I currently have the following Fourier transformation that I need to compute \begin{equation} \int \frac{d^3q}{(2\pi)^3}\frac{\log(\mathbf{q^2})}{(\mathbf{q})^4}e^{i\mathbf{q}\cdot \mathbf{r}} \end{equation} where I know that if I had instead just the $\log(\mathbf{q}^2)$ I would get $-(2\pi r^3)^{-1}$ and for $1/|\mathbf{q}|$ becomes $(2\pi^2 r^2)^{-1}$ and $1/|\mathbf{q}|^2$ would be $(4\pi r^2)^{-1}$. I even know that if I had a $\mathbf{q}^{-4}$ that the Fourier transform would give $-\pi^2 r$ (which would be dependent upon the regularization scheme) but I don't know how to handle the one I first stated. I listed all the previous ones in hope that there may be a way to use them.
Here he how I would start \begin{align} \int\frac{d^3q}{(2\pi)^3}\frac{\log(\mathbf{q}^2)}{\mathbf{q}^4}e^{i\mathbf{q}\cdot\mathbf{r}} & = \frac{1}{(2\pi)^3}\int_0^\infty dq\frac{\log(q^2)}{q^4}\cdot q^2\int_{-1}^1 d\cos(\theta) \int_0^{2\pi} d\phi e^{iqr\cos(\theta)}\\ & = \frac{1}{(2\pi)^2}\int_0^\infty dq \frac{1}{iqr}{\frac{\log(q^2)}{q^2}}(e^{iqr} - e^{-iqr})\\ \end{align} and here is where I do not know how to handle the analyticity of the integral. Perhaps I should add a small $i\delta$ into the denominator and then perform the contour integration (where $\delta>0$) which would only include one of the exponential terms (I think). Anyways, any sort of suggestions are helpful, thanks!