# A finite difference construction of the spheroidal wave functions

## Description

Abstract A fast and simple finite difference algorithm for computing the spheroidal wave functions is described. The resulting eigenvalues and eigenfunctions for real and complex spheroidal bandwidth parameter, c, agree with those in the literature from four to more than eleven significant figures. The validity of this algorithm in the extreme parameter regime, up to ^(c2) =10 ^(14) , ... Title of program: SWF_8thOrder Catalogue Id: AEQE_v1_0 Nature of problem The problem is to construct the angular eigenfunctions of the Laplacian in three dimensional, spheroidal coordinates. These are the prolate, oblate and generalized spheroidal wave functions and to compute the corresponding eigenvalues. Equivalently, the task can be seen as generating the angular functions which arise when solving the Helmholtz wave equation by separation of variables in three dimensional, spheroidal coordinates: [Δ Η (1 - Η 2 )Δ Η + (-c 2 Η 2 - m 2 /(1-Η 2 ))] S l m = λ ml (c)S ... Versions of this program held in the CPC repository in Mendeley Data AEQE_v1_0; SWF_8thOrder; 10.1016/j.cpc.2013.07.024 This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2019)