Poisson's 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).

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References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485-486,
905, and 912, 1985.
Cite this as:
Weisstein, Eric W. "Green's Function--Poisson's Equation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/GreensFunctionPoissonsEquation.html

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