Calculate the winding number of a triangle

Question

My question is about calculating the winding number of a triangle, i.e. $$ Ind_\gamma(z) = \frac{1}{2i\pi} \int_\gamma\frac{1}{w-z}dw, $$ where $\gamma$ is the path given by the border of a triangle $\Delta \subset \mathbb C$.

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What I have done so far is calculating the value of the function outside of the triangle, and that is $Ind_\gamma(z) = 0 \quad \forall z \notin\Delta$ $\;$, but I don't know how to calculate it inside $\Delta$, intuitively I guess that its value is $Ind_\gamma(z) = 1 \quad \forall z \in\Delta.$

Answer 1

You can use Cauchy's Integral Formula, with $\;f(z)=1\;$ , and get

$$\frac1{2\pi i}\oint_\Delta\frac1{w-z}dw=f(z)=1$$

and we're done.

Another way is trying to parametrize the triangle's perimeter curve, say: take the triangle whose vertices are $\;(0,0),\,(0,1),\,(1,0)\;$, then the horizontal side is parametrized by $\;r(t)=t(1,0)+(1-t)(0,0)=(t,0),\,0\le t\;$ , and etc.