Defining single-valued branches

Question

These questions come from 2.2 of Ahlfor's famous text. I admit that defining branches of power functions and log functions in $\mathbb C$ has been conceptually difficult for me, and I think having a few more classical examples spelled out could help me in my understanding.

1. Give a precise definition of a single-valued branch of $\sqrt{1+z}+\sqrt{1-z}$ in a suitable region, and prove that it is analytic.

2. Same problem for $\log {\log{z}}$.

Unfortunately, I have no work to show here, since I have difficulty gaining a solid, geometric grasp of the detailed analysis in the discussion preceding these problems in Ahlfors' text. A kick in the right direction would be the approach most appreciated here.

I'm doing my qualifier studying from Greene and Krantz (having long abandoned Conway) but Ahlfors has been a good supplement so far. I appreciate your input as I continue to work through these problems.

Answer 1

A general comment: First of all, a branch of $logz$ is a "partial inverse" of the exponential function $e^z:=expz$. The exponential function is not $1-1$, so that it does not have a global inverse; it does have local inverses in regions where it is $1-1$, and these local inverses are what we call the branches of $logz$ ; the maximal regions where $e^z$ is $1-1$ are strips in the $y$-axis of the complex plane, with height $2\pi^{-}$ , i.e., regions of the form {$x+iy:x$ in $\mathbb R ; 2(k-1)\pi< y \leq 2k\pi$}, where a local inverse can be defined.

If you understand the concept of branch point, then you want to define a region of the plane where no curve winds around the branch point, which is, somewhat-informally, a point so that when you wind around it, you do not come back to the original value of the function. In the standard case of $Logz$, you want to avoid winding around the branch point $z=0$ , since, as you wind around $0$, your function changes values, i.e., $argz$ changes by $2\pi$ every time you wind around. Removing , e.g., the nonpositive Real axis, i.e., removing $(-\infty,0]$ prevents any curve from winding around the origin. Something similar is the case for $z^{1/n}; n>1$ , since $e^{i\theta}$ and $e^{i(\theta+2\pi)}$ will have different values. Since you define square roots in terms of $Logz$ (or some other branch of logz) , you want to define your branch in a similar way, i.e., $z^{1/n}; n>1:= e^{(1/n)logz}$. Basically, you want to define you domain for $z$ so that $g(z)$ does not fall in the region where $logz$ is not defined, i.e., $g(z)$ is not in the branch cut of $logz$.

In your case, once you define a branch of logz , you define $e^{(1/2)log(z-1)}+e^{(1/2)log(z+1)}$ , you want to choose $z$, so that the sum of the two expressions avoids the branch cut.