorem 9. Also as in Theorem 9, the bound involves
†
RC (0,2 z) :
†
0 < -r < 32 xy( )- 1
2 r-1 t- 12 t +
2z
Ê
Ë Á
ˆ
¯ ˜
-1
dt0
•
Ú
= 32 xy( )- 1
2 r-1pz2
Ê
Ë Á
ˆ
¯ ˜
12
.
This is the bound claimed in (4.22). n
Since
†
RD (x,y,z) = RJ (x, y,z,z) and
†
RQ(x,z) = RM (x,z,z) , Theorem 12 has the following corollary:
35
†
RD (x,y,z) = 3p4 RQ (x,z) + r, (4.24)
where
†
0 < -r < 3p2 2 z- 3
2 x- 12 y
12 . (4.25)
Finally, we show the behavior of
†
RM (x,y,r) as
†
y Æ 0. It does not appear that the following theorem is a special case of any previous theorem, and its proof provides an illustration of how Wong’s method can be used for small arguments as well as large arguments.
Theorem 13. If x, y, and r are positive, then
†
RM (x,y,r) = 2p r-1 x- 1
2 log 16xy
Ê
Ë Á
ˆ
¯ ˜ - 2RC (x,r)
Ê
Ë Á
ˆ
¯ ˜ + r, (4.26)
where
†
0 < r < 12p yx- 3
2 r-2 r + 2x( ) log 1+4xr
y 2x + r( )
Ê
Ë Á
ˆ
¯ ˜ +1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ . (4.27)
Proof. Let
†
w =1 y . Then as
†
y Æ 0+,
†
w Æ +•, and we can apply Theorem 1 to
†
RM (x,1 w ,r):
†
RM (x,1 w ,r) = 2p t x + t( ) 1
w+ t
Ê
Ë Á
ˆ
¯ ˜
Ê
Ë Á
ˆ
¯ ˜
- 12
0
•
Ú r + t( )-1dt
= 2p w
12 t x + t( )( )- 1
2 r + t( )-1{ } 1+ wt( )- 12{ }dt.
0
•
Ú(4.28)
The quantities in curly brackets in (4.28) are of the form required of
†
f (t) and
†
h(wt) in Theorem 1. Here,
†
h(t) ~ t- 12 as
†
t Æ +• , and
†
f (t) ~ x- 12 r-1t- 1
2 as
†
t Æ 0+ . So
†
u = v = 12 , and the
second case of Theorem 1 applies:
†
RM (x,1 w ,r) = 2p c0( 1
2) + x- 12 r-1 log w + w
12d1(w)( ) , (4.29)
To find
†
c0( 12) , terms with similar coefficients can be combined, as was done in the proof of
Theorem 2. The quantity
†
a0j1 + b0y1 has the same functional form for all t > 0, and we obtain
36
†
c0( 12) = x- 1
2r-1 1+tx
Ê
Ë Á
ˆ
¯ ˜
- 12
t-1 1+tr
Ê
Ë Á
ˆ
¯ ˜ + t 1+ t( )( )- 1
2 -1t
Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt.
0
•
Ú
In the first term of the integrand, we write
†
t-1 1+tr
Ê
Ë Á
ˆ
¯ ˜
-1
= t-1 - r-1 1+tr
Ê
Ë Á
ˆ
¯ ˜
-1
.
It follows from (1.11) that
†
c0( 12) = x- 1
2r-1J - 2r-1RC (x,r),
where
†
J = t-1 1+tx
Ê
Ë Á
ˆ
¯ ˜
- 12
+ t 1+ t( )( )- 12 -
1t
Ê
Ë Á Á
ˆ
¯ ˜ ˜ dt
0
•
Ú . (4.30)
From Gradshteyn and Ryzhik, [6, 2.224 and 2.261], we find
†
J = limb Æ•e Æ0
log 1+ t x( )- 12 -1
1+ t x( )- 12 +1
Ê
Ë Á Á
ˆ
¯ ˜ ˜ + log t
12 1+ t( )
12 + t + 1
2( ) - log tÏ Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô e
b
= log 16x( ).
Therefore, since
†
w =1 y ,
†
c0( 12) + x- 1
2 r-1 logw = -2r-1RC (x,r) + x- 12 r-1 log 16x
yÊ
Ë Á
ˆ
¯ ˜ .
This is the approximation given in (4.25).
