On extremal properties of Jacobian elliptic functions with complex modulus

A thorough analysis of values of the function $m\mapsto\mbox{sn}(K(m)u\mid m)$forcomplexparameter$m$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 $\mathrm{\mid }z-1\mathrm{\mid }<1$ and $\mathrm{\mid }z\mathrm{\mid }>1$. The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that, if $u\le 1\mathrm{/}2$, then the global maxim is located at $m=1$ with the value equal to $1$. While if $u>1\mathrm{/}2$, 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.