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
本文提出一种用于计算椭球波函数（spheroidal wave functions）的快速简洁有限差分算法（finite difference algorithm）。针对实、复椭球带宽参数$c$，所得到的本征值与本征函数与已有文献结果的吻合度可达4位乃至11位以上有效数字。该算法在极端参数区间（最大至$c^2=10^{14}$）内的有效性得以验证……

## 程序名称：SWF_8thOrder
## 目录编号：AEQE_v1_0

## 问题本质
本问题旨在构建三维椭球坐标系下拉普拉斯算子（Laplacian）的角向本征函数，即长椭球、扁椭球以及广义椭球波函数，并求解对应的本征值。等价而言，该任务可视为在三维椭球坐标系中通过分离变量法（separation of variables）求解亥姆霍兹波动方程（Helmholtz wave equation）时所产生的角向函数生成问题，其控制方程为：
$[(1 - eta^2)
abla^2_eta + (-c^2eta^2 - m^2/(1-eta^2))] S_{lm} = lambda_{ml}(c) S_{lm} dots$

## 收录版本
孟德尔莱数据（Mendeley Data）中的计算机物理通讯（Computer Physics Communications, CPC）程序库所收录的本程序版本为：AEQE_v1_0；SWF_8thOrder；DOI: 10.1016/j.cpc.2013.07.024

本程序源自贝尔法斯特女王大学维护的CPC程序库（1969-2019年）