Understanding where branch cuts live

Question

I understand how we define $e^z = e^x e^{iy}$ and use this to define the multi-valued function $\log(z) = \ln(r) + i(\theta + 2\pi n)$. I think I also understand how we may take any branch by setting an arbitrary real value, call it $\alpha$, and then restrict $\theta+2\pi n$ to the interval between $\alpha$ and $\alpha+2\pi$, so that the function becomes single-valued and still defined for all nonzero $z$.

What I'm particularly unclear on is what a branch cut is here. It is defined to be a curve used to describe a branch. (Brown and Churchill: "A branch cut is a portion of a line or curve that is introduced in order to define a branch F of a multiple-valued function $f$.") But where does this curve live? It doesn't seem to be in the domain or the range of the function.

For instance, any branch of $\log(z)$ is defined on the complex plane except the origin, so we cannot be restricting the domain. The image of any branch is the horizontal band with complex parts between $\alpha$ and $\alpha+2\pi$, and where the real part is strictly positive.

I might understand if the branch cut were some edge of this band. But later in Brown and Churchill they say that the branch cut is the origin and the ray $\theta=\alpha$. As far as I can tell, the points on this ray do not correspond to points in the domain or the range. For one thing, many different values of $\alpha$ pick out the same set of points, and yet every distinct value of $\alpha$ is (I think) supposed to correspond to a different branch.

Maybe that's ok because the definition of a branch cut did not specify where the cut must live, and just seems to use vague language about the cut being used to define a branch, but doesn't specify exactly how it must be used to accomplish this. But then that seems super open-ended and not usually how we define things in math.

I figure it's just more likely that I'm misunderstanding something.

Answer 1

$\newcommand{\Cpx}{\mathbf{C}}$tl; dr: Branch cuts live in the domain.

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Ultimately the definition of a branch of logarithm requires careful attention. In my experience, if $G$ is a plane region, i.e., a non-empty connected open set in $\Cpx$, a branch of logarithm is a holomorphic function $\log:G \to \Cpx$ satisfying $\exp(\log z) = z$ for all $z$ in $G$. (In other contexts, one speaks of a right inverse of $\exp$, or a section of $\exp$.)

With this definition, there is no branch of logarithm in the punctured plane: There do exist "logarithm functions" $L:\Cpx^{\times} \to \Cpx$ satsfying $\exp(L(z)) = z$ for all $z \neq 0$, but no such function is continuous, much less holomorphic.

If $G$ is the complement of an arc $A$ from $0$ to $\infty$ (i.e., the image of a simple, piecewise continuously-differentiable path approaching $0$ in one direction and approaching $\infty$ in the other), there exists a branch of logarithm in $G$ in the sense above, and $A$ is the corresponding branch cut. Common examples include rays from $0$. If $\alpha$ is real and $G$ is the complement of the ray at polar angle $\alpha$, there exist countably many branches of logarithm in $G$, all of the form $$ \log_{\alpha,k} z = \log|z| + (\theta + 2\pi k)i,\quad \alpha < \theta < \alpha + 2\pi $$ for some integer $k$.

There also exist more exotic arcs and branches. If we let $A$ be a logarithimic spiral about $0$ together with $0$, for example, the corresponding branches of logarithm have unbounded imaginary part.

In the sense above, there exist "non-maximal" branches of logarithm in every non-empty open set $G$ contained in a simply-connected subset of $\Cpx^{\times}$. Speaking of branch cuts, however, would be anomalous if $\Cpx \setminus G$ were not a simple arc.