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I was asked a question that I am pretty sure has a catch. I was asked how would the series representations of certain complex functions be affected by the choice of branch. My understanding is that the series representation, where it exists, is unique, so it really doesn't matter which branch one chooses, but I don't think that is what they are looking for. Consider the familiar function $\ln (1+z)$ for example? Thank you.

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for $\ln(1+z)\ $ the branch will add an offset of $2 k \pi i\ $ for the k-th branch. In other cases it may be more complicated : the [Lambert W]( branches $W_0$ and $W_{-1}$ are rather different – Raymond Manzoni Feb 19 '12 at 12:08
@RaymondManzoni: So the uniqueness theorem is only applicable to within a branch? – tycho Feb 19 '12 at 12:14
Well Taylor and Laurent series are unique in the disk or annulus where they are valid. Perhaps that this Wikipedia entry on branch points will help you. – Raymond Manzoni Feb 19 '12 at 12:40

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