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?

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the asymptitics of the trace of the resolvent for some $n\in\mathbb{N}$,i.e. $\text{tr}(\Delta-\mu)^{-n}$ and then using Cauchy formula $\text{tr}e^{-t\Delta}=\frac{(n-1)!}{2\pi i}\int_{\Gamma}e^{-t\mu}\text{tr}(\Delta-\mu)^{-n}d\mu$ find the desired asymptotics. On a compact Riemannian manifold asymptotics of the trace of the resolvent is given by power series so the contour integral is easy to calculate, but on manifold with singularities some logarithmic terms might occur and I do not know how to deal with the corresponding integrals. The exact statement of the problem is the following.

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?