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**ThePerfectHacker** You should be careful how to parametrize the circle. You see the typical definition for $\displaystyle \log z$ is $\displaystyle \log r + i\theta$ where $\displaystyle z = re^{i\theta},r>0,\theta \in (-\pi,\pi]$. Thus, parametrize $\displaystyle C_R$ as $\displaystyle \gamma: [-\pi,\pi ] \to \mathbb{C}$ with $\displaystyle \gamma(t) = Re^{it}$. So the integral will look exactly like before. Just not from $\displaystyle 0$ to $\displaystyle 2\pi$ but instead $\displaystyle -\pi$ to $\displaystyle \pi$. However, this is not an issue here because if you replace $\displaystyle t$ by $\displaystyle t+\pi$ and use substitution it does not change the integral.