PlanetPhysics/Legendre Polynomials

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....