Integral representation of cosh x On Wolfram math world, there's apparently an integral representation of $\cosh x$ that I'm unfamiliar with. I'm trying to prove it, but I can't figure it out. It goes \begin{equation}\cosh x=\frac{\sqrt{\pi}}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \frac{ds}{\sqrt{s}}\,e^{s+\frac{1}{4}\frac{x^{2}}{s}} \gamma >0\end{equation}
The contour is taken along a vertical line with positive real part. I thought at first glance to use the residue theorem but it seems to be of no use here.

It will almost certainly help to know that 
$$
\mathcal L^{-1}\{F(s)\}= \frac1{2\pi i} 
\int_{\gamma - i\infty}^{\gamma + i\infty} e^{st}F(s)\,ds \quad \gamma>0
$$
Gives you the Inverse Laplace transform of $F(s)$ (where $\gamma$ is taken to be in the region of convergence of $F(s)$).  Your formula looks very much like some permutation of the above.
