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,(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int_C \frac{f'(z)}{f(z)}dz =\Delta Arg(f(z)) = 0[/tex] 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.

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# Property of the index of a function

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