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+---
+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).