My book on complex analysis introduces $\arg(z)$ and $\arg(z-a)$ after the complex logarithm is introduced. It shows that the two are just the oriented angles between $z$ and the point $z_0$ of the complex logarithm that a contour makes that goes from $z_0$ to $z$.
Now it introduces also the complex logarithm of $F(z)$, which is a holomorphic function in $\Omega$ and which has no zero points in $\Omega$ by:
$$(\log F)(z) = w_0 + \int_{z_0}^z{\frac{f'(\zeta)}{f(\zeta)}d\zeta} $$
In this way they can also define $\arg F(z)$ by saying that this is the imaginary part of $\log F(z)$. They start using a similar property to that of $\arg(z)$, without specifying it. So I wonder if anyone can tell how this $\arg F(z)$ can be calculated, probably it is in the same way as $\arg(z)$ or something similar, but I don't see why that should be true.