In conical coordinates, Laplace's
equation can be written
|
(1)
|
where
(Byerly 1959). Letting
|
(4)
|
breaks (1) into the two equations,
|
(5)
|
|
(6)
|
Solving these gives
|
(7)
|
|
(8)
|
where are ellipsoidal
harmonics. The regular solution is therefore
|
(9)
|
However, because of the cylindrical symmetry, the solution is an th degree spherical harmonic.
See also
Conical Coordinates,
Helmholtz
Differential Equation
Explore with Wolfram|Alpha
References
Arfken, G. "Conical Coordinates ." §2.16 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118-119,
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, p. 263, 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. 39-40, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 659,
1953.
Cite this as:
Weisstein, Eric W. "Laplace Equation--Conical Coordinates." From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/LaplaceEquationConicalCoordinates.html
Subject classifications