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

• Published in: Journal of computational and applied mathematics Vol. 206; no. 1; pp. 454 - 472
• Main Author: Paris, R.B.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2007; Elsevier
• Subjects: Asymptotics; Exact sciences and technology; Hadamard expansions; Hyperasymptotics; Laplace-type integrals; Mathematical analysis; Mathematics; Measure and integration; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Hyperasymptotics; Asymptotics; Hadamard expansions; Laplace-type integrals; Integration; Numerical analysis; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2006.08.016

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.