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I'm trying to solve this integral.

$$ \int_0^\infty dx' \int_{-\infty}^\infty dy' \frac{1}{((x-x')²+(y-y')² +z²)^{3/2}} $$ I wasn't able to come up with a proper substitution yet.

This integral is an attempt to solve the Potential of a point charge in the half space V := {$\textbf{r} \in \mathbb{R}^3| z \geq 0$} on the surface $\partial V$ = {$ \textbf{r} \in \mathbb{R}^3|z = 0 $} with Dirichlet Boundary Condition for the Green function with the method of images.

Where \begin{equation} \phi(\textbf{r})= \begin{cases} 0, & \textbf{r} \in \partial V, x<0\\ \Phi, & \textbf{r} \in \partial V, x\geq0 \end{cases} \end{equation}

Thanks in advance

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1 Answer 1

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Since $$ \forall A>0,\qquad \int_{-\infty}^{+\infty}\frac{dy'}{\left(A+(y-y')^2\right)^{3/2}}=\int_{-\infty}^{+\infty}\frac{dy'}{\left(A+y^2\right)^{3/2}}=\frac{2}{A}\tag{1} $$ the given integral equals $$ 2\int_{0}^{+\infty}\frac{dx'}{(x-x')^2+z^2}=\frac{2}{|z|}\int_{-x/|z|}^{+\infty}\frac{dx'}{x'^2+1^2}=\frac{2}{|z|}\left(\frac{\pi}{2}+\arctan\frac{x}{|z|}\right).\tag{2}$$

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