Fundamental solution of the Laplacian

Summary

The fundamental solution of the Laplacian is a radial function $\Phi$ on $\R^n \setminus \set{0}$ with $-\Delta \Phi = \delta$, the Dirac delta at the origin. It equals $-\frac{1}{2\pi} \log \abs{x}$ in the plane and $\frac{1}{n(n-2)\omega_n} \abs{x}^{2-n}$ in dimension $n \ge 3$, where $\omega_n$ is the volume of the unit ball. It is the fundamental solution of the Laplace operator that is used throughout potential theory, up to sign. Convolving it with a source $f$ solves Poisson's equation, and it is the building block of Green's functions and of the Newtonian and logarithmic potentials.