---
tag: NB0NAQSKJZ
kind: concept
name: Spheroidal wave function
msc: ["33E10", 34B30]
sources:
  - url: https://en.wikipedia.org/wiki/Spheroidal_wave_function
same_as:
  wikidata: Q7576728
  wikipedia: Spheroidal wave function
---

## Summary

Spheroidal wave functions are solutions of the equations obtained by [[separation of variables|47NFRVB564]] in the [[Helmholtz equation|MNXKTXJ35G]] in prolate or oblate spheroidal coordinates.
The angular spheroidal wave functions $S_{mn}(c, x)$ are the solutions bounded on $[-1, 1]$, and they exist only for special eigenvalues $\lambda_{mn}(c)$.
For $c = 0$ they reduce to the [[associated Legendre polynomials|VPJCSVRY6S]], so they deform the spherical case to the spheroidal one.
The prolate functions with $m = 0$ are also central in signal processing.
There they are the functions of a given bandwidth that are most concentrated in a time interval.

## Explanations

### L1 human

A spheroid is a sphere stretched or squashed along one axis, like a rugby ball or a lentil.
Waves around a sphere are described with Legendre functions; waves around a spheroid need a deformed family, the spheroidal wave functions.
The parameter $c$ measures how far the problem is from the spherical one, and for $c = 0$ the deformation disappears.
As with Legendre polynomials, well-behaved solutions exist only for a discrete list of separation constants.