Definition(Branch): A branch of a multiple-valued function $f$ is any single-valued function $F$ that is analytic in some domain at each point $z$ of which the value $F(z)$ is one of the possible values of $f.$
This is the definition of branch of multi valued functions.
Clearly $$Log(z)=ln(r)+i\theta (r>0,-\pi<\theta<\pi)$$ is a branch of the multi valued function $log(z).$
Now my question is according to the above definition $$f(z)=ln(r)+i\theta (r>0,0<\theta<\pi)$$ is also a branch of the multi valued function $log(z)?$ I am thinking so because $f$ is also single valued analytic(in upper half domain) assuming exactly one of various possible values of $Log.$ Am i right? Please suggest me. Thanks in advance.