Explicit error term of the elliptic asymptotics for the first Painlev\'e transcendents

TL;DR

This paper provides an explicit error bound for the elliptic asymptotics of the first Painlevé transcendents, refining the understanding of their behavior near infinity with a precise order estimate.

Contribution

It introduces a concrete error bound for the elliptic asymptotics of the first Painlevé transcendents, improving previous asymptotic representations.

Findings

Explicit error bound of order $-1$ for the elliptic asymptotics

Refined asymptotic representation near infinity

Enhanced precision in Painlevé transcendents analysis

Abstract

For the first Painlev\'e transcendents Kitaev established an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips near the point at infinity. We present an explicit error bound of this asymptotic expression, which leads to the order estimate of exponent $-1$.