Let $m\in\mathbb{N},t>0$ how to compute the integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$,
where $\gamma$ is contour $\{|\arg(\mu+1)|=\pi/4\}$ transversed upward?

Here are my thoughts:

If I try to use the Residue theorem I need a closed contour. Assume 0 is in the contour then we have troubles with $\log$ because the singularity at 0 is not a pole! And if 0 is not in the contour the the integral is zero... Is there any contour such that the integral is not zero?