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Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex logarithm function injective on U? Also is its derivative there is the function $1/z$?

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Yes, it is injective because $\exp\ln z=z$ no matter what branch of the logarithm you use. And yes, the derivative of any branch of the complex logarithm is $1/z$. One way to see this is to differentiate the formula above.

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