I have some questions regarding what in my textbook is defined as the "variation of the argument" of a holomorphic function $f$ over a non self-intersecting contour $C \subseteq \mathbb{C}$.
First, the author claims that given a contour $C$ parameterized by a surjective curve $\phi : [0, 1] \rightarrow C$ and a function $f$ which is holomorphic on a neighborhood of C not vanishing on C, there is an open, simply connected domain $\Omega$ such that $\Omega \cap C = C \setminus\{\phi(0), \phi(1)\}$ with $f(s) \neq 0$, $\forall s \in \Omega$. Is such a domain simply the neighborhood of $C$ on which the function $f$ is defined? How can such a domain be derived?
Second, the author defines the "variation of the argument of $f$ along $C$" as
$$\Delta_{C}arg f(s) = \mathcal{Im} \Big [ log \big [f(\phi(1^{-}))\big ] - log \big [f(\phi(0^{+}))\big ] \Big ]$$
And claims this is equivalent to the logarithmic derivative of $f$,
$$\mathcal{Im} \Big [ \oint\limits_{C} \frac{f^{'}(s)}{f(s)} ds \Big ]$$
I'm not sure how these two forms are equal, I can kind of see intuitively from just the fundamental theorem of calculus, but I'm not sure how this would be made rigorous. This is actually a number theory book, and I don't have as strong a background in complex analysis, so I'm not entirely sure how to reason about this.