Fundamental solution of the Laplacian

Remarks

The function $\Phi$ is locally integrable on $\R^n$, and in the sense of distributions it satisfies $-\Delta \Phi = \delta$, where $\Delta$ is the Laplacian and $\delta$ is the Dirac delta at $0$. So $-\Phi$ is a fundamental solution of $\Delta$ in the general sense; many texts use this sign convention so that $\Phi > 0$ near the origin. Example: for $n = 3$, $\omega_3 = \frac{4\pi}{3}$ and $\Phi(x) = \frac{1}{4\pi\abs{x}}$. For a radial function $f(r)$ with $r = \abs{x}$ one has $\Delta f = f'' + \frac{n-1}{r} f'$, and $f = r^{2-n}$ gives $(2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$, so $\Phi$ is harmonic on $\R^n \setminus \set{0}$. Non-example: $\abs{x}^{-1}$ on $\R^4$ is not a multiple of $\Phi$, and it is not harmonic there, since $\Delta \abs{x}^{-1} = -\abs{x}^{-3}$. The constant is chosen so that the flux of $-\nabla \Phi$ out of every sphere about $0$ equals $1$; here $n\omega_n$ is the area of the unit sphere. Other normalisations, such as $\abs{x}^{2-n}$ or $\log(1/\abs{x})$ without constants, are common in potential theory.