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---
title: "Legendre polynomials"
firstLetter: "L"
publishDate: 2021-09-08
categories:
- Mathematics
date: 2021-09-08T17:00:53+02:00
draft: false
markup: pandoc
---
# Legendre polynomials
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:
$$\begin{aligned}
P_\ell(x)
&= \frac{1}{2^\ell \ell!} \dv[\ell]{x} (x^2 - 1)^\ell
\end{aligned}$$
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 $x \in [-1, 1]$:
$$\begin{aligned}
\boxed{
\braket{P_m}{P_n}
= \int_{-1}^{1} P_m(x) \: P_n(x) \dd{x}
= \frac{2}{2 n + 1} \delta_{nm}
}
\end{aligned}$$
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:
$$\begin{aligned}
\boxed{
f(x)
= \sum_{\ell = 0}^\infty a_\ell P_\ell(x)
= \sum_{\ell = 0}^\infty \frac{\braket{P_\ell}{f}}{\braket{P_\ell}{P_\ell}} P_\ell(x)
}
\end{aligned}$$
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 $m \ge 0$:
$$\begin{aligned}
P_\ell^m(x)
= (-1)^m (1 - x^2)^{m/2} \dv[m]{x} P_\ell(x)
\end{aligned}$$
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).
|
