Asymptotics of the first Appell function F-1 with large parameters II

作者：Lopez Jose L^*; Pagola Pedro; Perez Sinusia Ester

来源：Integral Transforms and Special Functions, 2013, 24(12): 982-999.

DOI：10.1080/10652469.2013.798658

• first Appell hypfunction
• asymptotic expansion
• Laplace%26apos
• s method

In this paper we continue the investigation initiated in Lopez et al. (Asymptotics of the first Appell function F-1 with large parameters. Integral Transforms Spec Funct. in press. Available from: http://www.tandfonline.com/doi/pdf/10.1080/10652469.2012.753071#.UZs_0xWbvmR). We consider the asymptotic behaviour of the first Appell function F-1(a, b, b, c; x, y) when several of its parameters a, b, b, c are large. It is shown in Lopez et al. (Asymptotics of the first Appell function F-1 with large parameters. Integral Transforms Spec Funct. in press) that there are 23 possible combinations of large parameters that require an asymptotic analysis. In Lopez et al. (Asymptotics of the first Appell function F-1 with large parameters. Integral Transforms Spec Funct. in press) we have analysed three of them. We analyse here other four asymptotic regions: (i) large positive b and b, (ii) large negative b and b, (iii) large positive b, b and c, and (iv) large negative b and b and large positive c. We derive complete asymptotic expansions of the F-1 function for every one of these four regions, applying Laplace%26apos;s method and using Wojdylo%26apos;s formula for the computation of Laplace%26apos;s coefficients. We consider as a starting point a one-dimensional integral representation of F-1(a, b, b, c; x, y). Numerical experiments are given for the four asymptotic regions analysed.

• 出版日期2013-12-1