Legendre polynomials
LegendrePolynomials
2006-05-22 23:05:06
2006-05-25 02:56:10
Definition
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The Legendre polynomials generate the power series that solves Legendre's differential equation:
$$ \left ( 1 - x^2 \right ) P''(x) - 2x P'(x) + n(n+1)P(x) = 0.$$
This ordinary differential equation with variable coefficients is named in honor of Adrien-Marie Legendre (1752-1833). While quite literally following in the footsteps of Laplace, he developed the Legendre polynomials in a paper on celestial mechanics. In a strange tangled web of fate, the Legendre polynomials are heavily used in electrostatics to solve Laplace's equation in spherical coordinates
$$ \nabla^2 \Phi_{sph} = 0 $$
The series can be easily generated using the Rodrigues' formula
$$ P_n(x) = \frac{1}{ 2^n n!} {d^n \over dx^n } (x^2 -1)^n. $$
The first six polynomials are:
$P_0(x) = 1$\\
$P_1(x) = x$\\
$P_2(x) = \frac{1}{2} \left ( 3x^2 - 1 \right )$\\
$P_3(x) = \frac{1}{2} \left ( 5x^3 - 3x \right )$\\
$P_4(x) = \frac{1}{8} \left ( 35x^4 - 30x^2 + 3 \right )$\\
$P_5(x) = \frac{1}{8} \left ( 63x^5 - 70x^3 + 15x \right )$\\
Not yet done....
\subsection{References}
[1] Lebedev, N. "Special Functions \& Their Applications." Dover Publications, Inc., New York, 1972.
[2] Jackson, J. "Classical Electrodynamics." John Wiley \& Sons, Inc., New York, 1962.
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