We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
We set $R(q) = \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta$. Prove that $q \mapsto \frac{R^{\prime}(q)}{q}$ is integrable on $]0, +\infty[$ and that
$$f(0,0) = -\frac{1}{\pi} \int_0^{+\infty} \frac{R^{\prime}(q)}{q}\,\mathrm{d}q$$
One may, to compute this last integral, use the change of variable $q = \operatorname{sh}(u)$.