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Let $\Omega$ be the complex plane minus two paths. Do some closed paths $\Gamma$ in $\Omega$ satisfy assumption $(1)$ of Theorem 10.35 of Rudin's Real and Complex Analysis without being null-homotopic in $\Omega$?

Theorem 10.35. Suppose $f \in H(\Omega)$, where $\Omega$ is an arbitrary open set in the complex plane. If $\Gamma$ is a cycle in $\Omega$ that satisfies$$\text{Ind}_\Gamma(\alpha) = 0 \text{ for every }\alpha \text{ not in }\Omega,\tag*{$(1)$}$$then$$f(z) \cdot \text{Ind}_\Gamma(z) = {1\over{2\pi i}} \int_\Gamma {{f(w)}\over{w - z}}\,dw \text{ for }z \in \Omega - \Gamma^*\tag*{$(2)$}$$and$$\int_\Gamma f(z)\,dz = 0.\tag*{$(3)$}$$If $\Gamma_0$ and $\Gamma_1$ are cycles in $\Omega$ such that$$\text{Ind}_{\Gamma_0}(\alpha) = \text{Ind}_{\Gamma_1}(\alpha) \text{ for every }\alpha \text{ not in }\Omega,\tag*{$(4)$}$$then$$\int_{\Gamma_0} f(z)\,dz = \int_{\Gamma_1} f(z)\,dz.\tag*{$(5)$}$$

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Let $\Omega = \mathbb C \setminus\{a,b\}$. Let $\Gamma$ be a closed curve in $\Omega$ so that $\Gamma$ is homotopic to $$\alpha* \beta *\alpha^{-1} *\beta^{-1},$$

where $\alpha$, $\beta$ are simple closed curves which wrap around $a$ and $b$ respectively. Then

$$\text{Ind}_\Gamma(a) = \text{Ind}_\Gamma (b) = 0$$

and $\Gamma$ is not null homotopic. Indeed, $\pi_1(\Omega)$ is freely generated by $\alpha$ and $\beta$ by the Van Kampen Theorem, so $\alpha* \beta *\alpha^{-1} *\beta^{-1}$ is not a trivial element in $\pi_1(\Omega)$.

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    $\begingroup$ It would be helpful to add (for those not familiar with the fundamental group of $\mathbb C\setminus\{a,b\}$) that $\alpha$ and $\beta$ do not commmute. $\endgroup$ – Yiorgos S. Smyrlis Apr 4 '16 at 10:00
  • $\begingroup$ @YiorgosS.Smyrlis : Some remarks are added. $\endgroup$ – user99914 Apr 4 '16 at 10:07
  • $\begingroup$ Is there a purely analytic way to do this? I ask because the question is Exercise 10.30 in Rudin's RCA, and there is not even a mention of the fundamental group in that chapter. $\endgroup$ – user363464 Jan 18 '18 at 7:10

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