## Green's Function--Poisson's Equation

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.