Lamé's Differential Equation

The ordinary differential equation

(1)

(Byerly 1959, p. 255). The solution is denoted  and is known as an ellipsoidal harmonic of the first kind, or Lamé function. Whittaker and Watson (1990, pp. 554-555) give the alternative forms

	(2)
	(3)
	(4)
	(5)
	(6)

(Whittaker and Watson 1990, pp. 554-555; Ward 1987; Zwillinger 1997, p. 124). Here,  is a Weierstrass elliptic function,  is a Jacobi elliptic function, and

			(7)
			(8)
			(9)

Two other equations named after Lamé are given by

(10)

and

(11)

(Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 124).

REFERENCES:

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.

Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961.

Ward, R. S. "The Nahm Equations, Finite-Gap Potentials and Lamé Functions." J. Phys. A: Math. Gen. 20, 2679-2683, 1987.

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

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997.