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

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

来源：Integral Transforms and Special Functions, 2013, 24(9): 715-733.

DOI：10.1080/10652469.2012.753071

• first Appell hypergeometric function
• asymptotic expansion
• Laplace%26apos
• s method

We consider the asymptotic behaviour of the first Appell function F-1(a, b, b, c; x, y) when several of the parameters a, b, b, c are large. We reduce to 23 the possible combinations of large parameters that require an asymptotic analysis. In this paper, we analyse 3 of those 23 asymptotic regions: (i) large positive c, (ii) large positive b and (iii) large positive c and large negative b. We derive complete asymptotic expansions of the F-1 function for every one of these three regions. The starting point is a single integral representation of F-1(a, b, b, c; x, y). We use three different asymptotic techniques: Laplace%26apos;s method and two variants of Laplace%26apos;s method. Analytical and numerical aspects of these formulas are discussed.

• 出版日期2013-9-1