dourtuntellorvl

2022-06-30

We know the Hankel transform of order 0 is defined as
${F}_{0}\left(k\right)={\int }_{0}^{\mathrm{\infty }}f\left(r\right){J}_{0}\left(kr\right)\phantom{\rule{thinmathspace}{0ex}}r\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}r.$
In this regard, I am now trying to calculate the Hankel transform of the function
$f\left(r\right)=\frac{{e}^{-a\sqrt{{r}^{2}+{z}^{2}}}}{\sqrt{{r}^{2}+{z}^{2}}},$
with $a\in \mathbb{R}$ . Unfortunately, I was only able to obtain the solution for $a=0$ . Any thoughts on how to solve this?

Paxton James

Expert

Insert into left-hand side of (1) and interchange $\int .$ The innermost integral has a closed form,

which Mathematica knows. So we now have

where from line 2a to 2b we substituted ${t}^{2}\to t$ and in the last step the limits of the integrand have been shifted with a subsequent change of the parameter. Now use the integral relationship
${\int }_{0}^{\mathrm{\infty }}{J}_{0}\left(c\sqrt{u}\right)\left(u+p{\right)}^{-3/2}du=2\frac{\mathrm{exp}\left(\phantom{\rule{negativethinmathspace}{0ex}}-c\sqrt{p}\right)}{\sqrt{p}}.$
Mathematica knows this integral with $c=1,$ and it is easy to work in the scaling factor. Algebra completes the proof.

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