A convergent and asymptotic Laplace method for integrals.

Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F(z)≔∫0∞e−zf(t)g(t)dt for large values of the parameter z. They are useful tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where, by the use of inverse factorial series, a new asymptotic as well as convergent expansion of F(z), for the particular case f(t)=t, was derived. In this paper we go some steps further and investigate a modification of the Laplace’s method for F(z), with a general phase function f(t), to derive asymptotic expansions of F(z) that are also convergent, accompanied by error bounds. An analysis of the remainder of this new expansion shows that it is convergent under a mild condition for the functions f(t) and g(t), namely, these functions must be analytic in certain starlike complex regions that contain the positive axis [0,∞). In many practical situations (in many examples of special functions), the singularities of f(t) and g(t) are off this region and then this method provides asymptotic expansions that are also convergent. We illustrate this modification of the Laplace’s method with the parabolic cylinder function U(a,z), providing an asymptotic expansions of this function for large z that is also convergent.