Lamé function

ellipsoidal harmonic function

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

$$ \tag{* } \frac{d ^ {2} w }{d \xi ^ {2} } + \frac{1}{2} \left ( \frac{1}{\xi - e _ {1} } + \frac{1}{\xi - e _ {2} } + \frac{1}{\xi - e _ {3} } \right ) \frac{dw}{d \xi } = $$

$$ = \ \frac{A + n ( n + 1 ) \xi }{4 ( \xi - e _ {1} ) ( \xi - e _ {2} ) ( \xi - e _ {3} ) } w , $$

where $ n $ is natural number and $ e _ {1} $, $ e _ {2} $, $ e _ {3} $, and $ A $ are constants, has a solution of one of the following forms:

$$ P ( \xi ) , $$

$$ \sqrt {\xi - e _ {i} } P ( \xi ) ,\ i = 1 , 2 , 3 , $$

$$ \sqrt {\xi - e _ {i} } \sqrt {\xi - e _ {j} } P ( \xi ) ,\ i , j = 1 , 2 , 3 ,\ i \neq j , $$

$$ \sqrt {\xi - e _ {1} } \sqrt {\xi - e _ {2} } \sqrt {\xi - e _ {3} } P ( \xi ) , $$

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

For fixed even $ n $ there are always values of $ A $( eigen values) such that there are $ ( n + 2 ) / 2 $ Lamé functions of the first form and $ 3 n / 2 $ of the third form, with polynomials $ P ( \xi ) $ of degree $ n / 2 $ and $ ( n - 2 ) / 2 $, respectively. For fixed odd $ n $ there are always values of $ A $ such that there are $ 3 ( n + 1 ) / 2 $ Lamé functions of the second form and $ ( n- 1)/2 $ of the fourth form, with polynomials $ P( \xi ) $ of degree $ ( n- 1)/2 $ and $ ( n - 3 ) / 2 $, respectively. For a given natural number $ n $ there are altogether $ 2 n + 1 $ 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.