Asymptotic properties of special function solutions of Painlev\'e III equation for fixed parameters

TL;DR

This paper derives the small and large x asymptotics of special solutions to the Painlevé-III equation using Toeplitz determinants and contour integrals, providing new insights into their behavior across the complex plane.

Contribution

It introduces a novel method for asymptotic analysis of Painlevé-III solutions via Toeplitz determinants and contour integrals, extending results to the entire complex plane.

Findings

Asymptotics for small x derived using elementary methods.

Asymptotics for large x obtained through analytic continuation.

Representation useful for numerical computation of solutions.

Abstract

In this paper, we compute the small and large $x$ asymptotics of the special function solutions of Painlev\'e-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained in arXiv:nlin/0302026. Toeplitz determinants are rewritten as multiple contour integrals using Andr\`eief's identity. The small and large $x$ asymptotics are obtained using elementary asymptotic methods applied to the multiple contour integral. The asymptotics is extended to the whole complex plane using analytic continuation formulas for Bessel functions. The claimed result has not appeared in the literature before. We note that Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlev\'e-III equation.