On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. II: Complex variable
This paper considers the application of the Hadamard expansions to the hyperasymptotic evaluation of Laplace integrals of the form \[ I(x)\equiv\int_C e^{zp(t)}f(t)\,dt \] for large \(z\), where \(z\) is a complex variable and the integration path \(C\) is a finite or semi-infinite interval in the complex plane. In Part I of this work [J. Comput. Appl. Math. 167, No. 2, 293-319 (2004; Zbl 1052.41016)] the author improves the convergence properties of the standard Hadamard expansion for real \(z\). In this paper (part II) the author extends his analysis to the case of complex \(z\). The author shows how the resulting Hadamard expansions can be employed in the neighbourhood of a Stokes line. Applying this modification of the Hadamard method, the author obtains a fast convergent expansion of the the Airy function. Several numerical examples illustrate the accuracy and the computational improvement achieved with this new method.
scientific article; zbMATH DE number 2085544
scientific article