Small perturbations of a loop in $\mathbb C$ does not change its winding number

From Visual Complex Fuctions by Elias Wegert, Lemma 2.7.19, it says that

Let $\gamma_0 : [0,1] \to \mathbb C - \{0\}$ be a loop, i.e. a continous function with same endpoints. Denote by $d$ the distance of $\gamma_0([0,1])$ from the origin. Then for all loops $\gamma : [0,1] \to \mathbb C - \{0\}$ with $|\gamma(t)-\gamma_0(t)| < d$ for all $t $, the winding number of $\gamma$ is equal to the winding number of $\gamma_0$.

Proof: Since the image of $\gamma_0$ is a compact subset of $\mathbb C - \{0\}$, its distance $d$ from the origin is positive. Then $|\gamma(t)-\gamma_0(t)| < d$ implies that $\frac{\gamma(t)}{\gamma_0(t)}$ lies in the right half plane. Let $a_0$ be a continuous branch of the argument along $\gamma_0$. If we chose a continous branch of the argument along $\gamma$ such that $|a(0)-a_0(0)| < \pi/2$ then $|a(t)-a_0(t)| < \pi/2$ for all $t \in [0,1]$. Invoking the triangle inequality w see that $|(a(1)-a(0)) - (a_0(1)-a_0(0))| < \pi/2$ and since this number is an integral multiple of $2\pi$ it must be zero.

Here, the winding number of a loop $\gamma$ whose image does not contain $0$ is defined to be $\frac{1}{2\pi}(a(1)-a(0))$ where $a$ is a continuous branch of the argument along $\gamma$. Intuitively, this lemma formalizes the notion that small perturbations of a loop does not change its winding number.

I do not get why "Let $a_0$ be a continuous branch of the argument along $\gamma_0$. If we chose a continous branch of the argument along $\gamma$ such that $|a(0)-a_0(0)| < \pi/2$ then $|a(t)-a_0(t)| < \pi/2$ for all $t \in [0,1]$" holds true, or why the fact that "$\frac{\gamma(t)}{\gamma_0(t)}$ lies in the right half plane" is relevant. Can anyone help me understand the proof of this lemma?

Solution 1:

The condition $\operatorname{Re}\frac{\gamma(t)}{\gamma_{0}(t)} > 0$ means $\gamma(t)$ lies in the same (light gray) open half-plane from the origin as $\gamma_{0}(t)$.

The condition $$ |\gamma(t) - \gamma_{0}(t)| < d = \min \{|\gamma_{0}(t)| : 0 \leq t \leq 1\} $$ means $\gamma(t)$ lies in the (darker gray) disk, which is contained in the open half-plane above.

If $|a(0) - a_{0}(0)| < \pi/2$, i.e., the arguments of $\gamma(0)$ and $\gamma_{0}(0)$ agree within $\pi/2$, then since $\gamma_{0}(t)$ and $\gamma(t)$ lie in the same half-plane through $0$ for all $t$, we have $|a(t) - a_{0}(t)| < \pi/2$ for all $t$.