Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 107, 19 pages      arXiv:1804.00563      https://doi.org/10.3842/SIGMA.2018.107
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Large $z$ Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane

Alfredo Deaño
School of Mathematics, Statistics and Actuarial Science, University of Kent, UK

Received April 17, 2018, in final form September 22, 2018; Published online October 03, 2018

Abstract
In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as $n\times n$ Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as $n$-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane.

Key words: Painlevé equations; asymptotic expansions; Airy functions.

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