How to compute the following integral :
\begin{equation} I = \int \frac{e^{-q|\mathbf r|}}{|\mathbf r|}e^{i\mathbf k\cdot\mathbf r}. d\mathbf r \end{equation}
where this 3D volume integral is considered carried out on an unbound domain. This looks like a simple Fourier transform of the Coulomb potential $\frac{e^{-q|\mathbf r|}}{|\mathbf r|}$ .
Integrating in spherical coordinates choosing $\mathbf k$ as the z axis, the integration on $\phi$ provides a factor $2\pi$ leaving the angular integral:
\begin{equation} I_{\theta} = \int_0^{\pi} e^{ikr\cos\theta}\sin\theta d\theta = \left[ \frac{e^{ikrx}}{ikr}\right]_{-1}^{+1} = \frac{\sin kr}{kr} \end{equation}
and radial integral :
\begin{equation} I_r = \int_0^{\infty} \frac{e^{-qr}}{r} I_{\theta}(kr)rdr = \Im_m(\int_0^{\infty} \frac{e^{(ik-q)r}}{kr}dr) \end{equation}
Using the method of residue could give :
\begin{equation} I_r = \frac{1}{2k}\Im_m(2i\pi) Res(z\frac{e^{iQz}}{z},0)=\frac{\pi}{k} \end{equation} where $Q = k+iq$ is a complex number.
I know the result should eventually give something of the form : $I \propto\frac{1}{\mathbf k^2 + q^2}$ there must be a mistake in my application of the theorem of residue but can not figure out where.
