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

Deaño, Alfredo (2018) Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane. Symmetry, Integrability and Geometry: Methods and Applications, 14 (107). ISSN 1815-0659. (doi:10.3842/SIGMA.2018.107)

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https://doi.org/10.3842/SIGMA.2018.107

## 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 Painlev´e II differential equation. Using the fact that these tau functions can be written as n × 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.

Item Type: Article 10.3842/SIGMA.2018.107 Painlev´e equations; asymptotic expansions; Airy functions Q Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Alfredo Deano-Cabrera 12 Oct 2018 08:25 UTC 29 May 2019 21:16 UTC https://kar.kent.ac.uk/id/eprint/69536 (The current URI for this page, for reference purposes) https://orcid.org/0000-0003-1704-247X