A finite difference construction of the spheroidal wave functions

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)## 摘要 本文描述了一种用于计算椭球波函数（spheroidal wave functions）的快速简便有限差分算法。针对实部与复部形式的椭球带宽参数$c$，所得到的本征值与本征函数与文献结果的吻合度可达4位乃至11位以上有效数字。该算法在极端参数范围（最高满足$c^2=10^{14}$）下的有效性得以验证…… ## 程序名称：SWF_8thOrder ## 目录号：AEQE_v1_0 ### 问题本质 本任务旨在构造三维椭球坐标系下拉普拉斯算子的角向本征函数，即长椭球、扁椭球与广义椭球波函数，并计算对应的本征值。等价地，该任务可视为生成通过分离变量法在三维椭球坐标系中求解亥姆霍兹波动方程时得到的角向函数： $$\left[\frac{d}{d\eta}\left(1-\eta^2\right)\frac{d}{d\eta} + \left(-c^2\eta^2 - \frac{m^2}{1-\eta^2}\right)\right] S_{lm}(\eta) = \lambda_{ml}(c) S_{lm}(\eta)$$ ## Mendeley数据中的CPC资源库存档版本 AEQE_v1_0; SWF_8thOrder; 10.1016/j.cpc.2013.07.024 本程序源自贝尔法斯特女王大学托管的CPC程序库（1969-2019年）