the Legendre equation
/ luh-ZHAHND /
When you solve a physical problem with spherical symmetry — the gravitational potential around a planet, the temperature on a ball, the electron in a hydrogen atom — and you separate variables, the angular part keeps producing the same equation again and again. That recurring angular equation is the Legendre equation. It is one of the most important variable-coefficient equations in physics precisely because spheres are everywhere.
It is written (1 - x^2) y'' - 2x y' + l(l+1) y = 0, where the constant l (often a parameter coming from the separation) controls everything, and x usually stands for cos(theta), the cosine of a polar angle, so the natural domain is -1 <= x <= 1. The points x = 1 and x = -1 — the north and south poles of the sphere — are regular singular points, while x = 0 is an ordinary point where a power-series solution applies. Building that series and matching coefficients gives a two-term recurrence; the magic is that when l is a non-negative integer, the series terminates into a polynomial and that polynomial stays finite at the poles, whereas the non-terminating solution blows up there.
This termination is the whole point: physical solutions on a sphere must be finite at the poles, and that requirement forces l to be a non-negative integer and selects the polynomial solutions — the Legendre polynomials. So the Legendre equation is not just an exercise in series; it is the gateway through which quantized angular momentum and the familiar multipole expansion enter physics.
Take l = 2: the equation (1 - x^2) y'' - 2x y' + 6 y = 0. The terminating series solution is the polynomial P_2(x) = (3x^2 - 1)/2, which is perfectly finite at x = +-1. The other, non-polynomial solution diverges logarithmically at the poles and is discarded on physical grounds.
Only integer l gives a polynomial finite at the poles — the physical boundary condition quantizes l, just as it quantizes angular momentum.
The 'general solution' for non-integer l exists mathematically but blows up at x = +-1; the physically admissible solutions are a much smaller set, selected by demanding finiteness at the poles — a boundary-condition effect, not an algebraic accident.