WKB approach to evaluate series of Mathieu functions in scattering problems

The scattering of a wave obeying Helmholtz equation by an elliptic obstacle can be described exactly using series of Mathieu functions. This situation is relevant in optics, quantum mechanics and fluid dynamics. We focus on the case when the wavelength is comparable to the obstacle size, when the most standard approximations fail. The approximations of the radial (or modified) Mathieu functions using WKB method are shown to be especially efficient, in order to precisely evaluate series of such functions. It is illustrated with the numerical computation of the Green function when the wave is scattered by a single slit or a strip (ribbon).