Find all branches of $e^{\sqrt z}$ in the domain $D=\Bbb C \setminus (-\infty,0]$.
Is there some kind of relation with the square root and complex logarithms? I've been studying about the branches for the complex logarithm, but I haven't seen them used with square roots. Searching online I found that the main branch of the square root in a domain $D$ is the map $g:D \to \Bbb C$ defined as $$g(re^{i\theta})=\sqrt r e^{\frac{\theta i}{2}}$$ where $\theta \in (-\pi, \pi)$, but I don't know where this is coming from.