  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.