The last term in (4.29) is the error term, r; by Theorem 1,
†
r = 2p x- 1
2 r-1 1- 1+tx
Ê
Ë Á
ˆ
¯ ˜
- 12
1+tr
Ê
Ë Á
ˆ
¯ ˜
- 12Ï
Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô
1t
Ê
Ë Á
ˆ
¯ ˜ 1- 1+
1wt
Ê
Ë Á
ˆ
¯ ˜
- 12Ï
Ì Ô
Ó Ô
¸ ˝ Ô
˛ Ô dt.
0
•
Ú (4.31)
37
By inequality (3.52), the first term in curly brackets is bounded by
†
1 2x( ) +1 r( )t for all t > 0. With this bound, the rest of the proof is similar to that for Theorem 2: in equations (3.11)!–!(3.16), replace z by w and
†
12 x + y( ) by
†
1 2x( ) +1 r . n
By using homogeneity, Theorems 2 – 13 can be used to derive asymptotic behavior when more than one variable tends to zero or infinity. For example,
†
RF (x, y,z) = z- 12 RF (x z ,y z ,1), so the
case of both x and y ending to zero with
†
x y fixed is essentially the same as z tends to infinity, which is the case described by Theorem 2. Many additional formulas are therefore easily derived from the theorems here together with (1.5). Still other cases of asymptotic behavior admittedly are not considered here at all, such as when two arguments tend to zero at different rates.
38
ACKNOWLEDGMENTS
The author is indebted to Professor B. C. Carlson for his guidance, direction, and many hours of consultation in the preparation of this work. The author also wishes to thank his wife, Denise, without whose patience, suggestions, and assistance the preparation of the manuscript would not have been possible.
39
LITERATURE CITED
1. Byrd, P. F. and Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Physicists. Berlin, Germany: Springer-Verlag, 1954.
2. Carlson, B. C. “Computing elliptic integrals by duplication.” Numerische Mathematik 33, (1979), 1–16.
3. !!!!!!!!!!. “Lauricella’s hypergeometric function
†
FD .” Journal of Mathematical Analysis and Applications, 7 (1963), 452–470.
4. !!!!!!!!!!. Special Functions of Applied Mathematics. New York, New York: Academic Press, 1977.
5. !!!!!!!!!! and Notis, E. M. “Algorithms for incomplete elliptic integrals.” ACM Transactions on Mathematical Software, 3 (1981), 398–403.
6. Gradshteyn, I. S. and Ryzhik, I. W. Table of Integrals, Series, and Products. New York, New York: Academic Press, 1965.
7. Gröbner, W. and Hofreiter, N. Integraltafel. 2nd ed. Vienna, Austria: Springer, 1958.
8. Kaplan, E. L. “Auxiliary table for the incomplete elliptic integrals.” Journal of Mathematics and Physics, 27 (1948), 11–36.
9. Legendre, A. M. Traite des Fonctions Elliptiques. Vol. I. Paris, France: Imprimerie de Huzard-Courcier, 1825.
10. Nellis, W. J. “Tables of elliptic integrals.“ Master’s thesis. Iowa State University, Ames, Iowa, 1965.
11. !!!!!!!!!! and Carlson, B. C. “Reduction and evaluation of elliptic integrals.” Mathematics of Computation, 20 (1966), 223–231.
12. Wong, R. “Error bounds for asymptotic expansions of integrals.” SIAM Review, 4 (1980), 401–435.
13. !!!!!!!!!!. “Explicit error terms for asymptotic expansions of Mellin convolutions.” Journal of Mathematical Analysis and Applications, 72 (1979), 740–756.
14. Zill, D. G. “Elliptic integrals of the third kind.” Ph.D. dissertation. Iowa State University, Ames, Iowa, 1967.
15. !!!!!!!!!! and Carlson, B. C. “Symmetric elliptic integrals of the third kind.” Mathematics of Computation, 109 (1970), 199–214.
40
APPENDIX A:NUMERICAL EXAMPLES
Behavior for Large Arguments
The following examples illustrate use of the asymptotic formulas, letting one argument grow large and leaving the others fixed. The error and the bound on the error are scaled by an appropriate function of the argument for clarity. Examples were originally calculated (1982) on a TRS-80 Model III microcomputer using the subroutines in Appendix B. [Note: recalculated to slightly higher precision in 2002; those results are shown here.]
Example 1.
†
RF (x, y,z) as
†
z Æ • (See equations (3.1) and (3.2)).
