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Date: 13-6-2018
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Date: 11-6-2018
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Date: 30-5-2018
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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).
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485-486, 905, and 912, 1985.
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