Helmholtz Differential Equation--Conical Coordinates

In Conical Coordinates, Laplace's Equation can be written

$${\partial^2V\over\partial\alpha^2}+{\partial^2V\over\partial... ...} \left({\lambda^2{\partial V\over\partial\lambda}}\right)=0,$$ (1)

where

$$\alpha = \int_a^\mu {d\mu\over\sqrt{(\mu^2-a^2)(b^2-\mu^2)}}$$ (2)

$$\beta = \int_0^\nu {d\nu\over\sqrt{(a^2-\nu^2)(b^2-\nu^2)}}$$ (3)

(Byerly 1959). Letting

$$V=U(u)R(r)$$ (4)

breaks (1) into the two equations,

$${d\over dr}\left({r^2{dR\over dr}}\right)=m(m+1)R$$ (5)

$${\partial^2U\over\partial\alpha^2}+{\partial^2U\over\partial\beta^2}+m(m+1)(\mu^2-\nu^2)U=0.$$ (6)

Solving these gives

$$R(r)=Ar^m+Br^{-m-1}$$ (7)

$$U(u)=E_m^p(\mu)E_m^p(\nu),$$ (8)

where $E_m^p$ are Ellipsoidal Harmonics. The regular solution is therefore

$$V=Ar^mE_m^p(\mu)E_m^p(\nu).$$ (9)

However, because of the cylindrical symmetry, the solution $E_m^p(\mu)E_m^p(\nu)$ is an $m$th degree Spherical Harmonic.

References

Arfken, G. ``Conical Coordinates $(\xi_1, \xi_2, \xi_3)$.'' §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.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 659, 1953.