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Give an example of a closed rectifiable curve $\gamma$ in $\mathbb{C}$ such that for every $k\in\mathbb{Z}$ there is some $a$ out of the curve such that $n(\gamma;a)=k$.

Here, $n(\gamma;a)$ is the index of a curve $\gamma$ around $a$ defined by $\dfrac{1}{2\pi i}\displaystyle\int_{\gamma}\dfrac{dz}{z-a}$.

If we only care for $k\ge 0$, I have thought something like a "spiral" converging to $0$. Would this work?:

$\gamma(t)=te^{i/t}$ if $0<t\le 1$, and $\gamma(0)=0$

My idea is you can always choose a point between the arcs, so that the curve travels a fixed $k$ times counterclockwise around the point. But I don't know how to choose these points formally.

This is a fine example or is there something easier?

Tanius
  • 1,195
  • you can think of something like infinite many nested figure-eight. The points of positive index are on the the right of $(0,0)$ and the ones of negative index the left with the usual parametrization $t \rightarrow cos(t - \pi/2) + i sin(2t)$. The points of index $0$ are the ones outside of this curve.

    I failed to find a way to write it formally, so I can't prove that its actually rectifiable, but it should be because the radius of each "flower" tend to 0 (just make it tends to 0 quick enough).

     – sure Feb 14 '17 at 14:33

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