PlanetPhysics/Legendre Polynomials

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

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

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:

$P_{0}(x)=1$ \\ $P_{1}(x)=x$ \\ $P_{2}(x)={\frac {1}{2}}\left(3x^{2}-1\right)$ \\ $P_{3}(x)={\frac {1}{2}}\left(5x^{3}-3x\right)$ \\ $P_{4}(x)={\frac {1}{8}}\left(35x^{4}-30x^{2}+3\right)$ \\ $P_{5}(x)={\frac {1}{8}}\left(63x^{5}-70x^{3}+15x\right)$ \\