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, $1^1=\exp(2k\pi i)=1$.