What are the branches of $z^b$?

Question

In the text I am currently reading through, I am not sure I understand correctly how they are defining a branch of $z^b$ where $b\in \mathbb{C}$ is fixed.

This is my understanding:

A branch of $z^b$ is a continuous function $g:G\rightarrow \mathbb{C}$ -- here $G$ is a region in $\mathbb{C}$ where there is a branch of $\log(z)$ -- such that $g(z) = \exp(bf(z))$ for some branch $f$ of $\log(z)$.

In other words the branches of $z^b$ are given by the continuous functions $$g(z) = \exp(b (\log|z| + i(\arg(z) +2\pi k))), k\in \mathbb{Z}$$

Is this resoning correct?

Answer 1

I'm not sure it makes sense to want "a banch" to be a particular thing. We can say that $z^{2/5}$ "has $5$ branches", but it is less meaningful to speak about a particular set of $5$ things that are the branches.

For example, here is a red dot and a shakily drawn curve that makes three windings around the dot:

And we can easily agree that there are indeed three windings, but that doesn't mean that there's a definite answer to "what exactly is a winding" that makes the diagram contain exactly three instances of "a winding".

It is better to understand "a branch" to be a mostly informal and intuitive concept. If you're looking at the situation locally, with a particular simply-connected subset of the domain of the function in mind, then you can speak of branches of the function on that subset, and they will be well-defined things. But you cannot do the same for the entire domain, because they thing about branches is that they turn smoothly into each other as you go around a branch point.

Answer 2

Some preliminary: given $z\in\Bbb C,\;\;z\neq0$, we define $\arg z$: it is called the principal argument and it's defined to be the unique $\theta\in]-\pi,\pi]$ such that $z=|z|e^{i\theta}$.

There are infinitely many different argument functions: once you have fixed $\theta_0\in\Bbb R$, you can define $$ \arg_{\theta_0}z:=\theta_0+\arg(ze^{-i\theta_0})\;\;. $$

The principal argument introduced at the beginning is the $\theta_0$-argument with $\theta_0=0$.

Now $z^b$ is defined by $$ z^b:=\exp\left[{{b(\log|z|+i(\arg_{\theta_0} z+2k\pi))}}\right] $$ with $\theta_0\in\Bbb R$ and $k\in\Bbb Z$, so the power function is defined up to a choice of the two parameter above $\theta_0$ and $k$, whose choice define the branch you are taking.

When $\theta_0=k=0$ you have the principal branch of $z^b$.

Answer 3

Given the definition used and the other post fosho made here (If $f$ and $g$ are branches of $z^a$ and $z^b$ respectively show that $fg$ is a branch of $z^{a+b}$) I think the text in question is Conway's Functions of One Complex Variable I.

The definition of a branch of $z^b$ is given with respect to a branch of log. So if f is a branch of log, we can define a corresponding branch of $z^b$ by the formula $exp(b f(z))$. In your original post, you list all of the branches of $z^b$ on the connected set $\mathbb{C} - \{z \in R : z \le 0\}$. These correspond to all of the branches of log on this set.

Reviewing Conway's definition, a branch of log on a connected open set G is a continuous function f that satisfies $z = exp(f(z))$ for all z in G. In Joe's answer, he notes that in addition to the choice of $2\pi i k$, there is also a set of branches of log associated with each argument function. It's worth noting that as he defines these argument functions, they are discontinuous along a ray of angle $\theta_0 + \pi$ going from 0 to infinity. Therefore, the branch of log associated with each one will only be defined on the complex plane excluding this ray. That said, when I usually see someone define a branch cut it does take this form, just removing some ray that emanates out from the branch point.

It's probably worth noting that using Conway's definition, there are a large number of other choices for a branch of log that you could make. To give the next simplest example that occurs to me, consider this diagram.

Our choice of the domain, G, is the entire plane except the arrowed curve that consists of a line segment from 0 to -1 and -1 to $-1+i\infty$. Let $f(z) = Log z$, the principal log value on the unshaded region, and $Log z - 2 \pi i$ on the shaded region, $\{Im z > 0, Re z < -1\}$. As you pass across the negative real axis for a real value below -1, this function is continuous. For z just below the negative real axis, $f(z)$ will have imaginary part slightly above $-\pi$. For $f(z)$ just above the negative real axis and with real part below $-1$, it will have imaginary part just below $(\pi - 2\pi) = -\pi$, so the imaginary part does not jump as you go across the negative real axis to the left of $-1$.

At this point in Conway, he hasn't yet defined the contour integral, but if you're familiar with it, one way to describe a large variety of the branches of log would be to say:

1. Choose a domain, G, that contains no curves that wind completely around the branch point at 0, and where any two points are connected by a sufficiently regular contour that the contour integral along the curve can be defined.
2. Choose a point $z_*$ in G.
3. Choose a value for the branch at the point $z_*$ from the possible values, $log z_* + 2\pi i k$ for some integer k.
4. The branch will then be well-defined if $f(z) = f(z_*) + \int_\gamma z^{-1} dz$, where $\gamma$ is any curve in G that goes from $z_*$ to z. That this definition does not depend on the choice of $\gamma$ is due to $z^{-1}$ being analytic away from 0 and $\gamma$ not being able to wind around 0, but this shouldn't be clear yet at this point in the book.

At this point in the book, the best thing you can really say is what Conway says, that any two branches defined on the same connected domain G will differ by a constant factor of $2\pi i k$ for some integer $k$. Therefore any two branches of $z^b$ will differ by a multiplicative factor of the form $exp(i 2 \pi k b)$. When b is a rational number p/q and p and q share no common factors, there will be q different branches with multiplicative factors distributed evenly around the unit circle. When b is irrational, the multiplicative factors will correspond to a countable dense subset of the unit circle. When b is imaginary the branches are real scalar multiples of each other.