11.097410.00330.0008198739460.000819874130100000000
10.8792!9.62890.0022285983930.002228603301 10000000
10.5883!9.12990.0058961543670.005896280502 1000000
10.1812!8.43480.0150045705360.015007641418 100000
!9.5733!7.42070.0359356927430.036004039969 10000
!8.5970!5.93500.0772315713630.078528034442 1000
!7.1319!4.06180.1290984181310.147803766237 100
!7.3123!2.56260.0441743736990.230768363604 10
Bound/
†
z- 32 log zError/
†
z- 32 log zAsymp. Approx.
†
RF (1,100,z)z
0.14120.13020.0010596634730.001059663476100000000
0.14350.13100.0029868794580.002986879525 10000000
0.14660.13200.0082940496410.008294051463 1000000
0.15090.13340.0225873811890.022587429752 100000
0.15740.13550.0599146454710.059915893450 10000
0.16820.13910.1530596778890.153090053461 1000
0.18980.14670.3688879454110.369563736298 100
0.25590.17590.8024554388690.815264309589 10
Bound/
†
z- 32 log zError/
†
z- 32 log zAsymp. Approx.
†
RF (0,1,z)z
The last of the three examples illustrates the approximation as applied to the complete elliptic integral
†
RK (see equation (1.10a)).
41
Example 2.
†
RD (x,y,z) as
†
z Æ • (See equations (3.20) and (3.21)).
1.86901.68562.71855042906D-112.71855045195D-11100000000
1.89491.68537.50459925404D-107.50460011304D-10 10000000
1.92941.68502.02777489387D-082.02777722085D-08 1000000
1.97781.68455.32017523398D-075.32023656151D-07 100000
2.05041.68381.33699936511D-051.33715445287D-05 10000
2.17151.68413.13575121391D-043.13942995963D-04 1000
2.41471.69686.46223837212D-036.54037984912D-03 100
3.16621.85949.51327193853D-021.08671998114D-01 10
Bound/
†
z- 52 log zError/
†
z- 52 log zAsymp. Approx.
†
RD (0.5,1,z)z
99.876885.9178!2.15962183809D-112.15962342754D-11100000000
97.912881.9603!5.73711188297D-105.73715365807D-10 10000000
95.294376.6891!1.46884631019D-081.46895226012D-08 1000000
91.630469.3576!3.55268786291D-073.55521296800D-07 100000
86.159858.7513!7.78070782293D-067.83481978704D-06 10000
77.373443.7253!1.36826384284D-041.46377845226D-04 1000
64.187426.3830!8.72952543947D-042.08793585282D-03 100
65.810714.1025-8.16160177221D-022.10703863273D-02 10
Bound/
†
z- 52 log zError/
†
z- 52 log zAsymp. Approx.
†
RD (1,100,z)z
1.27071.13152.87899041993D-112.87899044771D-11100000000
1.29161.13248.01195507875D-108.01195565593D-10 10000000
1.31931.13362.18821489203D-082.18821645819D-08 1000000
1.35821.13545.82753105868D-075.82757239384D-07 100000
1.41651.13801.49743936413D-051.49754418184D-05 10000
1.51371.14333.64310703862D-043.64560451730D-04 1000
1.70861.16168.06663836234D-038.12013278171D-03 100
2.30301.29551.45868301856D-011.55301287972D-01 10
Bound/
†
z- 52 log zError/
†
z- 52 log zAsymp. Approx.
†
RD (0,1,z)z
The latter case represents the complete elliptic integral,
†
RQ(y,z) (see equation (1.10b)).
42
Example 3.
†
RD (x,y,z) as
†
y Æ • (See equations (3.29) and (3.30)).
-0.7915-0.73091.75735931288D-041.75735917823D-04100000000
-0.7975-0.72825.55725809601D-045.55725438440D-04 10000000
-0.8054-0.72461.75735931288D-031.75734930267D-03 1000000
-0.8164-0.71955.55725809601D-035.55699615292D-03 100000
-0.8331-0.71191.75735931288D-021.75670359241D-02 10000
-0.8608-0.70015.55725809601D-025.54196522483D-02 1000
-0.9171-0.68331.75735931288D-011.72589208589D-01 100
-1.1010-0.70105.55725809601D-015.04685378040D-01 10
Bound/
†
y- 32 log yError/
†
y- 32 log yAsymp. Approx.
†
RD (0.5,y,1)y
Example 4.
