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