Categories: Mathematics.

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) 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 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).


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