A refinement of the WKB method and its application to the electromagnetic wave theory

When dealing with the problem of diffraction of waves, certain special functions appear which are defined by ordinary differential equations of the second order; for example, Bessel functions, Legendre functions, and Mathieu functions appear for the case of a circular cylinder, a sphere, and an elliptic cylinder, respectively. Exact solutions are obtained in the form of infinite series of such functions, which are, in general, poorly convergent when the wavelength is comparable with or smaller than the dimension of the body. In such a case the series can be transformed into contour integrals and then evaluated by the method of steepest descents or by forming the residue series. For this purpose asymptotic expressions for the special functions are needed. A similar situation arises also in the problem of propagation of short radio waves in a horizontally stratified atmosphere. In this paper a refinement of the WKB method is presented which enables one to obtain very accurate and compact expressions for such functions, which are particularly suited for the evaluation of zeros. The application of the method is illustrated for the case of Bessel functions, parabolic cylinder functions, Coulomb wave functions, etc.