Poisson's Equation equation is
|
(1) |
where is often called a potential function and a density function, so the differential operator in this case is
. As usual, we are looking for a Green's function
such that
|
(2) |
But from Laplacian,
|
(3) |
so
|
(4) |
and the solution is
|
(5) |
Expanding
in the Spherical Harmonics gives
|
(6) |
where and are Greater Than/Less Than Symbols. This expression simplifies
to
|
(7) |
where are Legendre Polynomials, and
. Equations
(6) and (7) give the addition theorem for Legendre Polynomials.
In Cylindrical Coordinates, the Green's function is much more complicated,
|
(8) |
where and are Modified Bessel Functions of the First
and Second Kinds (Arfken 1985).
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485-486, 905, and 912, 1985.
© 1996-9 Eric W. Weisstein
1999-05-25