I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone could explain the steps from left to right. I guess it is related to the branch cuts where my knowledge is limited.
$$
\frac{1}{i}\int_{\epsilon-i\infty}^{\epsilon+i\infty} dt \frac{\exp\left( a t \right)}{\sin^{2\mu} \left( \pi t \right)}= \int_{0}^{\infty} dx \frac{\exp\left( i a x - i \pi \mu \right)}{\sinh^{ 2 \mu} \left( \pi x \right)} + \mbox{C.C.}
$$
Thank you!