I have been struggling finding the domain of analyticity in complex analysis, my teacher finds them with a drawing like the one I've attatched (in the pic they were asking for $\sqrt{z-1}=e^{1/2 \log(z-1)}$ if it helps):
So I would appreciate if it was solved that way. I've tried to do it and got the image ($\{x+iy\in \mathbb{C}\ | \ x,y\geq 0\}$) but can't find the domain of $|z|$? Should I try to do it with the $$|z|=\sqrt{x^2+y^2}$$ or some sort?