Complex integration of $\log(z)$ over a closed counter-clockwise curve containing the origin only once What's the integral of $\log(z)$ in $C$, where $C$ is a closed curve enclosing the origin only once (counter-clockwise)?
I tried to use the circle with radius $r$, $\{z=re^{\theta i} : \theta \in (0,2\pi) \}$, but then I obtain the result $2 \pi r i$, and I think that the result should not depend on the radius.
 A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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You can choose any curve that encloses the origin. In particular,
  $\ds{\braces{z \mid z = \expo{\theta\,\ic} \mid \theta \in \pars{0,2\pi}}}$ which indeed is evaluated as a limiting case which 'takes care' of the, for example, $\ds{\ln}$-Principal Branch.

Namely,
\begin{align}
&\lim_{\epsilon \to 0^{+}}\braces{%
-\int_{-1}^{-\epsilon}\bracks{\ln\pars{-x} + \pi\ic}\dd x -
\int_{\pi}^{-\pi}\bracks{\ln\pars{\epsilon} + \theta\,\ic}\epsilon\expo{\theta\,\ic}\ic\,\dd\theta
-\int_{-\epsilon}^{-1}\bracks{\ln\pars{-x} - \pi\ic}\dd x}
\\[5mm] = &
-\int_{0}^{1}\pi\ic\,\dd x + \int_{0}^{1}\pars{-\pi\ic}\dd x =
\bbx{-2\pi\ic}
\end{align}
