Property of the index of a function

[Jim Kata]

Hi, I'm working through a paper and I am quite stupid so some things that maybe obvious are not obvious to me. Say you have some have some complex analytic function that is defined on some simply closed curve, and the index of this function defined on this curve is zero,

\int_C \frac{f'(z)}{f(z)}dz =\Delta Arg(f(z)) = 0 Then there exists for a specific branch of the logarithm a g(z) such that exp(g(z))=f(z). My question is why do you need the index of f(z) to be zero for this curve for g(z) to be unique on the curve for a specific branch.

 

[Simon Bridge]

Um. Because that's how the math works out.
It sounds a bit like asking if x-a=0 then why does x have to be equal to a to make the relation true?

... what happens when the index is not zero?

 

[Jim Kata]

I'll rephrase the question. Let f(z)=R(z)exp(i\theta(z)) . Why is it when

ind_C(f(z))=0 then \alpha < \theta (z) < \alpha +2\pi for \alpha \in \mathbb{R}. When z \in C

 

[Simon Bridge]

Same answer. That's how the math works out.
Can you think of some way that it shouldn't work out that way?

It can help to understand these things by considering the converse:
What does it mean when ind_C(f(z)) ≠ 0 ?

What's the paper you are working through?