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A similar question can be found in this post. I apologize for any inconvenience.

I can't understand the geometric interpretation of Rouché's theorem in Wikipedia. The standard formulation of Rouche's Theorem is

Theorem: Suppose $f$ and $g$ are holomorphic functions inside and on the boundary of some closed contour $\mathcal{C}$. If $$ |g(z)|<|f(z)| $$ on $\mathcal{C}$, then $f$ and $f+g$ have the same number of zeros on the interior of $\mathcal{C}$.

I copy in full what is written on this web page Ngô Quốc Anh

Geometric explanation

It is possible to provide an informal explanation on why the Rouche’s theorem holds. First, we need to rephrase the theorem a little bit. Let $h(z) = f(z) + g(z)$. Notice that $f$, $g$ holomorphic implies $h$ holomorphic too. Then, with the conditions imposed above, Rouche’s theorem says that

If $|f(z)| > |h(z)-f(z)|$ then $f(z)$ and $h(z)$ have the same number of zeros on the interior of $\mathcal{C}$.

Notice that the condition $|f(z)| > |h(z)-f(z)|$ means that for any $z$, the distance of $f(z)$ to the origin is larger than the length of $h(z)- f(z)$, which in the following picture means that for each point on the blue curve, the segment joining to the origin is larger than the green segment associated to it. Informally we can say that the red curve $h(z)$ is always closer to the blue curve $f(z)$ than to the origin.

Rouché theorem

But the previous paragraph shows that since $f(z)$ winds exactly once around $0$, so must $h(z)$, and by the argument principle, the index of both curves around zero is the same, which means that $f(z)$ and $h(z)$ have the same number of zeros.

One popular, informal way to summarize this argument is as follows: If a person were to walk a dog on a leash around and around a tree, and the length of the leash is less than the radius of the tree, then the person and the dog go around the tree an equal number of times. (Indeed, one may see that the converse of Rouche’s theorem is false, insofar as the leash need only be less than the circumference of the tree.)

Questions

  1. What is the relationship between the theorem and its proof?
  2. Why: The index of a holomorphic function is equal to the number of roots?
  3. I can't figure out what's in italics.

Thank you for your help.

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