Green's function for the three-variable Laplace equation

In physics, the Green's function (or fundamental solution) for Laplace's equation 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

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 ${\displaystyle f(\mathbf {x} )}$:

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