Could you help me compute Abel transform of Gauss function. I need
$$A_g[\sigma](x) = \int_x^\infty \frac{r}{\sqrt{r^2-x^2}} e^{-(\frac{r}{\sigma})^2} \, d\mathrm{r}, \,\,\,\,\,\, x\geq0, $$ where $\sigma > 0$ and eventually transform of other bell shaped curves $$A_h[a,b,c](x) = \int_x^\infty \frac{r}{\sqrt{r^2-x^2}} \frac{1}{(a+br^2)^c} \, d\mathrm{r}, \,\,\,\,\,\, x\geq0, $$ where $a>0$, $b>0$, $c>0$.
Theory behind
There are particular relationships between point spread function PSF, line spread function LSF and edge spread function ESF. In particular $$\frac{d ESF(r)}{dr} = LSF(r),$$ and $$Abel(PSF(x)) = LSF(x).$$ When estimating PSF of the system, you typically estimate edge spread function by something as the slanted edge method. This function is noisy so you denoise it and fit something smooth to it, typically sigmoid function. But you are looking for PSF so your fit shall survive all that transforms and still be a good representation of optical system. Thus I need some bell shaped analytical function $PSF$, which has well defined Abel transform $LSF$, which in turn can be interpreted as a derivative of another well behaved sigmoid function $ESF$, which I would like to fit to my edge spread function, because there are these relationships
It is cumbersome to do these transforms numerically, because I would be loosing precision but at the same time I would like to have some flexibility to fit the shape properly.
Similar work was done Wood 2014, which implicitly assumes
$$A_h[1,b,3/2]=\frac{1}{\sqrt{b} + b^{3/2}r^2}$$
and
$$A_g[\sigma](x) = \frac{\sqrt{\pi} \sigma}{2} e^{-(\frac{r}{\sigma})^2}.$$
I have confirmed this using Wolfram alpha and substitution pointed by @eyeballfrog $y= \sqrt{r^2-x^2}$. In Dribinski et.al. (15) formula is slightly different but up to a constant scaling.
If you know any other bell shaped functions with a nice Abel transform, or general formula for $A_h[a,b,c](x)$ let me know.