On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals: II. complex variable

摘要

In this sequel to Paris (On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals: I. Real variable, submitted for publication), we extend the discussion of the application of Hadamard expansions to the hyperasymptotic evaluation of Laplace-type integrals ∫ from x=C of e~(zp(t))f(t)dt (|z|→∞) to complex values of the variable z. The integration contour C can be either a finite or an infinite path in the complex plane. We consider examples of linear, quadratic and cubic phase functions p(t) and show how the resulting Hadamard expansions can be employed in the neighbourhood of a Stokes line. Numerical examples are given to illustrate the accuracy that can be achieved with this new procedure.