# Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry.

Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, and $\chi_{\Omega}(y)$ is the characteristic funtion of $\Omega$

In particular, for n=3 and only need explicit expression of exterior Newtonian potential.

Or more precisely what is the explicit expression of the following integration as a function of $x_0,y_0,z_0$

$u(x_0,y_0,z_0)=\int_{B(0,1)}\frac{1}{\sqrt{a^2(x-x_0)^2+b^2(y-y_0)^2+c^2(z-z_o)^2}}$