---
tag: 69XMERM6KJ
kind: concept
name: Legendre functions
msc: [33C45, 33C05]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre_function
same_as:
  wikidata: Q17098449
  wikipedia: Legendre function
---

## Summary

Legendre functions are the solutions of the general Legendre differential equation, which has two parameters, a degree $\nu$ and an order $\mu$.
The solutions of the first kind, $P_\nu^\mu$, and of the second kind, $Q_\nu^\mu$, are the standard basis of solutions.
When $\mu = 0$ and $\nu$ is a nonnegative integer, the first kind gives the [[Legendre polynomials|QFYGJVA996]].
They arise when the Laplace equation or the Helmholtz equation is separated in spherical coordinates.

## Explanations

### L1 human

Legendre polynomials solve [[Legendre's equation|W6VE30V10Y]] only for special integer values of a parameter.
If the parameter is any number, the equation still has solutions, but they are no longer polynomials.
These solutions are the Legendre functions.
For example, $Q_0(x) = \frac12 \ln \frac{1 + x}{1 - x}$ on $(-1, 1)$ solves the equation with parameter $0$, and it blows up at both ends of the interval.