I have been trying to evaluate the following integral using contour method \begin{equation} G(\tau)=\oint\frac{dz}{2\pi i} \frac{e^{{-z \tau}}}{z+ sign(Imz)}\frac{1}{e^{\beta z}+1}, \end{equation}
where $\tau \in (-\beta,\beta)$, and the contour is (as determined by the discontinuity of the sign function) the semi-circle enclosing the whole upper plane plus the semi-circle that encloses the lower half plane. The real axis is cut out as a branch cut.
The integral arises from trying to evaluate the infinite sum that can be written as a Lerch Zeta function \begin{equation} G(\tau)=\frac{1}{\beta}\sum_{-\infty}^{\infty} \frac{e^{{-i\omega_n \tau}}}{-i \omega_n-\text{sign}{(\omega_n})}, \end{equation}
where where $\omega_n$=$(2n+1)\pi$/$\beta$.