On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals—IV: Poles

This paper is one of a series considering the application of Hadamard expansions in the hyperasymptotic evaluation of Laplace-type integrals of the form ∫ C exp { - z ψ ( t ) } f ( t ) d t for large values of | z | . It is shown how the procedure can be employed to deal with the case when the amplitude function f ( t ) possesses poles which may coalesce with a saddle point of the integrand or approach the integration path C. A novel feature introduced here is the reverse-expansion procedure. This results in contributions at each exponential level (after the first) of the expansion in the form of rapidly convergent series, thereby enabling the high-precision evaluation of the above integral in coalescence problems. Numerical examples are given to illustrate the procedure.