Branch cuts have been asked about and discussed on MSE extensively. That is, every answer to something along the lines of "What is a branch cut?" is... extensive.

I'm looking for a quick, intuitive description of what a branch cut is.

I know that:

  • The logarithm of a complex number can have different values (making it not a function)
  • A branch of the logarithm is a chosen set of values such that it is a function

What is the branch cut, though?

Do I say $$\ln{R} + i\theta,~~\theta \in [0, 2\pi) \\ \ln{R} + i\theta, ~~ \theta \in [2\pi, 4\pi)$$ are two distinct branches? Do these two branches have a branch cut between them? Or do they have distinct branch cuts?

I understand that some people know a lot of information regarding this subject, but reading the answers that describe this the same way that my textbook does hasn't helped. (What helped me at least start to get the idea of branches most was someone's comment: "It involves the 'non-function-ness' of the function. I.e. when a function can be mutli-valued. You have to decide which "branch" you are going to work on. — It's simple, to the point, not overly elaborate, etc.

  • $\begingroup$ You can not explain it simple if you do not understand it well. $\endgroup$ – science Mar 11 '15 at 4:37
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    $\begingroup$ I would say that a branch cut for a multivalued function (like a logarithm or fractional power of $x$) is a subset $S$ of the complex plane such that the function in question is a well-defined analytic function on $\Bbb C\setminus S$. Typical branch cuts for the mentioned functions are rays emanating from $0$, but a spiral $\{re^{ir}\colon r\ge0\}$ would also be a branch cut for these functions. $\endgroup$ – Greg Martin Mar 11 '15 at 4:46

A branch cut itself is a curve in the complex plane where we cut the multivalued function so that we get a single valued branch. In the logarithm example you gave, it is a multivalued function that can be visualized as a spiral:

enter image description here

The vertical dimension is the imaginary part, as a function in the complex plane (horizontal dimensions). The two branches you gave are separated from each other by a cut along the positive real axis. If you imagine taking scissors to the spiral and making cuts everywhere the spiral crosses the positive real axis, then each piece left is a branch, and each piece doesn't overhang itself (it is single valued).

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  • $\begingroup$ I'm sorry, man. Maybe it's too early in the morning—but I can't figure out what that picture is supposed to represent. I could tell you "The horizontal part is the complex plane and the vertical part is the imaginary part of the logarithm," but I'm just not putting two and two together. Changing the magnitude of either the real or imaginary parts when the other is 0 shouldn't change the angle—and therefore shouldn't change the imaginary part of the logarithm, right? It looks like the graph is spiralling downward vertically along those lines, though. $\endgroup$ – AmagicalFishy Mar 11 '15 at 9:16
  • $\begingroup$ I can definitely see the singularity at the origin. $\endgroup$ – AmagicalFishy Mar 11 '15 at 9:16
  • $\begingroup$ You are right that changing the magnitude shouldn't change the angle; the grid lines on the graph are not lines of constant "height" which is why they are curved. The graph is formed by taking the positive real axis and sweeping it around the origin while raising its height at a constant rate. Only the imaginary part is shown because the real part is quite boring (it's just the log of the magnitude, so it's rotationally symmetric about the origin and already single-valued). $\endgroup$ – Victor Liu Mar 11 '15 at 13:12

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