Abel transform

Summary

The Abel transform of a function $f$ of a radius $r \ge 0$ is $F(y) = 2 \int_y^\infty \frac{f(r)\, r}{\sqrt{r^2 - y^2}}\, dr$. It gives the integral of a circularly symmetric function of the plane along a line at distance $y$ from the centre. So it is the Radon transform of a radial function. It can be inverted by a formula of the same type, which solves an integral equation first studied by Abel. It is used to recover radial profiles of emission or density from line-of-sight measurements, for example in plasma physics and astronomy.