Integral representation of incomplete gamma function

[patric44]

TL;DR

verifying the integral representation of incomplete gamma function

hi guys
I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by
$$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$
i was thinking about taking substitutions in order to reach the generating function of Bessel, but it took me nowhere.
i will appreciate any help

 

[fresh_42]

I'm not sure I understand what you want to do. Are you sure your formula is correct? Where did you get it from? Nevertheless, here are some formulas:
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Bestimmte_Integrale:_Form_R(x,BesselJ)

 

[patric44]

the formula is correct but I guess it is not that famous

 

[jasonRF]

I would be tempted to start with an integral representation of the Bessel function then switch the order of integration. You might need to try several different ones. It might not lead anywhere, but might be worth a try. My go-to online resource for formulas
https://dlmf.nist.gov

your formula is Equation 8.6.2
https://dlmf.nist.gov/8.6

jason