Integral of something like Gamma Function

Question

I am currently reading a paper that uses this integral as being less that one automatically. I am just wondering how do I know this to be true? Any help would be appreciated. Kindest regards, Catherine

\begin{equation} \int^{\varepsilon^{-2}cT}_0 e^{-\eta} \eta^{-\beta/m} \, d\eta \end{equation}

Everything apart from $\eta$ is constant.

Answer 1

You can take a look at the incomplete Gamma function. Indeed, your integral can be expressed in terms of the lower incomplete gamma function $\gamma(s, x)$: $$\int_0^{\epsilon^{-2}cT} e^{-\eta}\eta^{-\beta/m}\,d\eta=\gamma\left(-\beta/m+1,\epsilon^{-2}cT\right)$$ Right now, I don't think there is a way to show that this integral will be less than one, unless contraints are placed on the parameters.