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**garunas** **Question:**

Show using a suitable contour that the complex logarithm function is defined by:

$\displaystyle Log(z) = \int_{[1,z]} \frac{1}{\zeta}\hspace{2mm} d\zeta = ln|z| + i\hspace{1mm}Arg(z)\hspace{5mm} for\hspace{3mm} z \in \mathbb{C}_{\pi} = \{z : Arg(z) \neq \pi \}$

What I did was i used the contour $\displaystyle \gamma (t) = (1-t) + zt \hspace{3mm} 0 \leq t \leq 1$ so that:

$\displaystyle \int_{[1,z]} \frac{1}{\zeta} \hspace{2mm}d\zeta = \int_{0}^1 \frac{\gamma'(t)}{\gamma(t)} \hspace{2mm} dt = \int_{0}^1 \frac{z-1}{1-t+zt}\hspace{2mm} dt = ln|z|$

I've reached here and now I'm absolutely stuck. What do i do next?