---
tag: 4ST57PN8K7
kind: definition
name: Spheroidal wave function
msc: ["33E10", 34B30]
defines: NB0NAQSKJZ
uses: [0QQRP17Z7S, VPJCSVRY6S, 7WV6B2502K]
sources:
  - url: https://en.wikipedia.org/wiki/Spheroidal_wave_function
---

## Definition

Let $m$ be a nonnegative integer and let $c \ge 0$.
The *prolate spheroidal wave equation* is the [[linear differential equation|0QQRP17Z7S]]
$$(1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + \left( \lambda - c^2 x^2 - \frac{m^2}{1 - x^2} \right) y = 0, \qquad -1 < x < 1,$$
with a real parameter $\lambda$.
Call $\lambda$ an eigenvalue if the equation has a nonzero solution that is bounded on $(-1, 1)$.
The eigenvalues form an increasing sequence $\lambda_{m, m}(c) < \lambda_{m, m + 1}(c) < \lambda_{m, m + 2}(c) < \cdots$.
For each eigenvalue, the bounded solution is unique up to a constant factor.
The *angular prolate spheroidal wave function* $S_{mn}(c, x)$, for $n \ge m$, is that bounded solution for $\lambda = \lambda_{mn}(c)$, with a chosen normalisation.
The *oblate* functions are defined in the same way with $c^2$ replaced by $-c^2$.

## Remarks

The eigenvalue problem is a singular [[Sturm–Liouville|7WV6B2502K]] problem on $(-1, 1)$, which explains the discrete increasing sequence of eigenvalues.
Example: for $c = 0$ the equation is the associated Legendre equation, $\lambda_{mn}(0) = n(n + 1)$, and $S_{mn}(0, x)$ is a multiple of the [[associated Legendre polynomial|VPJCSVRY6S]] $P_n^m(x)$.
For $c = 0$, $m = 0$ and $n = 0$: $y = 1$ is bounded and satisfies $-2x \cdot 0 + (0 - 0) \cdot 1 = 0$ with $\lambda = 0$.
Non-example: for $c = 0$, $m = 0$ and $\lambda = 0$, the function $y = \frac12 \log \frac{1 + x}{1 - x}$ also solves the equation, but it is unbounded near $x = \pm 1$, so it is not a spheroidal wave function.
The radial spheroidal wave functions solve the same equation on $x > 1$ and describe the dependence on the other coordinate.
Normalisations differ between authors; a common one makes $S_{mn}(c, x)$ tend to $P_n^m(x)$ as $c \to 0$.