I would like to prove that this function is analytic. It is really hard for me to see.

We suppose that f is analytic and zero free in a domain D, the function I wish to prove is analytic is:

$\displaystyle \int_{z_0}^z \frac{f'(\zeta )}{f(\zeta )} ~d\zeta$

My professor tells me there is a standard argument where I take a $\displaystyle z_1 \in D$ and a small disk $\displaystyle D_r(z_1) \subset D$ then take a fixed path from $\displaystyle z_0$ to $\displaystyle z_1$ and then a line segment from $\displaystyle z_1$ to $\displaystyle z$, but I am not sure where to go from there.

Any help is greatly appreciated!