I want to solve the Poisson equation with the following charge density: $$ \rho(\vec{r})=q\delta(\vec{r}-\vec{r_i})-q\frac{1}{4\pi r^2_D} \frac{e^{-|\vec{r}-\vec{r_i}|/r_D}}{|\vec{r}-\vec{r_i}|} $$ So I think the first step is to let $\vec{r_i}$ be zero and in the end transfer it back to $\vec{r_i}$ being free to choose again. My problem is that the density has the position as an argument and the potential should only have the radius as an argument. Transformation into spherical coordinates doesn't work here. I’ve also tried using some Fourier transformation with $$ \delta(\vec{r})=\frac{1}{(2\pi)^3}\int{}e^{i\vec{k}\vec{r}}d^3k=\frac{1}{(2\pi)^3}\int{}e^{ikr\cos\alpha}d^3k $$ but... the $\alpha$. I hope somebody has a tip about how to tackle this problem cause I’m really out of ideas.
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