  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).
