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A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function
Author(s) -
López José L.,
Pagola Pedro J.
Publication year - 2011
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00510.x
Subject(s) - saddle point , mathematics , saddle , hypergeometric function , exponential function , laplace transform , asymptotic expansion , gauss , mathematical analysis , sequence (biology) , function (biology) , generalized hypergeometric function , asymptotic analysis , confluent hypergeometric function , algebraic number , geometry , physics , mathematical optimization , quantum mechanics , evolutionary biology , biology , genetics
In recent works [1] and [2], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [1] and [2] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function 2 F 1 ( a , b , c ; z ) for large b and c with c > b + 1 .