Contour Integral of Gamma Function

Theorem
Let $\Gamma \left({z}\right)$ denote the gamma function.

Let $y$ be a positive number.

Then for any positive number $c$
 * $\dfrac{1}{2\pi i} \displaystyle \int_{c-i\infty}^{c+i\infty} \Gamma \left({t}\right) y^{-t} \rd t = e^{-y}$

Proof
We will take the Contour Integral of $\Gamma \left({t}\right) y^{-t}$ about the rectangular contour, L, defined by the corners $c\pm iR$, $-N-\frac 1 2 \pm iR$. By the Residue Theorem, this contour is equal to $2\pi i$ times the residues of the poles it encloses, and from Poles of Gamma Function we see these poles are located at the nonnegative integers. Thus we have:
 * $\displaystyle \oint_L \Gamma \left({t}\right) y^{-t} \, \mathrm d z = 2 \pi i \displaystyle\sum_{k\mathop = 0}^N \operatorname{Res} \left({-k}\right)$

and thus we obtain
 * $ \displaystyle \lim_{N \to \infty} \displaystyle \lim_{R \to \infty} \displaystyle \oint_L \Gamma \left({t}\right) y^{-t} \mathrm d z = 2 \pi i\displaystyle \sum_{k\mathop = 0}^{\infty} \operatorname{Res} \left({-k}\right)$

From Residues of Gamma Function, we see that
 * $ \operatorname{Res} \left({-k}\right) =  \dfrac{ \left(-1\right)^k y^{k} }{k!} $

Which gives us:

Our goal from here is to show that the all but the righthand side of the rectangular contour go to 0 as we take these limits, as our result follows readily from this. $\gamma\left(t\right)= c\pm iR - t$ where $0<t<c+N+\frac{1}{2}$. thus the absolute value of the contour integral is given by
 * The top and bottom portions of the contour can be parameterized by:

From Bound on Complex Values of Gamma Function, we have that:

for all $|R|\geq 1$. Because $|R|\geq 1$, we have that


 * Combining the two inequalities we obtain:

We see that

as the poles of Gamma are at the nonpositive integers, which means that the integral is a definite integral of a continuous function. The above is enough to allow for the interchange of limits by the Dominated Convergence Theorem, thus we have

But using Equation $(1)$ from above we see:

Thus by the Squeeze Theorem we have
 * $ \displaystyle \lim_{R \to \infty} \left\vert \Gamma \left({c\pm iR - t}\right) \right\vert = 0$

Which means we have

Thus we have that the top and bottom of the contour go to 0 in the limit.
 * The left-hand side of the contour may be parameterized by
 * $\gamma\left(t\right)= -N-\dfrac 1 2 -it$

where t runs from -R to R. Thus the absolute value of integral of the left-hand side is given

Thus we have

Which gives us

Thus we have the left, top, and bottom of the rectangular contour go to 0 in the limit, which gives us