This is a step in Rouche Theorem proof.
and are analytic on and inside a closed path .
for all on
Let
I need to prove
Please help - not able to follow this step atall.
Also, cannot pass through any zeros of f or g. Now for any function analytic in except for poles:
where is the winding number right? That's how many times the contour goes around the origin.
Now, by Rouche's theorem:
Since the contour never passes through any zero of either function, we can divide that expression by and end up with:
That means is inside the unit circle centered at one right? What then does that say about the expression:
Thanks. The inequality guarantees that the zeros don't lie on C.
Sorry but I have no clue of winding numbers etc. Isn't there a simpler explanation using maybe just Laurent Series, Cauchy Formula etc?
I follow your logic - never circles 0, hence the winding number is zero. My question would then be how do you get -
Let me try. I need to understand it better too.
So the winding number is just the number of times a contour goes around a point in the complex plane. For the contour around the origin, we can express it as:
and for the contour traced by , we could write:
but if and then: