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.


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.

  • $\begingroup$ Thank you for your response! Based on your explanation, my immediate intuition is to define $\log{\log{z}}$ over $\mathbb C - (-\infty,1]$. That image would leave the infinite strip from $-\pi * i$ to $\pi * i$ without the nonpositive real axis, preventing the "wind around" there. Is this the correct intuition? $\endgroup$ – Darrin Dec 24 '13 at 5:06
  • $\begingroup$ @Darrin: Here is a problem worked out in detail:math.stackexchange.com/questions/481185/… Re your question: if you use the standard branch of Logz , then, yes, you want to avoid having loglogz from falling on $[0, \infty)$, so that you want to avoid logz from falling on the negative real axis. Like Some Number said, given log(g(z)), you want g(z) to not fall in the chosen branch cut for log(z). $\endgroup$ – DBFdalwayse Dec 24 '13 at 5:15
  • $\begingroup$ DBF, thank you. Based on the analysis to which you linked, I'd infer that a definition of the branch of $\sqrt{z+1}+\sqrt{z-1}$ could be the sum of the respective roots with positive real parts, over the domain $\mathbb C - {(\infty,-1] \bigcup [1, \infty)}$. Such a function would be analytic, because it is well-defined and a sum of compositions of functions that are analytic on that region. Does it appear I am getting the drift? $\endgroup$ – Darrin Dec 24 '13 at 6:15

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