
complex analysis
Let $\displaystyle \gamma$: [0,1] $\displaystyle \rightarrow C$ be any $\displaystyle C^1$ curve. Define $\displaystyle f(z)=\oint_{\gamma} \frac{1}{\zetaz}d\zeta$
Prove that $\displaystyle f$ is holomorphic on $\displaystyle C$\$\displaystyle \tilde{\gamma}$, where $\displaystyle \tilde{\gamma}=\{ \gamma (t):0\leq t \leq 1\}.$

You can compute the difference quotient $\displaystyle [f(z+w)f(z)]/w$ as $\displaystyle w\rightarrow 0$. The limit is (what a surprise) $\displaystyle f'(z)=\int_{\gamma} \frac{1}{(\zetaz)^2}d\zeta$.
ps. what's with the \oint? I believe that notation is,
only to signify counterclockwise integration along a closed curve (Thinking)