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Uniform asymptotic expansions of symmetric elliptic integrals. (English) Zbl 1086.41015

The article under review deals with complete convergent expansions of the three symmetric standard elliptic integrals

R F (x,y,z)=1 2 0 dt (t+x)(t+y)(t+z),R D (x,y,z)=3 2 0 dt (t+x)(t+y)(t+z) 3 ,R J (x,y,z,p)=3 2 0 dt (t+x)(t+y)(t+z)(t+p),

for x,y,z,(p) nonnegative, distinct real numbers.

Using the distributional approach [R. Wong, Asymptotic Approximations of Integrals. (Academic Press, NY) (1989; Zbl 0679.41001)] seven convergent expansions for the above elliptic integrals are proved. A typical example of such an expansion reads

R F (x,az,bz)=1 2π abx k=0 n-1 (k-1)!A k (a,b)x k Γ(k+1 2)z k + (1 2) k πbx k+1/2 k!a k z k+1/2 F k+1 2,1 2 1 | 1 - b a+R n (x,az,bz),(1)

for 0xazbz, 0<az and n a positive integer. Here F= 2 F 1 is the Gauss hypergeometric function, (a) n =a(a+1)(a+n-1)=Γ(a+n)/Γ(a) is a shifted factorial,

A k (a,b)=- j=0 k-1 (1 2) j (1 2) k-j-1 j!(k-j-1)!a j b k-j-1

(A 0 (a,b)=0), and

0-R n (x,az,bz)π(n-1)!|A n (a,b)|x n 2abxΓ(n+1 2)z n ·

The seven expansions derived in the paper generalize earlier first-order approximations of B. C. Carlson and J. C. Gustafson [SIAM J. Math. Anal. 25, 288–303 (1994; Zbl 0794.41021)] and complement expansions by B. C. Carlson [Rend. Semin. Mat., Torino, Fasc. Spec., 63–89 (1985; Zbl 0606.33004)].

Remark: It is somewhat unfortunate that the author has not always simplified his main results. For example, the expansion (1) does of course only depend on a, b and z through the products az and bz. Moreover, in the second term in square brackets the hypergeometric function has been used but not in the first term, despite the fact that for k1

A k (a,b)=-b 1-k (1 2) k-1 (k-1)!F1-k,1 2 3 2-k | b a=-b 1-k Γ(k+1 2) π(k-1)!(k-1 2)F1-k,1 2 3 2-k | b a,

cancelling several terms in (1).

MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33E05Elliptic functions and integrals