# Branch of a complex multi valued function.

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.

• I think "some domain" is intended to imply an open set. So if you said $0 < \theta < \pi$ then it would be a "branch" of $\log z$.. – GEdgar Oct 23 '15 at 12:30
• but is is not usually consider as branch...i don't know why. – neelkanth Oct 23 '15 at 12:46
• Definition(Branch): A branch of a multiple-valued function $f$ is $\color{teal}{any}$ single-valued function $F$ that is $\underline {analytic}$ in some domain at each point $z$ $\color{teal}{of \ which}$ the value $F(z)$ is one of the possible values of $f.$ here I don't understand the word analytic This is the definition of branch of multi valued functions. Why does $$Log(z)=ln(r)+i\theta (r>0,-\pi<\theta<\pi)$$ is a branch of the multi valued function $log(z)?$ Is $ln(r)+i\theta$ a possible value of $log(z)$? – IggyPass Oct 23 '15 at 13:09
• yes $Log(z)$ is one of the infinite values of $log(z).$ – neelkanth Oct 23 '15 at 13:11

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 $\ln(1+x)$
• @neela No my example is a branch of $ln(1+z)$ with only the unit disc as domain. The point is that one is allowed to exclude much more than minimally required which sometimes is practical. – skyking Oct 23 '15 at 13:39