Lamé Function

Lamé functions, also called ellipsoidal harmonic functions, are the solutions to Lamé's equation, a second-order ordinary differential equation named after the French mathematician Gabriel Lamé, who introduced them in 1837. The equation takes the form of a standard second-derivative term plus a term built from the Weierstrass elliptic function, and in certain special cases its solutions can be written as polynomials known as Lamé polynomials. Lamé functions arise naturally when the method of separation of variables is applied to the Laplace equation in elliptic coordinates, making them a tool of applied mathematics and mathematical physics. They also appear in quantum mechanics, describing small fluctuations around classical solutions of the Schrödinger equation for periodic and anharmonic potentials, and their deeper study draws on Floquet theory and on connections to hyperelliptic curves in what is called the finite-gap case.