A generalised spheroidal wave equation

Abstract

1.1. The differential equation considered in this paper may be written in the form d / d ξ { (1-ξ²) d X/ d ξ } + { λ²ξ² - 2 p λξ - n 3²/1-ξ² + μ׳ } X = 0. (1) When p = 0 this equation becomes the equation giving the solution of ∇²X —λ²X = 0 in spheroidal co-ordinates. The equation may therefore be called a generalised spheroidal wave equation. In Part I of this paper we shall consider equation (1) when p ≠ 0, and in Part II, which consists of sections 6 to 10, we shall consider the equation with p = 0. The transformation X = (ξ² — l) n 3/² f reduces the equation to one with polynomial coefficients (1 - ξ²) d ² f / d ξ²- 2 (n3 + 1)ξ df / d ξ +{ λ²ξ² - 2 p λξ + μ׳ - n 3 ( n 3 + 1) } f = 0. (2)