†
RJ (x, y,z,r) as
†
r Æ • (See equations (3.43) and (3.44)).
42.743430.6887 4.74430608709D-094.74430685431D-09200000000
42.743430.6522 4.74070355714D-084.74071122190D-08 20000000
42.743430.5375!4.72931135717D-074.72938770827D-07 2000000
42.743430.1807!4.69328605771D-064.69404057569D-06 200000
42.743429.1097!4.57936405801D-054.58664148600D-05 20000
42.743426.2034!4.21911106336D-044.28461949663D-04 2000
42.743420.0503!3.07989106682D-033.58114814640D-03 200
42.743411.7466-5.22638880897D-032.41399893344D-02 20
Bound/
†
r-2Error/
†
r-2Asymp. Approx.
†
RJ (0.5,1,100,r)r
43
Example 5.
†
RJ (x, y,z,r) as
†
z Æ • (See equations (3.54) and (3.55)).
-0.6978-0.54924.22695173799D-064.22694162218D-06100000000
-0.6903-0.52051.33667950517D-051.33665297686D-05 10000000
-0.6804-0.48234.22695173799D-054.22628544705D-05 1000000
-0.6664-0.42931.33667950517D-041.33511662486D-04 100000
-0.6456-0.35354.22695173799D-044.19439695694D-04 10000
-0.6114-0.25011.33667950517D-031.28204386287D-03 1000
-0.5505-0.14024.22695173799D-033.58114814601D-03 100
-0.4842-0.06901.33667950517D-028.34460975704D-03 10
Bound/
†
z- 32 log zError/
†
z- 32 log zAsymp. Approx.
†
RJ (0.5,1,z,200)z
-0.8115-0.73712.03902145672D-042.03902132094D-04100000000
-0.8203-0.73536.44795200120D-046.44794825353D-04 10000000
-0.8320-0.73282.03902145672D-032.03901133247D-03 1000000
-0.8484-0.72946.44795200120D-036.44768645211D-03 100000
-0.8730-0.72432.03902145672D-022.03835434264D-02 10000
-0.9140-0.71656.44795200120D-026.43230107541D-02 1000
-0.9966-0.70702.03902145672D-012.00646439574D-01 100
-1.2536-0.74046.44795200120D-015.90885028621D-01 10
Bound/
†
z- 32 log zError/
†
z- 32 log zAsymp. Approx.
†
RJ (0.5,1,z,0.75)z
44
Behavior for Small Arguments
Example 6.
†
RF (x, y,z) as
†
z Æ 0+ (See equations (4.1) and (4.2)).
-1.5708-1.41411.8540746877301.8539332694500.00000001
-1.5708-1.41381.8540746877301.8536275987260.00000010
-1.5708-1.41291.8540746877301.8526618129700.00000100
-1.5708-1.41001.8540746877301.8496160032190.00001000
-1.5708-1.40081.8540746877301.8400662073000.00010000
-1.5708-1.37291.8540746877301.8106607481070.00100000
-1.5708-1.29191.8540746877301.7248857627940.01000000
-1.5708-1.09221.8540746877301.5086951632150.10000000
Bound/
†
z12Error/
†
z12Asymp. Approx.
†
RF (0.5,1,z)z
-0.1111-0.10003.6956373629903.6955373680690.00000001
-0.1111-0.10003.6956373629903.6953211860130.00000010
-0.1111-0.09993.6956373629903.6946378765020.00000100
-0.1111-0.09983.6956373629903.6924801546760.00001000
-0.1111-0.09953.6956373629903.6856878285000.00010000
-0.1111-0.09843.6956373629903.6645121825410.00100000
-0.1111-0.09523.6956373629903.6004039969000.01000000
-0.1111-0.08673.6956373629903.4215774848450.10000000
Bound/
†
z12Error/
†
z12Asymp. Approx.
†
RF (1,100,z)z
Example 7.
†
RD (x,y,z) as
†
z Æ 0+ (See equations (4.6) and (4.7)).
-14.697-8.1034.24264068712D+044.24183042805D+040.00000001
-14.697-8.1001.34164078650D+041.34083080248D+040.00000010
-14.697-8.0914.24264068712D+034.23454953303D+030.00000100
-14.697-8.0641.34164078650D+031.33357698579D+030.00001000
-14.697-7.9784.24264068712D+024.16285636205D+020.00010000
-14.697-7.7191.34164078650D+021.26444771544D+020.00100000
-14.697-6.9954.24264068712D+013.54313795592D+010.01000000
-14.697-5.3671.34164078650D+018.04918445710D+000.10000000
BoundErrorAsymp. Approx.
