 TOPICS  # Laplace Equation--Confocal Ellipsoidal Coordinates

Using the notation of Byerly (1959, pp. 252-253), Laplace's equation can be reduced to (1)

where   (2)   (3)   (4)   (5)   (6)   (7)

In terms of , , and ,   (8)   (9)   (10)

Equation (◇) is not separable using a function of the form (11)

but it is if we let   (12)   (13)   (14)

These give   (15)   (16)

and all others terms vanish. Therefore (◇) can be broken up into the equations   (17)   (18)   (19)

For future convenience, now write   (20)   (21)

then   (22)   (23)   (24)

Now replace , , and to obtain (25)

Each of these is a Lamé's differential equation, whose solution is called an ellipsoidal harmonic of the first kind. Writing   (26)   (27)   (28)

gives the solution to (◇) as a product of ellipsoidal harmonics of the first kind . (29)

Confocal Ellipsoidal Coordinates, Ellipsoidal Harmonic of the First Kind, Helmholtz Differential Equation

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## References

Arfken, G. "Confocal Ellipsoidal Coordinates ." §2.15 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 117-118, 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. 251-258, 1959.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 43-44, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.

## Cite this as:

Weisstein, Eric W. "Laplace Equation--Confocal Ellipsoidal Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplaceEquationConfocalEllipsoidalCoordinates.html