On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: real variable
We review and discuss the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals where, for simplicity, in this paper x is restricted to be a positive real variable. The integration path C can be taken over both finite and semi-infinite intervals in the complex plane. In general, these expansions take the form of compound expansions, each associated with a different exponential level, and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor. The early terms in each convergent expansion possess a rapid asymptotic-like decay (when the variable x is large) with late terms that can be transformed into a rapid decay comparable with that of the early terms. The Hadamard expansion of the above integral when the phase function p(t) is linear is shown to depend significantly on the singularity structure of the amplitude function f(t). The application of the theory to Laplace-type integrals with quadratic, cubic and nonpolynomial phase functions is considered; in addition to the amplitude function, the location of the saddle points satisfying p′(t)=0 also plays a role in the detailed structure of the different exponential levels in the resulting Hadamard expansion. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.