# Winding number of a point outside the curve is 0

I've been looking for the answer to the following question for a little while now:

Let $γ$ be a closed (C1-)curve whose image is contained in ${z: |z| < R}$ for some $R > 0$. Show that for any $z$ with $|z| > R$ we have $\operatorname{Ind}(γ,z) = 0$.

I think I am supposed to use the definition of the index of a winding number, but I have absolutely no idea of how to do it. To me if $z$ is outside the curve then the index is 0 by definition...

Any pointers would be greatly appreciated thanks!

• "Index of a winding number?" Commented Mar 20, 2014 at 21:56

$\mathrm{Ind}(\gamma,z_0) = \frac{1}{2\pi i} \int_{\gamma} \frac{1}{z-z_0} \textrm{d}z$.
Since $\left|z_0\right| > R$, the function $z \mapsto \frac{1}{z-z_0}$ is holomorphic on $\{z : \left|z\right| < R\}$ and the result follows by Cauchy's Theorem.
• There are a few different versions, but the simplest one (and what I'd imagine you saw first) states that if $f : U \rightarrow \mathbb{C}$ is holomorphic ($U \subset \mathbb{C}$ open connected) and $\gamma \subset U$ a closed curve, then $\oint_{\gamma} f(z) \textrm{d}z = 0$ Commented Mar 21, 2014 at 10:40