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

(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

Two other equations named after Lamé are given by

(10)

and

(11)

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

See also Ellipsoidal Wave Equation ,

Lamé's Differential Equation
Types ,

Wangerin Differential Equation
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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. Referenced on Wolfram|Alpha Lamé's Differential
Equation
Cite this as:
Weisstein, Eric W. "Lamé's Differential Equation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LamesDifferentialEquation.html

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