Help me understand a proof of Rouche's theorem

Question

I was preparing for the exam on complex analysis and looked up the proof of Rouche's theorem ( https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem#Geometric_explanation ).

It is said there:"The previous paragraph shows that h(z) must wind around the origin exactly as many times as f(z). The index of both curves around zero is therefore the same, so by the argument principle, f(z) and h(z) must have the same number of zeros inside C."

However, i don't understand how exactly the argument principle, which states that "if f(z) is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then

$\oint _{C}{f'(z) \over f(z)}\,dz=2\pi i(Z-P)$ where Z and P denote respectively the number of zeros and poles of f(z) inside the contour C, with each zero and pole counted as many times as its multiplicity and order, respectively, indicate." leads to the above conclusion.

All i'm seeing that we can apply to the argument principle from the conditions of the Rouche's theorem is that the curve $C$ for both functions must be the same, however i don't see how the integrals $\oint _{C}{f'(z) \over f(z)}\,dz$ and $\oint _{C}{h'(z) \over h(z)}\,dz$ will be equal, in order for us to say that the number of zeros is the same (there should be no poles, since both $f$ and $h$ are holomorphic from the conditions of the Rouche's theorem).

Also, the paragraph in Wikipedia is named "geometric explanation" but it seems to be a legitimate proof, when i understand it. Is that so?

Answer 1

To prove Rouché's theorem, by the argument principle we just need to show that $$ \frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\ dz=\frac{1}{2\pi i}\oint_C \frac{h'(z)}{h(z)}\ dz $$ where $h(z)=f(z)+g(z)$ and $|g(z)|<|f(z)|$ inside the domain bounded by $C$. You asked how we can show that the left side equals the right side. Well, the key idea is that (by the argument principle) both sides are integers, so if we can find a way to continuously interpolate between $f(z)$ and $h(z)$ in such that a way that all the functions in the interpolation have no zeroes on the contour, then the two integers will have to be equal (since there is no way to continuously modify one integer into another one - $\mathbb Z$ is discrete).

To get from the left side to the right, we can use a homotopy. The details are described on wikipedia but I agree with the commenter that better explanations can be found in most beginning complex analysis textbooks.