# complex analysis

• March 16th 2010, 11:53 PM
Kat-M
complex analysis
Let $\gamma$: [0,1] $\rightarrow C$ be any $C^1$ curve. Define $f(z)=\oint_{\gamma} \frac{1}{\zeta-z}d\zeta$
Prove that $f$ is holomorphic on $C$\ $\tilde{\gamma}$, where $\tilde{\gamma}=\{ \gamma (t):0\leq t \leq 1\}.$
• September 12th 2010, 09:43 PM
Rebesques
You can compute the difference quotient $[f(z+w)-f(z)]/w$ as $w\rightarrow 0$. The limit is (what a surprise) $f'(z)=-\int_{\gamma} \frac{1}{(\zeta-z)^2}d\zeta$.

ps. what's with the \oint? I believe that notation is,
only to signify counter-clockwise integration along a closed curve (Thinking)