Talk:PlanetPhysics/Legendre Polynomials

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%%% Primary Title: Legendre polynomials
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 The Legendre polynomials generate the \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} series that solves Legendre's \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html}:

$$   \left ( 1 - x^2 \right ) P''(x) - 2x P'(x) + n(n+1)P(x) = 0.$$

This \htmladdnormallink{ordinary differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} 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 \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}. In a strange tangled web of fate, the Legendre polynomials are heavily used in electrostatics to solve \htmladdnormallink{Laplace's equation}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} in spherical coordinates

$$ \nabla^2 \Phi_{sph} = 0 $$

The series can be easily generated using the Rodrigues' \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html}
$$   P_n(x) = \frac{1}{ 2^n n!} {d^n \over dx^n } (x^2 -1)^n. $$

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 \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} \& Their Applications." Dover Publications, Inc., New York, 1972.

[2] Jackson, J. "Classical Electrodynamics." John Wiley \& Sons, Inc., New York, 1962.

\htmladdnormallink{http://www-groups.dcs.st-and.ac.uk/\~{}history/Biographies/Legendre.html}{http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Legendre.html}
\htmladdnormallink{http://astrowww.phys.uvic.ca/\~{}tatum/celmechs.html}{http://astrowww.phys.uvic.ca/~tatum/celmechs.html}
\htmladdnormallink{http://www.du.edu/\~{}jcalvert/math/legendre.htm}{http://www.du.edu/~jcalvert/math/legendre.htm}
\htmladdnormallink{http://en.wikipedia.org/wiki/Legendre\_polynomials}{http://en.wikipedia.org/wiki/Legendre_polynomials}

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