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$\gamma _1$ and $\gamma _2$ are two closed curves in $\Bbb C\setminus\{0\}$ both with parameter interval $[a,b]$. We define the curve $\gamma(t)=\gamma_1(t)\gamma_2(t)$.

I've just shown that $\gamma$ is closed in $\Bbb C\setminus\{0\}$. Now I have to show that $w(\gamma ,0)=w(\gamma _1 ,0)+w(\gamma _2 ,0)$. I first used a theorem saying that under the given circumstances(or so I thought), $w(\gamma,0)=(1/2\pi i)\int_\gamma (dz/z)$, which easily solved the problem. It turned out, however, that for this to be true, $\gamma$ has to be piecewise $C^1$, which we don't know if it is, so I can't use that theorem. Any ideas what to do instead?

EDIT: My book defines winding number as $w(\gamma,0)=\frac{1}{2\pi}argvar(\gamma)$, and $argvar(\gamma)=\theta(b)-\theta(a)$, where $\theta$ is an arbitrary continuous argument function along $\gamma$.

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  • $\begingroup$ what exactly do you call $w$ ? $\endgroup$ – Glougloubarbaki May 30 '13 at 20:44
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    $\begingroup$ How are you defining the winding number and which textbook are you using? You are right that it is a simple consequence of the integral formula for winding number if the curve is piecewise smooth. However, for more general curves, you need to use a covering space argument. More context would allow me to provide a more appropriate answer for you situation. $\endgroup$ – Potato May 30 '13 at 20:50
  • $\begingroup$ Sorry, I've defines the winding number now. $\endgroup$ – MBrown May 30 '13 at 21:27
  • $\begingroup$ Which book? The point of defining winding number is that you can't define a single valued continuous argument function on $\mathbb C\backslash \{0\}$, so that definition needs some tweaking. $\endgroup$ – Potato May 30 '13 at 21:32
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Here's a more geometric description of what's going on.

Since all the curves avoid the origin, you can parametrize them in polar coordinates using Euler's formula (treating $\mathbb{C} \setminus 0 \cong \mathbb{R}^2 \setminus \{0,0\}$). $$ \gamma(t) = \begin{pmatrix} r(t) \cos \theta(t) \\ r(t) \sin \theta(t)\end{pmatrix}, $$ where $r$ and $\theta$ are continuous functions from $[a, b] \to \mathbb{R}$, and $r(t) > 0$ for all $t$. Now, normalize by projecting onto the unit circle: $$ \begin{array}{rccc} \epsilon = \frac{\gamma}{||\gamma||}: &[a,b] &\longrightarrow &S^1 \\ & t &\longmapsto &\begin{pmatrix} \cos \theta(t) \\ \sin \theta(t)\end{pmatrix} \end{array} $$ This map lifts to a map $\tilde{\epsilon}: [a, b] \to \mathbb{R}$ that is unique once you specify $\tilde{\epsilon}(a)$ (or the image of any one point, for that matter). In other words, $$ \epsilon = p \circ \tilde{\epsilon}, $$ where $p$ is the projection $\mathbb{R} \to S^1$, $u \to (\cos u, \sin u)$.

If $\gamma$ is a closed curve, then $\epsilon$ is too, so $$ \tilde{\epsilon}(b) - \tilde{\epsilon}(a) = 2\pi w $$ for some $w \in \mathbb{Z}$. This $w$ is the winding number of $\gamma$ around $0$.


Now, to address your actual question, in which you multiply the curves in the complex plane, first parametrize them as maps $[a, b] \to \mathbb{R}^2 \setminus \{0, 0\}$: $$ \gamma_1(t) = \begin{pmatrix} r_1(t) \cos \theta_1(t) \\ r_1(t) \sin \theta_1(t)\end{pmatrix} $$ and $$ \gamma_2(t) = \begin{pmatrix} r_2(t) \cos \theta_2(t) \\ r_2(t) \sin \theta_2(t)\end{pmatrix}. $$ Now, $$ \begin{align} \gamma(t) &= \gamma_1(t) \gamma_2(t) \\ &= r_1(t) r_2(t) \begin{pmatrix} \cos \theta_1(t) \cos \theta_2(t) - \sin \theta_1(t) \sin \theta_2(t) \\ \sin \theta_1(t) \cos \theta_2(t) - \cos \theta_1(t) \sin \theta_2(t) \end{pmatrix} \\ &= r_1(t) r_2(t) \begin{pmatrix} \cos \big(\theta_1(t) + \theta_2(t) \big) \\ \sin \big(\theta_1(t) + \theta_2(t) \big) \end{pmatrix}. \end{align} $$ From this you can see that the winding number is additive.

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