 |
(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
![phi(r)=intG(r,r^')[4pirho(r^')]d^3r^'=-int(rho(r^')d^3r^')/(|r-r^'|).](/images/equations/GreensFunctionPoissonsEquation/NumberedEquation5.svg) |
(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,
![G(r_1,r_2)=1/(2pi^2)sum_(m=-infty)^inftyint_0^inftyI_m(krho_<)K_m(krho_>)e^(im(phi_1-phi_2))cos[k(z_1-z_2)]dk,](/images/equations/GreensFunctionPoissonsEquation/NumberedEquation8.svg) |
(8)
|
where 
and 
are modified Bessel functions
of the first and second
kinds (Arfken 1985).
