It is well-known that $$\int_{-\infty}^{+\infty} \frac{\mathrm{d}x}{1+x^2}=\pi$$
Since $\pi$ is also the area of the unit disk, there may exist an area-preserving mapping between the unit disk and
$$D=\left\{(x,y)\in\mathbb{R}^2, 0 \leq y\leq \frac{1}{1+x^2}\right\}$$
What I mean by "area-preserving" is that for any measurable $C$ set of $D$, $C$ and its image in the disk have the same area.
I have two questions :
- Does there exist one such mapping ?
- If so, can we find an explicit expression ?