As shown by Morse and Feshbach (1953) and Arfken (1970), the Helmholtz differential equation is separable in prolate spheroidal coordinates.
Helmholtz Differential Equation--Prolate Spheroidal Coordinates
See also
Helmholtz Differential Equation, Prolate Spheroidal CoordinatesExplore with Wolfram|Alpha
References
Arfken, G. "Prolate Spheroidal Coordinates
." §2.10 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107,
1970.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, pp. 243-244, 1959.Moon, P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, p. 30, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.
Cite this as:
Weisstein, Eric W. "Helmholtz Differential Equation--Prolate Spheroidal Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationProlateSpheroidalCoordinates.html