Legendre Polynomials

Differential Equations · Axiom Academy

From Laplace's equation on a sphere to an orthogonal family of polynomials — the special functions behind gravity, charge, and the hydrogen atom. 1. Origin in Spherical Coordinates Start with Laplace's equation in spherical coordinates — the equation governing gravitational and electrostatic potentials with no interior sources. Separating variables with and assuming symmetry about the polar axis (no dependence) isolates the angular part. The angular equation, with substituted in Here , where is the polar angle measured from the z -axis — so as ranges over the sphere's surface ( 0 to ), x ranges over exactly [-1,1] . The constant n is a non-negative integer , forced by requiring the solution to stay finite at both poles (i.e. ). 2. Polynomial Solutions P_n(x) For each non-negative integer n , the Legendre equation has a solution that is a polynomial of degree n , denoted P_n(x) — the Legendre polynomial . (The other, independent solution blows up at and is discarded for any physical problem that must stay finite at the poles.) Two useful facts follow directly from the pattern above: P_n(1)=1 for every n , and P_n(-1) = (-1)^n . The degree- n polynomial P_n(x) has exactly n real zeros , all lying strictly inside (-1,1) . Rodrigues' formula — generates P_n(x) directly, no recurrence needed 3. Orthogonality and Applications

This is the written version of the interactive lesson above. See the full Differential Equations course.