Green's function for the three-variable Laplace equation

In physics, the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form

${\displaystyle \nabla ^{2}u(\mathbf {x} )=f(\mathbf {x} )}$

where ${\displaystyle \nabla ^{2}}$ is the Laplace operator in ${\displaystyle \mathbb {R} ^{3}}$, ${\displaystyle f(\mathbf {x} )}$ is the source term of the system, and ${\displaystyle u(\mathbf {x} )}$ is the solution to the equation. Because ${\displaystyle \nabla ^{2}}$ is a linear differential operator, the solution ${\displaystyle u(\mathbf {x} )}$ to a general system of this type can be written as an integral over a distribution of source given by ${\displaystyle f(\mathbf {x} )}$:

${\displaystyle u(\mathbf {x} )=\int _{\mathbf {x} '}G(\mathbf {x} ,\mathbf {x'} )f(\mathbf {x'} )d\mathbf {x} '}$

where the Green's function for Laplacian in three variables ${\displaystyle G(\mathbf {x} ,\mathbf {x'} )}$ describes the response of the system at the point ${\displaystyle \mathbf {x} }$ to a point source located at ${\displaystyle \mathbf {x'} }$:

${\displaystyle \nabla ^{2}G(\mathbf {x} ,\mathbf {x'} )=\delta (\mathbf {x} -\mathbf {x'} )}$

and the point source is given by ${\displaystyle \delta (\mathbf {x} -\mathbf {x'} )}$, the Dirac delta function.