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

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

The series can be easily generated using the Rodrigues’ formula

The first six polynomials are:

P₀(x) = 1
P₁(x) = x
P₂(x) =
P₃(x) =
P₄(x) =
P₅(x) =

Not yet done....

0.1 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