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Let $\Omega$ be a simply connected region in $\mathbb{C}$ and $\gamma$ a closed piecewise continuous differentiable path.

Is there an intuitive explanation why the winding number $\mathrm{ind}_\gamma(\alpha)=0$, $\alpha \in \mathbb{C}\backslash \Omega$, on simply connected regions?

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A simply connected open set is, by definition, is a set in which every closed chain is homologous to a point. I.e., an open set in which, by definition, every closed chain has winding number 0. There's not necessarily an intuitive reason, but the motivation for the definition is that these are exactly the open sets in which you can define the logarithm. – rotskoff Jun 6 '12 at 23:59
up vote 3 down vote accepted

I assume you mean simply connected rather than "simple", and also that you mean $\mathbb C\setminus \Omega$ rather than the other way around.

In that case it is because by definition of "simply connected" any closed path in a simply connected region is homotopic to a point. The winding number is a continuous function of the curve, and since for a closed curve it is always an integral multiple of $2\pi$ the winding number cannot change during a homotopy, provided that it exists at every intermediate step -- such as here when $\alpha\notin \Omega$.

Since a point (i.e., a constant curve) obviously has winding number 0, so has any closed curve that is homotopic to it.

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Great, thank you! – Chris Jun 6 '12 at 23:25

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