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I am trying to calculate the winding number of the curve $\gamma$ at point $a$. My strategy is $\oint_\gamma \frac{1}{z-a} \,dz = \oint _{\gamma_2} \frac{1}{z-a}\,dz = 2\pi i$ as the corridor gets closer. But this is wrong since the winding number should clearly be $2$ rather than $1$. Why is this argument wrong?

  • $\begingroup$ it is not true that $\int_\gamma \frac{dz}{z-a} = \int_{\gamma_2} \frac{dz}{z-a}$ (why do you think it is ?) and it is correct that $ \int_{\gamma} \frac{dz}{z-a} = 4 i \pi$ $\endgroup$ – reuns Oct 15 '16 at 19:22
  • $\begingroup$ By CIT for star-shaped regions. $\endgroup$ – Keith Oct 15 '16 at 19:32
  • $\begingroup$ your red contour $C$ is closed, and $\frac{1}{z-a}$ isn't holomorphic inside, that's why $\int_C \frac{dz}{z-a} = - 2i\pi$ $\endgroup$ – reuns Oct 15 '16 at 19:33
  • $\begingroup$ I am applying the CIT to the red contour plus the black contour. Together they are a closed contour, then $\oint_\gamma \frac{1}{z-a} - \oint _{\gamma_2} \frac{1}{z-a} = 0$, isn't this right? $\endgroup$ – Keith Oct 15 '16 at 19:40
  • $\begingroup$ No. Try with $\gamma = (e^{i t},t \in [0,4\pi])$ $\endgroup$ – reuns Oct 15 '16 at 19:45

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