Let $\displaystyle \gamma$: [0,1] $\displaystyle \rightarrow C$ be any $\displaystyle C^1$ curve. Define $\displaystyle f(z)=\oint_{\gamma} \frac{1}{\zeta-z}d\zeta$

Prove that $\displaystyle f$ is holomorphic on $\displaystyle C$\$\displaystyle \tilde{\gamma}$, where $\displaystyle \tilde{\gamma}=\{ \gamma (t):0\leq t \leq 1\}.$