# The Cluster's Legendre Polynomials

A page about Legendre polynomials — the angular solutions to Laplace's equation in spherical coordinates.

## The Legendre differential equation

The Legendre differential equation is:
(1 - x^2) d^2y/dx^2 - 2x dy/dx + l(l+1) y = 0
where x = cos(theta) and l is a non-negative integer. The solutions are the Legendre polynomials P_l(x).

For l = 0, 1, 2, 3:
P_0(x) = 1
P_1(x) = x
P_2(x) = (1/2)(3x^2 - 1)
P_3(x) = (1/2)(5x^3 - 3x)

The general solution is given by Rodrigues' formula:
P_l(x) = (1 / (2^l l!)) d^l/dx^l [(x^2 - 1)^l]

In the cluster, the edit Legendre differential equation has an edit solution.

## Properties

- Orthogonality: integral_{-1}^{1} P_l(x) P_{l'}(x) dx = (2 / (2l + 1)) delta_{ll'}
- Normalization: P_l(1) = 1
- Parity: P_l(-x) = (-1)^l P_l(x)
- Recurrence: (l+1) P_{l+1}(x) = (2l+1) x P_l(x) - l P_{l-1}(x)
- Derivative relation: dP_l/dx = (l / (x^2 - 1)) (x P_l - P_{l-1})

In the cluster, the edit properties give an edit orthogonality relation.

## The generating function

The generating function for Legendre polynomials is:
g(x, t) = 1 / sqrt(1 - 2xt + t^2) = sum_{l=0}^{infinity} P_l(x) t^l
For |t| < 1 and x in [-1, 1]. Setting x = cos(theta) and t = r_< / r_>:
1 / |r - r'| = sum_{l=0}^{infinity} (r_<^l / r_>^{l+1}) P_l(cos(theta))
This is the expansion of the Coulomb potential in spherical harmonics.

In the cluster, the edit generating function gives an edit expansion.

## The addition theorem

The Legendre addition theorem relates P_l(cos gamma) to products of spherical harmonics:
P_l(cos gamma) = (4 pi / (2l + 1)) sum_{m=-l}^{l} Y_{lm}*(theta', phi') Y_{lm}(theta, phi)
where gamma is the angle between two directions (theta, phi) and (theta', phi').

In the cluster, the edit addition theorem gives an edit relation between edit angles.

## Applications

- Multipole expansion of the Coulomb potential: V(r) = sum_l (Q_lm / r^{l+1}) Y_{lm}(theta, phi)
- Solution of Laplace's equation in spherical coordinates: V(r, theta) = sum_l (A_l r^l + B_l / r^{l+1}) P_l(cos theta)
- Gravitational potential of axisymmetric mass distributions
- Angular momentum theory (Wigner D-matrices)
- Quantum scattering theory (partial wave expansion)

In the cluster, edit applications include:
- edit Multipole expansion
- edit Solution of Laplace's equation
- edit Gravitational potential
- edit Angular momentum theory
- edit Quantum scattering theory

## This polynomial

This page is about Legendre polynomials. P_l(x) = (1/(2^l l!)) d^l/dx^l [(x^2-1)^l]. Orthogonality: integral P_l P_{l'} dx = 2/(2l+1) delta_{ll'}. The polynomial is real.