Helmholtz Differential Equation--Conical Coordinates

In conical coordinates, Laplace's equation can be written

(1)

where

			(2)
			(3)

(Byerly 1959). Letting

(4)

breaks (1) into the two equations,

(5)

(6)

Solving these gives

(7)

(8)

where  are ellipsoidal harmonics of the first kind. The regular solution is therefore

(9)

However, because of the cylindrical symmetry, the solution  is an th degree spherical harmonic.

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.