Mathieu functions for purely imaginary parameters

For the Mathieu differential equation y″(x)+[a−2qcos(x)]y(x)=0y″(x)+[a−2qcos(x)]y(x)=0 with purely imaginary parameter q=is, the characteristic value aa 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 ss is presented.