Complex integration of $\log(z)$ over a closed counter-clockwise curve containing the origin only once

Question

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.

Answer 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}