Ellipsoidal Harmonic of the First Kind

The first solution to Lamé's differential equation, denoted  for , ..., . They are also called Lamé functions. The product of two ellipsoidal harmonics of the first kind is a spherical harmonic. Whittaker and Watson (1990, pp. 536-537) write

			(1)
			(2)

and give various types of ellipsoidal harmonics and their highest degree terms as

1. 

2. 

3. 

4. .

A Lamé function of degree  may be expressed as

(3)

where  or 1/2,  are real and unequal to each other and to , , and , and

(4)

Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by , , , and ,

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)

REFERENCES:

Byerly, W. E. "Laplace's Equation in Curvilinear Coördinates. Ellipsoidal Harmonics." Ch. 8 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251-266, 1959.

Humbert, P. Fonctions de Lamé et Fonctions de Mathieu. Paris: Gauthier-Villars, 1926.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.