This is a paragraph from Ahlfors Complex Analysis
It states that for the function $z^{2 \alpha}$ where $0<\alpha<1$, it is possible to choose an analytical branch of the function whose argument lies between $-\pi \alpha$ and $3\pi \alpha$; that it is well defined and analytical within the region obtained by ommiting the negative imaginary axis.
My question is why can't a single valued analytical branch be defined on the negative imaginary axis? I suspect that it may have something to do with the possibility of defining a single valued analytical branch of $\ln z$ everywhere except on the non-positive real axis.