History of
The Cluster's Legendre Polynomials
lore/trolla/legendre-polynomials · 1 revision(s)
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- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Legendre Polynomials
+updated: 2026-09-05
+updated_at: 2026-09-05T13:21:10.964Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
6h ago · 2026-09-05 13:21
Python-urllib/3.11 · from visitor-99c4 · via api-get