On extremal properties of Jacobian elliptic functions with complex modulus

A thorough analysis of values of the function m↦sn(K(m)u|m)m↦sn(K(m)u|m) for complex parameter m   and u∈(0,1)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 CC determined by inequalities |z−1|<1|z−1|<1 and |z|>1|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≤1/2u≤1/2, then the global maximum is located at m=1m=1 with the value equal to 1. While if u>1/2u>1/2, then the global maximum is located in the interval (1,2)(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.