How to evaluate an integral involving incomplete gamma , exponential and power functions?

Question

I am attempting to solve the following integral:

\begin{equation} \int_0^\infty z^{\kappa^2 -2} \Gamma\left(-\frac{\kappa^2}{2}+n+1,\frac{(K+1) z^2}{A_0^2 \Omega }\right) \exp{\left(-\left(\frac{1}{2}\right)\left(\frac{w/z-\mu_R}{\sigma_R}\right)^2\right)} dz \end{equation}

This integral arises when I try to derive the expression for the probability density function (pdf) of the product of three random variables using the multiplicative convolution technique.

By making a few substitutions, abbreviating the constant terms by defining $\alpha=-\frac{\kappa^2}{2}+n+1$ and $\beta=\frac{(K+1)}{A_0^2 \Omega }$ and setting $\mu_R=0$, I was able to express it in a much cleaner way.

\begin{equation} \frac{1}{2}\left(\frac{\sqrt{2}\sigma_R}{w}\right)^{-\kappa^2+1}\int_0^\infty y^{\frac{\kappa^2}{2}} e^{-y} \Gamma\left(\alpha,\frac{\beta w^2}{2 \sigma_R^2 y} \right) dy \end{equation}

It looks like the final result might be a product of gamma functions, but I am still unable to figure out the final solution by using any techinique I am familiar with. Also, Wolfram Alpha doesn't have an answer for it.

Answer 1

Mathematica yield a result for

$$\int_0^{\infty } e^{-y} y^{\alpha } \Gamma \left(\beta ,\frac{\gamma }{y}\right) \, dy$$

$$\Gamma (\alpha +1) \Gamma (\beta )-\pi \csc (\pi (\alpha -\beta )) \left(\gamma ^{\alpha +1} \Gamma (\alpha +1) \, _1\tilde{F}_2(\alpha +1;\alpha +2,\alpha -\beta +2;\gamma )-\gamma ^{\beta } \Gamma (\beta ) \, _1\tilde{F}_2(\beta ;\beta +1,\beta -\alpha ;\gamma )\right)$$

$$\text{if}\quad \Re(\gamma )\geq 0\land \Re(\alpha )>-1\land (\Re(\alpha -\beta )>-3\lor \Re(\gamma )>0)$$

in Mathematica code

  Gamma[1+\[Alpha]] Gamma[\[Beta]]-
  \[Pi] Csc[\[Pi] (\[Alpha]-\[Beta])]
 (\[Gamma]^(1+\[Alpha]) Gamma[1+\[Alpha]] 
HypergeometricPFQRegularized[{1+\[Alpha]},{2+\[Alpha],2+\[Alpha]-
\[Beta]},\[Gamma]]-
\[Gamma]^\[Beta] Gamma[\[Beta]] 
HypergeometricPFQRegularized[
    {\[Beta]},{1+\[Beta],-\[Alpha]+\[Beta]},\[Gamma]]) 
if Re[\[Gamma]]>=0&&Re[\[Alpha]]>-1&&(Re[\[Alpha]-\[Beta]]>-3||Re[\[Gamma]]>0)