The article under review deals with complete convergent expansions of the three symmetric standard elliptic integrals
for 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
for , and a positive integer. Here is the Gauss hypergeometric function, is a shifted factorial,
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 , and through the products and . 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
cancelling several terms in (1).