This question was asked in my complex analysis assignment and I am confused on how to attempt it.
What does monodromy theorem tells about log z and $z^{1/2}$?
I have to choose a Domain D and then define it on a subdomain$D_0$ and then think if it can be analytically continued on D , but defining it on subdomain for log z it equals ${z-1} - (z-1)^2 /2 + (z-1)^3/3-...$ defined for 0<x<1 but it can't be extended for x=0. So, I think there doesn't exists a single valued function f(z) which equalslog z on 0<x<1 and is analytic on whole $\mathbb{C}$.
For $(z)^{1/2}$ it is not analytic at z=0 ( using it's taylor series expansion ) so I think the same can be said about it.
Am I right? If wrong can you please tell what are the right observations about these functions?
Both have a branch point at $0$, they are not the same when continued analytically along the unit circle, for $f(z)=\log z, |z-1|<1/2$ the analytic continuation sends $f(z)$ to $f(z)+2i\pi$, for $g(z)=z^{1/2}$ it is $g(z)\to -g(z)$. Thus clearly neither of them can be extended to an analytic function on $0<|z|<2$.