Uniform asymptotic behaviour of Jacobi-$\sn$ near a singular point. The Lost formula from handbooks for elliptic functions

TL;DR

This paper develops a uniform asymptotic expansion for the Jacobi elliptic function sn(t|m) as the parameter m approaches 1, valid over a significant portion of its period, and compares it with classical handbook formulas.

Contribution

It provides a new uniform asymptotic expansion for sn(t|m) near m=1, including the turning point, and clarifies differences with existing handbook formulas.

Findings

Asymptotic expansion valid over more than half the period

Inclusion of the turning point in the approximation interval

New asymptotic formula for elliptic integral of the first kind

Abstract

In this work we construct uniform asymptotic expansion of $\sn(t|m)$ - Jacobi when $m\to1-0$. The constructed expansion is valid over more than a half of period. The turning point is included into the interval of validity for the approximation. In addition we obtain the asymptotic formula for elliptic integral of the first kind and discuss the differences with the same formula from a handbook.