I need help calculating the following integral $$\int_0^\infty\!\mathrm{d}u\frac{u^{-s}(u+i\omega_n)^s}{u-w+i\omega_n+i\omega_m}$$ where $0<s<1$.
What I have thought so far is to do contour integration that involves two branch cuts. One of them is a "regular" one along the real axis, but the other one is not located on the real axis but at $[0+i\omega_n,\infty+i\omega_n)$ and I am not sure how to deal with it.
What I have so far according to my reasoning is $$ \begin{split} \oint_C \cdots =(1-e^{-i2\pi s})\int_0^\infty\!\mathrm{d}u\frac{u^{-s}(u+i\omega_n)^s}{u-w+i\omega_n+i\omega_m}\\ +(1-e^{i2\pi s})\int_{0+i\omega_n}^{\infty+i\omega_n}\!\mathrm{d}u\frac{u^{-s}(u+i\omega_n)^s}{u-w+i\omega_n+i\omega_m} =Res|_{u=w-i\omega_n-i\omega_m} \end{split} $$ but I have trouble relating the second term with the integral I'm looking for, and I'm not even sure if what I have is correct.