A doubt from proof of residue theorem from Tao's notes.
notes 4 Theorem 21 (Residue theorem) Let $U\subset \mathbb{C}$ be a simply connected open set, and let $f: U\setminus S\rightarrow \mathbb{C}$ be holomorphic outside of a closed discrete singular set $S$ (thus all singularities in $S$ are isolated singularities). Let $\gamma$ be a closed curve in $U\setminus S$. Then $$\frac{1}{2\pi i}\int_\gamma f(z) dz=\sum_{z_0\in S} W_{\gamma}(z_0) Res(f;z_0),$$ where only finitely many of the terms on the right-hand side are non-zero
Proof: Being simply connected, $\gamma$ can be contracted to a point inside $U$. This homotopy take values inside some compact subset $K$ of $U$, and thus only contains finitely many of the singularities in $S$. By Rouche's theorem, the winding number $W_{\gamma}(z_0)$ then vanishes for any singularity $z_0$ in $S\setminus K$ (since $\gamma$ can be contracted to a point without touching $z_0$). […]
So,
Being simply connected, $\gamma$ can be contracted to a point inside $U$
Let $\gamma_1$ be the zero curve (ie $Im(\gamma_1)$ be the point).
doubt 1: can the homotopy invariance (ref below) of winding number be used (as follows)? Since $\gamma_1$ is the zero curve, $\frac{1}{2\pi i}\int_{\gamma_1} \frac{1}{z-z_0} dz =0$. So $W_{\gamma_1}(z_0)=0$. $\gamma$ is homotopic to $\gamma_1$. So, $W_{\gamma}(z_0)=0$.
doubt 2: how is Rouche's theorem (ref below) applicable? There's probably no information about $\mid \gamma_1 (t)-\gamma (t)\mid$ and $\mid \gamma (t)-z_0\mid $
notes 3 lemma 41 (Homotopy invariance) Let $z_0\in \mathbb{C}$, and let $\gamma_0, \gamma_1$ be two closed curves in $\mathbb{C} \setminus \{z_0\}$ and are homotopic as closed curves upto reparameterization in $\mathbb{C} \setminus \{z_0\}$. Then $W_{\gamma_0}(z_0)=W_{\gamma_1}(z_0)$
note: author is referring to Rouche's theorem for winding number and not Rouche's theorem, since the latter is yet to be introduced.
notes 3 corollary 42 (Rouche's theorem for winding number) Let $\gamma_0:[a,b]\rightarrow \mathbb{C}$ be a closed curve, and let $z_0$ lie outside of the image of $\gamma_0$. Let $\gamma_1:[a,b]\rightarrow \mathbb{C}$ be a closed curve such that $$\mid \gamma_1 (t)-\gamma_0 (t)\mid < \mid \gamma_0 (t)-z_0\mid $$ for at $t\in [a,b]$. Then $W_{\gamma_0}(z_0)=W_{\gamma_1}(z_0)$
The notes.
Thanks
Best Answer
Regarding Doubt 1. The extra details you provide in your proposed proof are all correct.
Regarding Doubt 2. I think that the proof presented in the notes should probably be edited. In my opinion, at this stage in the progression of the notes you cite, it would be simpler and more natural to just appeal directly to the Homotopy Invariance of Winding Number (rather than the Rouche Principle for Winding Number, which is is just a corollary of homotopy invariance which is of less general applicability than the Homotopy Invariance Principle itself.) That is, I think your doubt is well-founded.