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Legendre polynomials (Definition)

The Legendre polynomials generate the power series that solves Legendre's differential equation:

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

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

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

The series can be easily generated using the Rodrigues' formula

$\displaystyle 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....

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.

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



"Legendre polynomials" is owned by bloftin.

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Cross-references: functions, formula, mechanics, ordinary differential equation, differential equation, power

This is version 4 of Legendre polynomials, born on 2006-05-22, modified 2006-06-12.
Object id is 177, canonical name is LegendrePolynomials.
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Classification:
Physics Classification: 02.30.Mv (Approximations and expansions)
 02.30.Hq (Ordinary differential equations)
 02.30.Gp (Special functions)
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