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Regarding question relating to branches as shown below:

Consider the multi-valued operation $z^z$. How do I find the branch points and all its possible value at $z=1$?

Thanks for helping me everybody

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The only possible value is $1$, since $1^1=\exp(1\times\log 1)$, where $\log(1)$ is some logarithm of $1$. But the logarithms of $1$ are the numbers of the form $2k\pi i$ ($k\in\mathbb Z$), and therefore, as I wrote, $1^1=\exp(2k\pi i)=1$.

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  • $\begingroup$ lets say if z is a complex number does your answer still holds??? $\endgroup$ – john Sep 26 '17 at 13:50
  • $\begingroup$ @john If $z=1$, yes. Otherwise, it depends upon the number. $\endgroup$ – José Carlos Santos Sep 26 '17 at 13:51
  • $\begingroup$ sorry, I start to get confused because I thought branch points are supposed to be points where it is zero or infinity?? So for this case if z=1, wouldn't it not fulfil the zero or infinity criteria?? $\endgroup$ – john Sep 26 '17 at 13:57
  • $\begingroup$ @john A branch point is a point such that if you go in a loop around it, you end elsewhere then where you started. A branch cut is what you use to make sense of this fact. $\endgroup$ – Kevin Sep 26 '17 at 13:59
  • $\begingroup$ apologies.. what I mean is branch cut $\endgroup$ – john Sep 26 '17 at 14:00

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