On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function:
$$ \int_0^\infty e^{-\pi tn^2}t^{(s+\varepsilon)/2} \frac{dt}{t} = \pi^{-(s+\varepsilon)/2} \Gamma\left(\frac{s+\varepsilon}{2}\right)n^{-s-\varepsilon}$$
where $\Re(s) > 1$ and $\varepsilon \in \{0, 1\}$.
The only thing I know about the Gamma function is its definition via the integral
$$\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}dx.$$
How do I prove the identity Bump cites?
A change of variable should work \begin{align} \int_0^\infty e^{-\pi tn^2}t^{(s+\varepsilon)/2-1} dt &=_{u=\pi n^2t} \int_0^\infty e^{-u}\left(\frac{u}{\pi n^2}\right)^{(s+\varepsilon)/2-1}\frac{du}{\pi n^2} \\ =& \pi^{-(s+\varepsilon)/2} \Gamma\left(\frac{s+\varepsilon}{2}\right)n^{-s-\varepsilon} \end{align}