We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Prove that the Radon transform of $f$ is defined on $\mathbb{R}^2$ and that
$$\forall q \in \mathbb{R}^+,\quad \forall \theta \in \mathbb{R} \quad \hat{f}(q,\theta) = 2\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$