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

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.