Possible values of $\int_\gamma{\frac{1}{z}}dz$ What are the possible values $\int_\gamma{\frac{1}{z}}dz$ where $\gamma$ is a path that starts at $z=-i$  and ends $z=2i$ and avoids the origin?
I pick a specific branch cut, for example negative x axis, so $-\pi \leq \theta \leq \pi$. $\mathbb{C} -$branch cut is a simply connected region so integral will be the same over all the paths. 
Then I have $\int_\gamma {\frac{1}{z}dz} = \int_\gamma{\frac{d}{dz}(\log(z))dz} = Log(2i) - Log(i) = \ln2 -i\frac{\pi}{2} - \ln(1) - i\frac{\pi}{2} = \ln(2) - i\pi$
Is this correct? I am confused about the 'possible values' in the problem statement
 A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

Since we are free to choose $\ds{\gamma}$, a 'nice candidate' is
  $\ds{\gamma \equiv \braces{{1 \over 2}\,\ic + {3 \over 2}\expo{\ic\theta}\ \left.\vphantom{\large A}\right\vert\
\theta \in \pars{-\,{\pi \over 2},{\pi \over 2}}}}$.

\begin{align}
\int_{\gamma}{\dd z \over z} & = \int_{-\pi/2}^{\pi/2}
{\pars{3/2}\expo{\ic\theta}\ic\,\dd\theta \over
\ic/2 + \pars{3/2}\expo{\ic\theta}} =
\left.\ln\pars{{1 \over 2}\,\ic + {3 \over 2}\expo{\ic\theta}}
\right\vert_{\ -\pi/2}^{\pi/2}
\\[5mm] & =
\ln\pars{2i} - \ln\pars{-\ic} =
\bracks{\ln\pars{2} + {\pi \over 2}\,\ic} -
\bracks{\ln\pars{1} + \pars{-\,{\pi \over 2}\,\ic}} =
\bbx{\ln\pars{2} + \pi\ic}
\end{align}

Another posibility is to go 'straight' from $\ds{-\ic}$ to $\ds{2\ic}$ where we consider the $\ds{\ln}$-Principal Branch. In such a case the integration is reduced to:

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