A simplification of the stationary phase method: application to the Anger and Weber functions

López, José L. ; Pagola, Pedro J.

Electronic transactions on numerical analysis, Tome 46 (2017), pp. 148-161.

Résumé

Summary: The main difficulty in the practical use of the stationary phase method in asymptotic expansions of integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is not explicitly known, and then the computation of those coefficients is cumbersome. Using the factorization of the exponential factor used in previous works of [Tricomi, 1950], [Erdélyi and Wyman, 1963], and [Dingle, 1973], we obtain a variant of the method that avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler and explicit. On the other hand, the asymptotic sequence is as simple as in the standard stationary phase method: inverse powers of the asymptotic variable. New asymptotic expansions of the Anger and Weber functions ${\bf J}_{\lambda x}(x)$ and ${\bf E}_{\lambda x}(x)$ for large positive $x$ and real parameter $\lambda\neq 0$ are given as an illustration.

Détail

Classification : 33B15, 41A60
Keywords: asymptotic expansions, oscillatory integrals, method of the stationary phase, anger and Weber functions