Green's Function--Poisson's Equation

Poisson's Equation equation is

$$\nabla^2\phi=4\pi\rho,$$ (1)

where $\phi$ is often called a potential function and $\rho$ a density function, so the differential operator in this case is $\tilde L = \nabla^2$. As usual, we are looking for a Green's function $G({\bf r}_1,{\bf r}_2)$ such that

$$\nabla^2 G({\bf r}_1,{\bf r}_2) = \delta^3({\bf r}_1-{\bf r}_2).$$ (2)

But from Laplacian,

$$\nabla^2\left({1\over \vert{\bf r}-{\bf r}'\vert}\right)= -4\pi\delta^3({\bf r}-{\bf r}'),$$ (3)

so

$$G({\bf r},{\bf r}') = - {1\over 4\pi \vert{\bf r}-{\bf r}'\vert},$$ (4)

and the solution is

$$\phi({\bf r})=\int G({\bf r},{\bf r}')[4\pi \rho({\bf r}')]\... ... {\rho({\bf r}')\,d^3{\bf r}'\over\vert{\bf r}-{\bf r}'\vert}.$$ (5)

Expanding $G({\bf r}_1,{\bf r}_2)$ in the Spherical Harmonics $Y_l^m$ gives

$$G({\bf r}_1,{\bf r}_2) = \sum_{l=0}^\infty \sum_{m=-l}^l {1\... ...r r_>^{l+1}} Y_l^m(\theta_1,\phi_1){Y_l^m}^*(\theta_2,\phi_2),$$ (6)

where $r_<$ and $r_>$ are Greater Than/Less Than Symbols. This expression simplifies to

$$G({\bf r}_1,{\bf r}_2) = {1\over 4\pi} \sum_{l=0}^\infty {r_<^l\over r_>^{l+1}} P_l (\cos\gamma),$$ (7)

where $P_l$ are Legendre Polynomials, and $\cos\gamma\equiv {\bf r}_1\cdot{\bf r}_2$. Equations (6) and (7) give the addition theorem for Legendre Polynomials.

In Cylindrical Coordinates, the Green's function is much more complicated,

$$G({\bf r}_1,{\bf r}_2) = {1\over 2\pi^2}\sum_{m=-\infty}^\in... ...\rho_<)K_m(k\rho_>)e^{im(\phi_1-\phi_2)}\cos[k(z_1-z_2)]\, dk,$$ (8)

where $I_m(x)$ and $K_m(x)$ 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.