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How to prove that a piecewise differentiable path integral is zero, if $\gamma[a,b]\cap[0,-\infty]$ is empty, and we have:

$\int_\gamma dz/z$

So that:

$$\int_\gamma dz/z=\sum\int_{x_i}^{x_{i+1}} \frac{\gamma'(t)}{\gamma(t)}dt$$

Possibly I need to show that an integral function exists?

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If Log denote teh principal logarithm that $\int_{\gamma} \frac 1 z\, dx= \int_{\gamma} \frac d {dz} Log\, z \, dz=Log (\gamma (b))-Log \, (\gamma (a))=0$.

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Hint: $\frac1z$ is holomorphic / analytic / entire on all of $\Bbb C\setminus (-\infty,0]$.

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