# usage of Rouche's theorem for winding number in proof of Residue theorem

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