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