†
RD (0.5,1,z)z
45
-0.603-0.3053.000000000000+032.99969523223D+030.00000001
-0.603-0.3059.48683298505D+029.48378595774D+020.00000010
-0.603-0.3043.000000000000+022.99695504634D+020.00000100
-0.603-0.3049.48683298505D+019.45644868982D+010.00001000
-0.603-0.3023.00000000000D+012.96982020527D+010.00010000
-0.603-0.2969.48683298505D+009.19132357410D+000.00100000
-0.603-0.2773.00000000000D+002.72275349479D+000.01000000
-0.603-0.2329.48683298505D-017.16729209556D-010.10000000
Bound/
†
y12Error/
†
y12Asymp. Approx.
†
RD (1,100,z)z
Example 8.
†
RD (x,y,z) as
†
y Æ 0+ (See equations (4.24) and (4.25)).
-4.712-4.2423.0205847775223.0201605690680.00000001
-4.712-4.2413.0205847775223.0192436927340.00000010
-4.712-4.2373.0205847775223.0163476919960.00000100
-4.712-4.2253.0205847775223.0072237691730.00001000
-4.712-4.1883.0205847775222.9787076094860.00010000
-4.712-4.0743.0205847775222.8917678394870.00100000
-4.712-3.7493.0205847775222.6456596725770.01000000
-4.712-2.9903.0205847775222.0751106804610.10000000
Bound/
†
y12Error/
†
y12Asymp. Approx.
†
RD (0.5,y,1)y
-0.00333-0.003008.12013278171D-038.11983279736D-030.00000001
-0.00333-0.003008.12013278171D-038.11918425284D-030.00000010
-0.00333-0.003008.12013278171D-038.11713435272D-030.00000100
-0.00333-0.003008.12013278171D-038.11066156214D-030.00001000
-0.00333-0.002988.12013278171D-038.09028821842D-030.00010000
-0.00333-0.002958.12013278171D-038.02679722520D-030.00100000
-0.00333-0.002858.12013278171D-037.83481978704D-030.01000000
-0.00333-0.002598.12013278171D-037.30147713135D-030.10000000
Bound/
†
y12Error/
†
y12Asymp. Approx.
†
RD (1,y,100)y
46
Example 9.
†
RJ (x, y,z,r) as
†
r Æ 0+ (See equations (4.12) and (4.13)).
-0.16994-0.1234242.75751635675242.7575163555180.00000001
-0.16994-0.1234237.11736505989937.1173650475570.00000010
-0.16994-0.1234231.47726819774231.4772680743200.00000100
-0.16994-0.1234225.83761677025325.8376154728120.00001000
-0.16994-0.1234220.20142980848820.2014174667150.00010000
-0.16994-0.1233814.59004290203414.5899195195900.00100000
-0.16994-0.12303 9.130975682636 9.1297453749410.01000000
-0.16994-0.11963 4.363920743258 4.3519576097630.10000000
Bound/rError/rAsymp. Approx.
†
RJ (0.5,1,0.75,r)r
-0.013028-0.00825012.02918937156212.0291893714800.00000001
-0.013028-0.00825010.48456977822110.4845697773960.00000010
-0.013028-0.008250 8.939961028244 8.9399610199950.00000100
-0.013028-0.008250 7.395441327534 7.3954412382580.00001000
-0.013028-0.008249 5.851618233293 5.8516174083510.00010000
-0.013028-0.008249 4.312830987027 4.3128227321180.00100000
-0.013028-0.008240 2.805575055074 2.8054926529360.01000000
-0.013028-0.008158 1.447433860331 1.4466181072050.10000000
Bound/rError/rAsymp. Approx.
†
RJ (0.5,1,10,r)r
Example 10.
†
RJ (x, y,z,r) as
†
z Æ 0+ (See equations (4.21) and (4.22)).
-0.02356-0.021212.62404569133D-022.62383357962D-020.00000001
-0.02356-0.021212.62404569133D-022.62337507412D-020.00000010
-0.02356-0.021192.62404569133D-022.62192640138D-020.00000100
-0.02356-0.021152.62404569133D-022.61735774558D-020.00001000
-0.02356-0.021012.62404569133D-022.60303363534D-020.00010000
-0.02356-0.020592.62404569133D-022.55893114141D-020.00100000
-0.02356-0.019372.62404569133D-022.43032153547D-020.01000000
-0.02356-0.016372.62404569133D-022.10646526959D-020.10000000
Bound/
†
z12Error/
†
z12Asymp. Approx.
