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Lithuanian Mathematical Journal

, Volume 54, Issue 1, pp 82–105 | Cite as

Exponentially small expansions of the Wright function on the Stokes lines

  • Richard B. Paris
Article

Abstract

We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given.

Keywords

asymptotics exponentially small expansions Wright function Stokes lines 

MSC

33C20 33C70 34E05 41A60 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Engineering, Computing and Applied MathematicsUniversity of Abertay DundeeDundeeUK

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