Lamé function


In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper. Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.

The Lamé equation

Lamé's equation is
where A and B are constants, and is the Weierstrass elliptic function. The most important case is when , where is the elliptic sine function, and for an integer n and the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points.
By changing the independent variable to with, Lamé's equation can also be rewritten in algebraic form as
which after a change of variable becomes a special case of Heun's equation.
A more general form of Lamé's equation is the ellipsoidal equation or ellipsoidal wave equation which can be written
where is the elliptic modulus of the Jacobian elliptic functions and and are constants. For the equation becomes the Lamé equation with. For the equation reduces to the Mathieu equation
The Weierstrassian form of Lamé's equation is quite unsuitable for calculation. The most suitable form of the equation is that in Jacobian form, as above. The algebraic and trigonometric forms are also cumbersome to use. Lamé equations arise in quantum mechanics as equations of small fluctuations about classical solutions—called periodic instantons, bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.

Asymptotic expansions

Asymptotic expansions of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of have been obtained by Müller.
The asymptotic expansion obtained by him for the eigenvalues is, with approximately an odd integer,
. Observe terms are alternately even and odd in and . With the following boundary conditions
as well as
defining respectively the ellipsoidal wave functions
of periods and for one obtains
Here the upper sign refers to the solutions and the lower to the solutions. Finally expanding about one obtains
In the limit of the Mathieu equation these expressions reduce to the corresponding expressions of the Mathieu case.