On extremal properties of Jacobian elliptic functions with complex modulus

TL;DR

This paper thoroughly analyzes the extremal behavior of Jacobian elliptic functions with complex modulus, identifying regions of maximum values and their locations, with applications in integrals and spectral analysis of complex matrices.

Contribution

It provides new extremal bounds and locations for Jacobian elliptic functions with complex modulus, including numerical insights and applications.

Findings

The absolute value of the function never exceeds 1 outside a specific complex region.

Maximum at u ≤ 1/2 occurs at m=1 with value 1.

For u > 1/2, the maximum lies in (1,2) and exceeds 1.

Abstract

A thorough analysis of values of the function $m\mapsto\mbox{sn}(K(m)u\mid m)$ for complex parameter $m$ and $u\in (0,1)$ is given. First, it is proved that the absolute value of this function never exceeds 1 if $m$ does not belong to the region in $\mathbb{C}$ determined by inequalities $|z-1|<1$ and $|z|>1$. The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that, if $u\leq1/2$, then the global maxim is located at $m=1$ with the value equal to $1$. While if $u>1/2$, then the global maximum is located in the interval $(1,2)$ and its value exceeds $1$. In addition, more subtle extremal properties are studied numerically. Finally, applications in a Laplace-type integral and spectral analysis of some complex Jacobi matrices are presented.