# Understanding where branch cuts live

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.

• $0, \infty$ are the singularities of the logarithm so if you cut any Jordan arc connecting them, you get a branch - the usual choices are rays where the argument stays bounded but you can use more complicated curves like a spiral where the argument doesn't stay bounded; in general branch cuts are done using Jordan arcs connecting the singularities of the function (a useful example is to look at $\sqrt {z^2-1}$ which has three singularities ($\pm 1, \infty$) May 12, 2022 at 20:36

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

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.

• Ok, so to check my understanding: A single branch cut does not necessarily correspond to a single branch? May 12, 2022 at 22:00
• Yes; more strongly, a single branch cut never corresponds to a single branch of log, it only delimits a domain where a holomorphic choice of log exists. (It's possible the terminology in this answer is not universal, but it's logically consistent.) May 12, 2022 at 23:19