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Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineering where the prolate or the oblate spheroidal coordinate system is used. In the evaluation of electromagnetic (EM) fields in spheroidal structures, spheroidal wave functions are frequently encountered, especially when boundary value problems in spheroidal structures are solved using full-wave analysis. | Spheroidal Wave Functions in Eledrotmignetic Theory Le-Wei Li Xiao-Kang Kang Mook-Seng Leong Copyright 2002 John Wiley Sons Inc. ISBNs 0-471-03170-4 Hardback 0-471-22157-0 Electronic 1 Introduction OVERVIEW Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineering where the prolate or the oblate spheroidal coordinate system is used. In the evaluation of electromagnetic EM fields in spheroidal structures spheroidal wave functions are frequently encountered especially when boundary value problems in spheroidal structures are solved using full-wave analysis. By applying the separation of variables to the Maxwell s equations satisfied by either an electric or magnetic field the spheroidal harmonics of electromagnetic waves corresponding to their spheroidal coordinate system can be obtained. With prolate or oblate spheroidal coordinates the separation of scalar variables results in three independent functions 1 the radial spheroidal function n c or Bmn - c i 2 the angular spheroidal function Smn c or Smn tc 77 and 3 the sine and cosine functions. This separability is analogous to that of solving the Laplace equation in spherical coordinates. Here the last pair of trigonometrical functions sine and cosine is well known but the first two are not so easily computed. In general spheroidal radial and angular functions in spheroidal coordinates are respectively the generalization of Legendre functions and spherical Bessel functions in the spherical polar coordinates 6 . Computation of the spheroidal radial or angular functions requires the eigenvalue computation and the forward and backward recur- 1 2 INTRODUCTION sion formulations. Theoretically the formulation of these harmonics was well documented by J. A. Stratton et al. in 1956 7 and C. Flammer in 1957 1 . So far there are only six coordinate systems in which the scalar Helmholtz equations are separable and solenoidal