I am trying to evaluate $\displaystyle \int_\gamma z^n dz$ where $\gamma$ is a circle not containing the origin with positive orientation ($n\in \mathbb{N}$).
I first calculated $\displaystyle \int_{\gamma_1} z^n dz$ over a circle around the origin and found the answer to be $0$.
Now, let us consider $\gamma$ to be the circle with center $w$ and radius $r$ such that $r<|w|$ i.e. $\gamma$ does not contain the origin. Consider a parametrization $z(\theta)=w+re^{i\theta}$ of the curve $\gamma$.
Thwn $\displaystyle \int_\gamma z^n dz=\int_{0}^{2\pi}(w+re^{i\theta}).(w+re^{i\theta})'d\theta=\sum_{k=0}^nw^{n-k}\binom{n}{k}\int_0^{2\pi}re^{i\theta}(re^{i\theta})'d\theta=\sum_{k=0}^nw^{n-k}\binom{n}{k}\int_{\gamma_1}z^k dz=0$.
Is this calculation correct and where am I using the fact that $\gamma$ does not contain the origin i.e. $r<|w|$? Thanks in advance.