In Conical Coordinates, Laplace's Equation can be written
![\begin{displaymath}
{\partial^2V\over\partial\alpha^2}+{\partial^2V\over\partial...
...}
\left({\lambda^2{\partial V\over\partial\lambda}}\right)=0,
\end{displaymath}](h_1006.gif) |
(1) |
where
(Byerly 1959). Letting
![\begin{displaymath}
V=U(u)R(r)
\end{displaymath}](h_1009.gif) |
(4) |
breaks (1) into the two equations,
![\begin{displaymath}
{d\over dr}\left({r^2{dR\over dr}}\right)=m(m+1)R
\end{displaymath}](h_1010.gif) |
(5) |
![\begin{displaymath}
{\partial^2U\over\partial\alpha^2}+{\partial^2U\over\partial\beta^2}+m(m+1)(\mu^2-\nu^2)U=0.
\end{displaymath}](h_1011.gif) |
(6) |
Solving these gives
![\begin{displaymath}
R(r)=Ar^m+Br^{-m-1}
\end{displaymath}](h_1012.gif) |
(7) |
![\begin{displaymath}
U(u)=E_m^p(\mu)E_m^p(\nu),
\end{displaymath}](h_1013.gif) |
(8) |
where
are Ellipsoidal Harmonics. The regular solution is therefore
![\begin{displaymath}
V=Ar^mE_m^p(\mu)E_m^p(\nu).
\end{displaymath}](h_1015.gif) |
(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.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 514 and 659, 1953.
© 1996-9 Eric W. Weisstein
1999-05-25