On extremal properties of Jacobian elliptic functions with complex modulus

A thorough analysis of values of the function m(mapping)sn( K( m) u| m) for complex parameter m and u∈ (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 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≤ 1/2, then the global maximum 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.