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Spheroidal Coordinates and Wave Functions 2.1 SPHEROIDAL COORDINATE SYSTEMS Prolate and oblate spheroidal coordinate systems are formed by rotating the two-dimensional elliptic coordinate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses, respectively [l] (shown in Figs. 2.1 and 2.2, where d is the interfocal distance). | Spheroidal Wave Functions in Electromagnetic 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 Spheroidal Coordinates and Wave Functions 2.1 SPHEROIDAL COORDINATE SYSTEMS Prolate and oblate spheroidal coordinate systems are formed by rotating the two-dimensional elliptic coordinate system consisting of confocal ellipses and hyperbolas about the major and minor axes of the ellipses respectively 1 shown in Figs. 2.1 and 2.2 where d is the interfocal distance . The prolate spheroidal coordinates shown in Fig. 2.1 are related to the rectangular coordinates by the following transformations z n2 2 l cos 2.1a y TjW -tW2 - i sM 2.1b z 2.1c where -1 t 1 l oo 0 2tt. 2.Id In a prolate spheroidal system the surface at constant 1 forms an elongated ellipsoid of revolution with major axis of length d and minor axis of length dy 2 1. The degenerate surface 1 is the straight line along the -axis from z to z d. The surface at t constant 1 forms a hyperboloid of revolution of two sheets with an asymptotic cone whose generating line passes through the origin and is inclined at the angle 0 cos-1 7 to the -axis. The degenerate surface t 1 is a part of the 13 14 SPHEROIDAL COORDINATES AND WAVE FUNCTIONS Fig. 2.1 Prolate spheroidal coordinate system. SPHEROIDAL COORDINATE SYSTEMS 15 Fig. 2.2 Oblate spheroidal coordinate .