Let $C$ be a simple closed positively oriented contour, and suppose $f$ is complex analytic both within and on $C$. Assume that the origin lies inside $C$.
How deal with contour integrals of the form
$$ \int_{C} \frac{f(z)}{z^\frac{1}{m}}dz,\qquad m\in \mathbb{N}.$$
Motivation: I need to solve the following integral $$ \int_{C_1} \frac{e^{-z^2}}{(z-\dot{\imath} t)^\frac{1}{3}}dz,\qquad t\in \mathbb{R}.$$
Obviously, I am interested in the case where $t$ lies in the domain $C_1$ encloses.