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 $x_0,y_0,z_0$



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