# Difficult integral, maybe multidimensional contour integration

I need to solve $$\int_{\mathbb{R}^3}d^3k\frac{e^{i\vec{k}\cdot\vec{\rho}}}{|\theta|+k^2}$$ I have a feeling that I should use contour integration, but in three variables I do not know how to employ it. Moreover, Mathematica cannot solve it for me.

I should say that I have some notes that say that I should pass from a form like $-i4\pi\int_{-1}^{1}d\cos\theta\int_0^\infty\frac{dk\sin(kr)}{|\theta|+k^2}$ but I do not know why.

I'd appreciate a help in solving that.

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Since you are a new user, you should explains what did you try. In this way, it would be easier for any user to help you. –  Felix Marin Jan 23 '14 at 15:42
In the integral, do you mean $\rho^2=k^2$ ? –  Tom-Tom Jan 23 '14 at 15:52
V.Rossetto, you are right. Corrected –  Rimon Jan 23 '14 at 16:45


\begin{align} \int_{\mathbb{R}^{3}} {\expo{\ic\vec{k}\cdot\vec{\rho}} \over \verts{\theta} + k^{2}}\,\dd^{3}\vec{k} &={2\pi \over \rho}\int_{-\infty}^{\infty}{x\sin\pars{x} \over k^{2} + \mu^{2}}\,\dd x = {2\pi \over \rho}\, \Im\int_{-\infty}^{\infty}{x\expo{\ic x} \over k^{2} + \mu^{2}}\,\dd x \\[3mm]&={2\pi \over \rho}\, \Im\bracks{2\pi\ic\,{\ic\mu\expo{\ic\pars{\ic\mu}} \over \ic\mu + \ic\mu}} ={2\pi^{2} \over \rho}\expo{-\mu} \end{align}

$$\color{#00f}{\large% \int_{\mathbb{R}^{3}} {\expo{\ic\vec{k}\cdot\vec{\rho}} \over \verts{\theta} + k^{2}}\,\dd^{3}\vec{k} = {2\pi^{2} \over \rho}\expo{-\rho\root{\verts{\theta}}}}$$

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Oh, wow, thanks! You spared me much time, I have never seen that substitution of the sine –  Rimon Jan 23 '14 at 16:50
@Rimon It's a usual one since $\large{\rm e}^{{\rm i}\theta} = \cos\left(\theta\right) + {\rm i}\sin\left(\theta\right)$. Thanks. –  Felix Marin Jan 23 '14 at 16:55
Actually I was referring to the first substitution (in polar coordinates?) –  Rimon Jan 23 '14 at 22:26