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Zbl 1086.41015
López, J. L.
Uniform asymptotic expansions of symmetric elliptic integrals. (English)
[J] Constructive Approximation 17, No. 4, 535-559 (2001). ISSN 0176-4276; ISSN 1432-0940/e

The article under review deals with complete convergent expansions of the three symmetric standard elliptic integrals $$ \align R_F(x,y,z)&={1\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)}}, \\ R_D(x,y,z)&={3\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)^3}}, \\ R_J(x,y,z,p)&={3\over 2} \int_0^{\infty} {dt \over \sqrt{(t+x)(t+y)(t+z)}(t+p)}, \endalign$$ for $x,y,z,(p)$ nonnegative, distinct real numbers. \par Using the distributional approach [{\it 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 $$\aligned R_F(x,az,bz)= &{1\over 2} \sqrt{{\pi \over abx}}\, \sum_{k=0}^{n-1}\biggl[ {(k-1)! A_k(a,b)x^k \over \Gamma(k+{1\over 2})z^k} +{({1\over 2})_k \sqrt{\pi b}\, x^{k+1/2} \over k!a^k z^{k+1/2}} F\biggl( {k+{1\over 2},{1\over 2}\atop 1}\biggm\vert 1-{b\over a}\biggr)\biggr]\\ &+R_n(x,az,bz), \endaligned \tag1$$ 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 $$ 0\leq -R_n(x,az,bz)\leq {\sqrt{\pi}(n-1)!\vert A_n(a,b)\vert x^n \over 2\sqrt{abx}\,\Gamma(n+{1\over 2})z^n}. $$ The seven expansions derived in the paper generalize earlier first-order approximations of {\it B. C. Carlson} and {\it J. C. Gustafson} [SIAM J. Math. Anal. 25, 288--303 (1994; Zbl 0794.41021)] and complement expansions by {\it 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 $k\geq 1$ $$ A_k(a,b)=-{b^{1-k}({1\over 2})_{k-1}\over (k-1)!} F\biggl({1-k,{1\over 2}\atop {3\over 2}-k}\biggm\vert {b\over a}\biggr) =-{b^{1-k}\Gamma(k+{1\over 2})\over \sqrt{\pi} (k-1)!(k-{1\over 2})} F\biggl({1-k,{1\over 2}\atop {3\over 2}-k}\biggm\vert {b\over a}\biggr), $$ cancelling several terms in (1).
[S. Ole Warnaar (Melbourne)]

MSC 2000:: *41A60 Asymptotic problems in approximation
33E05 Elliptic functions and integrals

Keywords: elliptic integrals; uniform asymptotic expansions; distributional approach

Citations: Zbl 0679.41001; Zbl 0794.41021; Zbl 0606.33004

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