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\begin{equation*}
  \Gamma_o = \{r e^{i \theta}: -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\}, \qquad
  \Omega_o = \{L(s): s \in \Gamma_o\} = \{L(r e^{i \theta}):  -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\}.
\end{equation*}
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