Let $D$ denote the unit disc , and $U$ is an open simple connected subset of $D-\{0\}$ , then we can define a square root function on $U$ by $$g(z)=e^{\frac12 \log z}$$ such that $g$ is an injective holomorphic mapping from $U \to D$
The statement above was in Stein's complex analysis Page $_{230}$ and I'm quite confused about the function $\log z$ defined here . Indeed , what we need here is $z^{\frac12}$ and we expect that whenever $z=re^{i \theta}$ we can have $|z^{\frac12}|=r^{\frac12}$ here . However , since I only know $U$ is simple connected , how could I define $g$ here ?
I am assuming that $0\notin U$. In that case, since $U$ is simply connected, $\frac1z$ has a primitive $\psi$ in $U$. Take $z_0\in U$ and choose $w_0\in\mathbb C$ such that $\psi(z_0)+w_0$ is a logarithm of $z_0$. Now, let $\log z$ be $\psi(z)+w_0$. Then, for each $z\in U$, $\log z$ is a logarithm of $z$. So $\left(e^{\frac12\log z}\right)^2=e^{\log z}=z$.