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# 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