if we set $H:=\Bbb C \setminus\{-ir:r \in \Bbb R, r>0\}$, and we wanted to define a complex logarithm $\log(z)$ which is holomorphic on $H$. Then is the following what we would do ?
We know that $\log(z)$ is not differentiable on the negative real line because the arg(z) is not continous on this line. We also know that $\log(z)=\ln|z|+iArg(z)$
So to define $\log(z)$ s.t. it is holomorphic on H we say $\log(z)=\ln|z|+i(Arg(z))$, where $Arg(z)\in(\pi,-\pi/2) \cup(-\pi/2,\pi) $ (i.e the angle goes from $\pi$ to $-\pi /2$ but doesnt include these points , then it goes from $-\pi/2$ to $\pi$ again not including the points).