†
RJ (0.5,1,z,200)z
47
-6.2832-5.65613.762536201583.761970595260.00000001
-6.2832-5.65443.762536201583.760748137960.00000010
-6.2832-5.64903.762536201583.756887247590.00000100
-6.2832-5.63193.762536201583.744726436080.00001000
-6.2832-5.57883.762536201583.706748206990.00010000
-6.2832-5.41693.762536201583.591237398600.00100000
-6.2832-4.96013.762536201583.266526783400.01000000
-6.2832-3.90733.762536201582.526927198960.10000000
Bound/
†
z12Error/
†
z12Asymp. Approx.
†
RJ (0.5,1,z,0.75)z
Example 11.
†
RM (x,y,r) as
†
y Æ 0+. (See equations (4.26) and (4.27)).
1.57231.555123.8100486825!23.8100489690!0.00000001
1.56951.549920.8783062870!20.8783087851!0.00000010
1.56951.543017.9465638914!17.9465852083!0.00000100
1.56071.533315.0148214959!15.0149980258!0.00001000
1.55301.518912.0830791042!12.0844780740!0.00010000
1.54071.4968 9.15133670490 9.161675961310.00100000
1.52301.4690 6.21959430938 6.287243181320.01000000
1.59161.5395 3.28785191386 3.642343394850.10000000
Bound/
†
y log yError/
†
y log yAsymp. Approx.
†
RM (1,y,0.5)y
0.020350.019931.2961942893091.2961942929800.00000001
0.020530.020051.1496071695331.1496072018450.00000010
0.020760.020211.0030200497571.0030203289020.00000100
0.021100.020430.8564329299810.8564352816790.00001000
0.021600.020760.7098458102050.7098649304540.00010000
0.022430.021320.5632586904290.5634059833640.00100000
0.024120.022530.4166715706530.4177092739220.01000000
0.029570.027070.2700844508770.2763184186760.10000000
Bound/
†
y log yError/
†
y log yAsymp. Approx.
†
RM (1,y,10)y
48
APPENDIX B:PROGRAMS USED TO CREATE EXAMPLES
The following programs are based on algorithms described in [2], and provide a method of obtaining numerical values for the basis functions
†
RF ,
†
RJ , and
†
RD . In all cases, the Duplication Theorem is used until all arguments are equal within some specified tolerance. The arguments are specified as assignment statements near the beginning of each program. Since
†
RD is easily obtained as a special case of
†
RJ by letting
†
r = z , there is no separate program for
†
RD . Minor changes in the initial arguments and the column headers enable these programs to generate all of the examples in Appendix A.
At the time of writing, Microsoft BASIC is perhaps the most widely implemented computer language in existence, owing to its near-universal use in personal computers; hence it is the language chosen here. Applications requiring speed or large numbers of function values should use a version of these algorithms written in a compiled language such as FORTRAN. Such a version is available as Algorithm 577 the ACM library [S21]; see reference [5].
49
Program 1. Computation of
†
RF (X 0,Y0,Z1) for
†
Z1 = 10, 100, ... 100000000.
100 '''''' Use double precision variables (14 decimals or better).110 x0 = 0.5 'Initial values of x and y.120 y0 = 1130 '''''' Print column headers.140 print " 3/2"150 print " z R (0.5,1,z) Asymp. Approx. Error and Bound/(z log z)"160 print " F"170 print "--------- -------------- -------------- ------ ------"180 for z0 = 1 to 8 '''''' z0 is the power of ten for z.190 z1 = 10^z0200 x1 = x0210 y1 = y0220 la = sqr(x1*y1)+sqr(x1*z1)+sqr(y1*z1)'''''' Form lambda230 mu = (x1+y1+z1)/3 '''''' Proceed with duplication theorem.240 x2 = (x1+la)/4250 y2 = (y1+la)/4260 z2 = (z1+la)/4270 x3 = 1-x1/mu280 y3 = 1-y2/mu290 z3 = 1-z2/mu300 s2 = (x3*x3+y3*y3+z3*z3)/4310 s3 = (x3*x3*x3+y3*y3*y3+z3*z3*z3)/6320 rf = (1+s2/5+s3/7+s2*s2/6+s2*s3*3/11+5*s2*s2*s2/26+3*s3*s3/26)/sqr(mu)330 x1 = x2 '''''' "New" arguments become "old".340 y1 = y2350 z1 = z2360 if abs((rf-r0)/rf) > 1.000000E-15 then r0 = rf : goto 220370 ''''''Relative error small; stop iterating.380 z1 = 10^z0 '''''' Reconstruct the z argument.390 '''''' Now compute the first term of the asymptotic expansion.400 t1 = sqr(z1)410 t2 = sqr(x0)420 es = log(4*t1/(t2+sqr(y0)))/sqr(z1)430 '''''' Print value, approximation, error, and bound.440 print using "######### ";z1;450 print using "##.############ ";rf;460 print using "##.############ ";es;470 print using "##.#### ";(rf-es)*z1^1.5/log(z1);480 print using "##.####";(x0+y0)/8*(log(1+4*z1/(x0+y0))+1)/log(z1)490 next z0500 end
50
The second program contains subroutines for
†
RC and
†
RJ .
