Find a branch of $\log (2z – 1)$ that is analytic at all points in the plane except those on the following rays
a) {$x + iy : x \leq \frac{1}{2}, y = 0$}
Definition: $F(z)$ is said to be a branch of a multiple valued function $f(z)$ in a domain $D$ if $F(z)$ is single valued and continuous in D and has the property that, for each $ z$ in $D$, the value $F(z)$ is one of the values of $f(z)$
Can someone please help me start this problem? I don't how to start, and there are no examples in the book. I would really appreciate it . Thanks
Best Answer
The best idea is probably to substitute variables, and investigate what happens to the given ray after the substitution:
Let, for example, $w=u+i\cdot v=2z-1=2(x+i\cdot y)-1$. Then we get that $z\in \mathbb{C}\setminus(-\infty,1/2]\Leftrightarrow w\in \mathbb{C}\setminus(-\infty,0]$.
Now, by definition, $\log(w)=\ln|w|+i\cdot \arg(w)$. Thus you should only investigate the argument. Basically, what we want to avoid is to have an argument branch that is continuous when $\Re(w)=u<0$.
Let us therefore create a branch for $\arg$, which we will call $\tilde\arg$. We define this branch by $\tilde\arg(w):=\theta(w)$, where $\theta$ is a continuosly varying argument for $w$ on one of the intervals $(-\pi+2n\pi,\pi+2n\pi)$, $n\in \mathbb{Z}$. This means that $\tilde\arg(w)$ is continuous for all $w\notin (-\infty,0]$.
Thus, we define the suitable branch of $\log$ with $\tilde\log(w):=ln|w|+i\cdot \tilde\arg(w)$. As the argument now varies continuously, this branch of the logarithm is now analytic on the desired set.
An example of a branch that works in this case is the principal branch of $\log$, which has an argument varying continuosly between $(-\pi,\pi)$.