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patric44

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- 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

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

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