I'm trying to evaluate what seems to be a straightforward contour integral:
$I=\int_{\gamma} \frac{dz}{\alpha + \beta z} $
where $\gamma (t) = e^{-it}$, $t \in \left[ 0,\pi\right]$, $\alpha, \beta \in \mathbb{C}$, and $|\alpha| \ne |\beta|$. Explicitly,
$I=\int_{0}^{\pi} \frac{(-ie^{-it})dt}{\alpha + \beta e^{-it}}$ $=-i\int_{0}^{\pi} \frac{dt}{\alpha e^{it} + \beta }$
I want to say that
$I=\frac{1}{\beta}\left(\log\left(\frac{\beta -\alpha}{\beta +\alpha}\right)-i \pi \right)$
but I feel like I'm neglecting some subtle branch cut issues. Any help or insight would be greatly appreciated.