A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

Abstract

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ ( z ) for large z and the Gauss hypergeometric function 2 F 1 ( a , b , c ; z ) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ ( z ) is also given.