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 formula_2 is the Weierstrass elliptic function. The most important case is when formula_3, where formula_4 is the elliptic sine function, and formula_5  for an integer "n" and formula_6 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 formula_7 with formula_8, 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 (observe we now write formula_10, not formula_11 as above) where formula_13 is  the elliptic modulus of the Jacobian elliptic functions and formula_14 and formula_15 are constants. For formula_16 the equation becomes the Lamé equation with formula_17. For formula_18 the equation reduces to the Mathieu equation The Weierstrassian form of Lamé's equation is quite unsuitable for calculation (as Arscott also remarks, p. 191). 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 formula_20 have been obtained by Müller. The asymptotic expansion obtained by him for the eigenvalues formula_21 is, with formula_22 approximately an odd integer (and to be determined more precisely by boundary conditions – see below), as well as (the prime meaning derivative) defining respectively the ellipsoidal wave functions of periods formula_30 and for formula_31 one obtains Here the upper sign refers to the solutions formula_33 and the lower to the solutions formula_34. Finally expanding formula_35 about formula_36 one  obtains In the limit of the Mathieu equation (to which the Lamé equation can be reduced) these expressions reduce to the corresponding expressions of the Mathieu case (as shown by Müller).