# Path integral $\int_\gamma dz/z$ equals zero if $\gamma[a,b]\cap[0,-\infty]=\emptyset$?

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?

If Log denote teh principal logarithm that $$\int_{\gamma} \frac 1 z\, dx= \int_{\gamma} \frac d {dz} Log\, z \, dz=Log (\gamma (b))-Log \, (\gamma (a))=0$$.
Hint: $$\frac1z$$ is holomorphic / analytic / entire on all of $$\Bbb C\setminus (-\infty,0]$$.