# Lamé function

*ellipsoidal harmonic function*

A function of special form satisfying the Lamé equation. If the Lamé equation in algebraic form,

(*) |

where is natural number and , , , and are constants, has a solution of one of the following forms:

where is a polynomial with leading coefficient one, then this solution is called a Lamé function of degree of the first kind and the first, second, third, or fourth form, respectively.

For fixed even there are always values of (eigen values) such that there are Lamé functions of the first form and of the third form, with polynomials of degree and , respectively. For fixed odd there are always values of such that there are Lamé functions of the second form and of the fourth form, with polynomials of degree and , respectively. For a given natural number there are altogether linearly independent Lamé functions.

Solutions of equation (*) that are linearly independent with the Lamé functions of the first kind and are obtained by means of the Liouville–Ostrogradski formula are called Lamé functions of the second kind.

For references see Lamé equation.

**How to Cite This Entry:**

Lamé function.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Lam%C3%A9_function&oldid=15271