Lamé function

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Lamé functions, also called ellipsoidal harmonics, are special solutions of Lamé's equation, a second‑order differential equation. They were introduced by Gabriel Lamé in 1837 and appear when solving Laplace’s equation in ellipsoidal coordinates. In some cases, the solutions are polynomials known as Lamé polynomials.

Lamé’s equation involves constants A and B and the Weierstrass elliptic function. A particularly important case is when B times the Weierstrass function equals a negative multiple of the square of the elliptic sine sn(x). If B℘(x) = -κ² sn²(x) with κ² = n(n+1)k² (n an integer and k the elliptic modulus), the corresponding solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points.

If you change the variable to t = sn(x), Lamé’s equation can be rewritten in an algebraic form and, after another change of variables, becomes a special case of Heun’s equation.

A more general form is the ellipsoidal (or ellipsoidal wave) equation, which uses the elliptic modulus k and constants κ and Ω. When Ω = 0, this reduces to Lamé’s equation with a parameter Λ. If, in addition, k = 0 and κ = 2h, the equation can further reduce to the Mathieu equation.

The Weierstrass form of Lamé’s equation is not convenient for calculations; the Jacobian form is preferred. The algebraic and trigonometric forms are more cumbersome to work with.

Lamé equations also appear in quantum mechanics as equations describing small fluctuations around periodic instantons (periodic solutions) of Schrödinger equations with periodic or anharmonic potentials.

Asymptotic expansions for large κ have been found. The eigenvalue Λ can be expanded in a parameter q, which is related to boundary conditions and is roughly an odd integer. Depending on the boundary conditions, you obtain ellipsoidal wave functions with different periods, leading to two families called Ec and Es. Expanding Λ(q) around a base q yields the asymptotic formulas; in the limit where the equation reduces to Mathieu’s equation, these formulas become the familiar Mathieu results.