Program 2.
†
RJ (X 0,Y0,Z0,P1) for
†
P1 = 1, 0.1, 0.01, ... , 0.000000001.
100 '''''' Use double precision variables and intrinsics (14 decimals or better).110 pi = 4*arctan(1)120 x0 = 0.5130 y0 = 1140 z0 = 0.75150 print " p RJ(.5,1,.75,p) Asymp. Approx. Error/p Bound/p"170 print "---------- --------------- --------------- ------- -------"180 for n = 1 to 8190 p0 = 10^-n200 x1 = x0210 y1 = y0220 z1 = z0230 p1 = p0240 gosub 610250 rg = rj260 l0 = sqr(x0*y0)+sqr(x0*z0)+sqr(y0*z0)270 x1 = x0+l0280 y1 = y0+l0290 z1 = z0+l0300 p1 = l0310 gosub 610320 p1 = p0330 al = (p0*(sqr(x0)+sqr(y0)+sqr(z0))+sqr(x0*y0*z0))^2340 be = p0*(p0+l0)^2350 gosub 470360 es = 2*rj+3*rc370 rj = rg380 print using "#.########";p0;390 print using " ##.############";rj;400 print using " ##.############";es;410 print using " ##.####";(rj-es)/p0;420 print using " ##.####";-6/5*l0^-2.5430 next n440 end450 '460 '''''' Compute RC(al,be):470 r1 = 1480 a1 = al490 b1 = be500 la = 2*sqr(a1*b1)+b1510 a2 = (a1+la)/4520 b2 = (b1+la)/4530 mv = (a1+2*b1)/3540 s1 = (b1-a1)/(3*mv)550 rc = (1+3*s1^2/10+s1^3/7+3*s1^4/8+9*s1^5/22+159*s1^6/208)/sqr(mv)560 a1 = a2570 b1 = b2580 if abs((rc-r1)/rc) > 1.000000E-15 then r1 = rc : goto 500 else return590 '600 '''''' Compute RJ(x1,y1,z1,p1):610 s = 0620 fa = 3630 rj = 0640 r0 = 1650 la = sqr(x1*y1)+sqr(x1*z1)+sqr(y1*z1)660 mu = (x1+y1+z1+2*p1)/5670 x2 = (x1+la)/4
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680 y2 = (y1+la)/4690 z2 = (z1+la)/4700 p2 = (p1+la)/4710 x3 = 1-x1/mu720 y3 = 1-y1/mu730 z3 = 1-z1/mu740 p3 = 1-p1/mu750 s2 = (x3^2+y3^2+z3^2+2*p3^2)/4760 s3 = (x3^3+y3^3+z3^3+2*p3^3)/6770 al = ((sqr(x1)+sqr(y1)+sqr(z1))*p1+sqr(x1*y1*z1))^2780 be = p1*(p1+la)^2 : gosub 470790 s = s+fa*rc800 fa = fa/4810 rj = 1+3*s2/7+s3/3+3*s2^2/22+s2*s3*3/13+3*s4/11+3*s5/13-s2^3/10820 rj = s+fa/4*(rj+3*s3^2/10+3*s2*s4/5)*mu^-1.5830 x1 = x2840 y1 = y2850 z1 = z2860 p1 = p2870 if abs((rj-r0)/rj) > 1.000000E-15 then r0 = rj : goto 650 else return
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