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.
Best Answer
Yes, you're correct. It fulfills all the requirements of the definition.
The definition does not put any requirements of the domain in which the branch is defined. For example this becomes handy if you analytically extend the real analytical function $\ln(1+x) = \sum x^n/n$, it would converge in a unit disc around $0$ which makes the McLaurin series a branch of $\ln(1+x)$