López, J. L. Uniform asymptotic expansions of symmetric elliptic integrals. (English) Zbl 1086.41015 Constructive Approximation 17, No. 4, 535-559 (2001). The article under review deals with complete convergent expansions of the three symmetric standard elliptic integrals \[ \begin{aligned} R_F(x,y,z)&={1\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)}}, \\ R_D(x,y,z)&={3\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)^3}}, \\ R_J(x,y,z,p)&={3\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)}(t+p)}, \end{aligned} \]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 \[ \begin{aligned} R_F(x,az,bz)= &{1\over 2} \sqrt{{\pi \over abx}}\, \sum_{k=0}^{n-1}\biggl[ {(k-1)! A_k(a,b)x^k \over \Gamma(k+{1\over 2})z^k} +{({1\over 2})_k \sqrt{\pi b}\, x^{k+1/2} \over k!a^k z^{k+1/2}} F\biggl( {k+{1\over 2},{1\over 2}\atop 1}\biggm| 1-{b\over a}\biggr)\biggr]\\ &+R_n(x,az,bz), \end{aligned} \tag{1} \]for \(0\leq x\leq az\leq bz\), \(0<az\) and \(n\) a positive integer. Here \(F={_2}F_1\) is the Gauss hypergeometric function, \((a)_n=a(a+1)\cdots (a+n-1)=\Gamma(a+n)/\Gamma(a)\) is a shifted factorial, \[ A_k(a,b)=-\sum_{j=0}^{k-1} {({1\over 2})_j({1\over 2})_{k-j-1} \over j!(k-j-1)! a^j b^{k-j-1}} \](\(A_0(a,b)=0\)), and \[ 0\leq -R_n(x,az,bz)\leq {\sqrt{\pi}(n-1)!| A_n(a,b)| x^n \over 2\sqrt{abx}\,\Gamma(n+{1\over 2})z^n}. \]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 \(k\geq 1\) \[ A_k(a,b)=-{b^{1-k}({1\over 2})_{k-1}\over (k-1)!} F\biggl({1-k,{1\over 2}\atop {3\over 2}-k}\biggm| {b\over a}\biggr) =-{b^{1-k}\Gamma(k+{1\over 2})\over \sqrt{\pi} (k-1)!(k-{1\over 2})} F\biggl({1-k,{1\over 2}\atop {3\over 2}-k}\biggm| {b\over a}\biggr), \] cancelling several terms in (1). Reviewer: S. Ole Warnaar (Melbourne) Cited in 11 Documents MSC: 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33E05 Elliptic functions and integrals Keywords:elliptic integrals; uniform asymptotic expansions; distributional approach Citations:Zbl 0679.41001; Zbl 0794.41021; Zbl 0606.33004 PDFBibTeX XMLCite \textit{J. L. López}, Constr. Approx. 17, No. 4, 535--559 (2001; Zbl 1086.41015) Full Text: DOI Digital Library of Mathematical Functions: §19.27(vi) Asymptotic Expansions ‣ §19.27 Asymptotic Approximations and Expansions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals