---
title: "Legendre polynomials"
date: 2021-09-08
categories:
- Mathematics
layout: "concept"
---

The **Legendre polynomials** are a set of functions that sometimes arise in physics.
They are the eigenfunctions $u(x)$ of **Legendre's differential equation**,
which is a ([Sturm-Liouville](/know/concept/sturm-liouville-theory/))
eigenvalue problem for $\ell (\ell + 1)$,
where $\ell$ turns out to be a non-negative integer:

$$\begin{aligned}
    \boxed{
        (1 - x^2) u'' - 2 x u' + \ell (\ell + 1) u = 0
    }
\end{aligned}$$

The $\ell$th-degree Legendre polynomial $P_\ell(x)$
is given in the form of a *Rodrigues' formula* by:

$$\begin{aligned}
    P_\ell(x)
    &= \frac{1}{2^\ell \ell!} \dvn{\ell}{}{x}(x^2 - 1)^\ell
\end{aligned}$$

The first handful of Legendre polynomials $P_\ell(x)$ are therefore as follows:

$$\begin{gathered}
    P_0(x) = 1
    \qquad \quad
    P_1(x) = x
    \qquad \quad
    P_2(x) = \frac{1}{2} (3 x^2 - 1)
    \\
    P_3(x) = \frac{1}{2} (5 x^3 - 3 x)
    \qquad \quad
    P_4(x) = \frac{1}{8} (35 x^4 - 30 x^2 + 3)
\end{gathered}$$

And then more $P_\ell$ can be computed quickly
using **Bonnet's recursion formula**:

$$\begin{aligned}
    \boxed{
        (\ell + 1) P_{\ell + 1}(x) = (2 \ell + 1) x P_\ell(x) - \ell P_{\ell - 1}(x)
    }
\end{aligned}$$

The derivative of a given $P_\ell$ can be calculated recursively
using the following relation:

$$\begin{aligned}
    \boxed{
        \dv{}{x}P_{\ell + 1}
        = (\ell + 1) P_\ell(x) + x \dv{}{x}P_\ell(x)
    }
\end{aligned}$$

Noteworthy is that the Legendre polynomials
are mutually orthogonal for $x \in [-1, 1]$:

$$\begin{aligned}
    \boxed{
        \Inprod{P_m}{P_n}
        = \int_{-1}^{1} P_m(x) \: P_n(x) \dd{x}
        = \frac{2}{2 n + 1} \delta_{nm}
    }
\end{aligned}$$

As was to be expected from Sturm-Liouville theory.
Likewise, they form a complete basis in the
[Hilbert space](/know/concept/hilbert-space/)
of piecewise continuous functions $f(x)$ on $x \in [-1, 1]$,
meaning:

$$\begin{aligned}
    \boxed{
        f(x)
        = \sum_{\ell = 0}^\infty a_\ell P_\ell(x)
        = \sum_{\ell = 0}^\infty \frac{\Inprod{P_\ell}{f}}{\Inprod{P_\ell}{P_\ell}} P_\ell(x)
    }
\end{aligned}$$

Each Legendre polynomial $P_\ell$ comes with
a set of **associated Legendre polynomials** $P_\ell^m(x)$
of order $m$ and degree $\ell$.
These are the non-singular solutions of the **general Legendre equation**,
where $m$ and $\ell$ are integers satisfying $-\ell \le m \le \ell$:

$$\begin{aligned}
    \boxed{
        (1 - x^2) u'' - 2 x u' + \Big( \ell (\ell + 1) - \frac{m^2}{1 - x^2} \Big) u = 0
    }
\end{aligned}$$

The $\ell$th-degree $m$th-order associated Legendre polynomial $P_\ell^m$
is as follows for $m \ge 0$:

$$\begin{aligned}
    P_\ell^m(x)
    = (-1)^m (1 - x^2)^{m/2} \dvn{m}{}{x}P_\ell(x)
\end{aligned}$$

Here, the $(-1)^m$ in front is called the **Condon-Shortley phase**,
and is omitted by some authors.
For negative orders $m$,
an additional constant factor is necessary:

$$\begin{aligned}
    P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x)
\end{aligned}$$

Beware, the name is misleading:
if $m$ is odd, then $P_\ell^m$ is actually not a polynomial.
Moreover, not all $P_\ell^m$ are mutually orthogonal
(but some are).