Exponentially small expansions related to the parabolic cylinder function

2019/12/30 by R. B. Paris, Paris, R B
Computer Science · Mathematics · #33C15 #34E05 #39E15 #41A60 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1912.12974

openalex publication_date 2019/12/30 · openalex created_date 2020/01/10 · openalex updated_date 2026/07/28

Abstract

The refined asymptotic expansion of the confluent hypergeometric function M(a,b,z) on the Stokes line arg z=π given in \it Appl. Math. Sci. \bf 7 (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function U(a,z) yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.

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