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If there is spherically symmetric gaussian charge density (http://en.wikipedia.org/wiki/Poisson's_equation)

[tex] \rho(\mathbf{r})=\frac{q_{i}}{(2 \pi)^{3/2} \sigma^3} e^{- \frac{\lvert r \rvert^2}{2 \sigma^2}}[/tex]

then it will have have potential [itex]\phi(r)[/itex] by solving Poisson equation [itex]\bigtriangledown^{2} \phi(r)=-4\pi\rho(\mathbf{r})[/itex]

[tex] \phi(r) = \frac{\mbox{erf} \bigg(\frac{r}{2\sigma}\bigg)}{r}[/tex]

But this is the case if gaussian is spherical symmetric which is diagonal element of convariance matrix are same [itex]\sigma_{x}^{2} = \sigma_{y}^{2} = \sigma_{z}^{2} = \sigma^2[/itex]

Since in general covariance matrix for tri-variate case give by

[tex]\varSigma=

\begin{bmatrix}

\sigma_{x}^{2} & \sigma_{x}\sigma_{y} & \sigma_{x}\sigma_{z} \\

\sigma_{y}\sigma_{x} & \sigma_{y}^{2} & \sigma_{y}\sigma_{z} \\

\sigma_{z}\sigma_{x} & \sigma_{z}\sigma_{y} & \sigma_{z}^{2} \\

\end{bmatrix}[/tex]

Now i want potential due to anisotropic gaussian distribution (i.e. if i have full covariance matrix). Will it have same analytic form of potential as give by spherically symmetric gaussian?

Is there article/ reference in literature where such problem solved. As i didn't found till now.

Any help will be highly thank full

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# Potential due to anisotropic gaussian distrubution

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