Integral including lower incomplete gamma

Question

I am trying to calculate the following integral:

$\int_0^\infty \frac{A}{\Gamma{(\alpha})}\gamma(\alpha,t) dt$

where A is a positive coefficient and all parameters are positive. Any help, Thanks!

Note that:

• $\gamma(\alpha,t) =\int_0^t x^{\alpha-1}e^{-x} dx$

• $\Gamma(\alpha)$ is the complete gamma function

Answer 1

We start by writing out the integral, and differently denoting the domain of integration.

\begin{align*} A\int_0^{\infty}\int_0^t x^{\alpha - 1}e^{-x}dxdt =& A \int_{0}^{\infty} \int_x^{\infty}x^{\alpha - 1}e^{-x}dtdx \\ \end{align*} This clearly does not converge, since the integrand does not tend to $0$ as $t \to \infty$.