Legendre Polynomials
A sequence of orthogonal polynomials satisfying Legendre's differential equation (1-x²) y'' - 2x y' + n(n+1) y = 0.
Governing formula
P_n(x) = (1 / (2^n n!)) (d^n / dx^n) ((x² - 1)^n)
SI unit
Dimensionless Orthogonal Polynomials
In depth
Rodrigues' formula defines P_n(x). Legendre polynomials form the angular solutions for central force problems in spherical coordinates (e.g. electrostatics and hydrogen atom orbitals).
Examples in the real world
Expanding 1/|r - r'| in multipole expansions for electrostatic and gravitational potentials.