Let $D \subseteq \mathbb{C} $ be bounded and simply connected domain, $\Gamma := \partial D \in C^2 $, $ g \in C^{0,\alpha}(\Gamma) $ be a complex-valued and nowhere vanishing function defined on the contour $\Gamma$ . The index of $g$ is defined by $$ \text{ind} \ g := \frac{1}{2\pi} \int_{\Gamma} d \text{arg} g. $$ Question1: Why it must be a integer?
My Thoughts: Since i think $g(z)$ may not go around the circle `by interger times when $z$ go around the $\Gamma $.
Let $\text{ind} g = \kappa $. Choose a point $a \in D $, define $G \in C^{0,\alpha}(\Gamma) $ by
$$
G(z) := (z-a)^{-\kappa} g(z), \quad z \in \Gamma.
$$
It has index $\text{ind} \ G = 0 $.
Question2: Why any branch of the logarithm can give a single-valued function $ \ln G \in C^{0,\alpha}(\Gamma) $?
My Thoughts: I suspect the possibility that the $G$ may turn counter-clockwise for some circles and then go back to the origin? This would destroy the single-valued property of some branch of logarithm?
Note: The latter question is one step in [Kress, Linear integral equations, P117].
