Mathieu functions for purely imaginary parameters

摘要

For the Mathieu differential equation y″(x)+[a−2qcos(x)]y(x)=0 with purely imaginary parameter q=is, the characteristic value a exhibits branching points. We analyze the properties of the Mathieu functions and their Fourier coefficients in the vicinity of the branching points. Symmetry relations for the Mathieu functions as well as the Fourier coefficients behind a branching point are given. A numerical method to compute Mathieu functions for all values of the parameter s is presented.