Green's function for the three-variable Laplace equation

The free-space Green's function for the three-variable Laplace equation is given in terms of the reciprocal distance between two points. That is to say the solution of the equation

:$abla^2\; G(mathbf\{x\},mathbf\{x\text{'}\})\; =\; delta(mathbf\{x\}-mathbf\{x\text{'}\})$

is

:

which has been written in terms of spherical coordinates $,!(r,\; heta,varphi)$. The less than (greater than) notation means, take the primed or unprimed spherical radius depending on which is less than (greater than) the other. The $,!gamma$ represents the angle between the two arbitrary vectors $(mathbf\{x\},mathbf\{x\text{'}\})$ given by

:$cosgamma=cos\; hetacos\; heta^prime\; +\; sin\; hetasin\; heta^primecos(varphi-varphi^prime).$

The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics text 3rd ed. pages 125-127. Using the Green's function for the three-variable Laplace equation one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation. There are many expansions in terms of special functions for the Green's function. In the case of a boundary put at infinity with the boundary condition setting the solution to zero at infinity then you have an infinite-extent Green's function. For the three-variable Laplace equation one can for instance expand it in the rotationally invariant coordinate systems which allow separation of variables. For instance:

:

where are the greater (lesser) variables $,!z$ and $,!z^prime$.Similarly, we can give the Green's function for the three-variable Laplace equation as a Fourier integral cosine transform of the difference of vertical heights whose kernel is given in terms of the order zero modified Bessel function of the second kind as

:

Rotationally invariant Green's functions for the three-variable Laplace equation

Green's function expansions exist in all of the rotationally invariant coordinate systems which are known to yield solutions to the three-variable Laplace equation through the separation of variables technique.

* cylindrical coordinates
* spherical coordinates
* Prolate spheroidal coordinates
* Oblate spheroidal coordinates
* Parabolic coordinates
* Toroidal coordinates
* Bispherical coordinates
* Flat-ring cyclide coordinates
* Flat-disk cyclide coordinates
* Bi-cyclide coordinates
* Cap-cyclide coordinates