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Questions tagged [branch-cuts]

A branch cut is curve in the complex extending from a branch point of the function.

A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always, taken between pairs of branch points.

Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function. (Wikipedia)

498 questions
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Constructing Branch Cuts for $\sqrt{z}$ Not on the Negative Real Axis

Can someone provide an example of different branch cut for the complex square root functioning than the classic one along the negative real axis? I'm a little hazy on the full purpose of branch cuts, is it to ensure a continuous domain is sent to a…
user82004
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Principal value for this question?

I have this question in my notes: Here is the answer: How in the world do they get that starting equation for -i?? THANK YOU
Amy
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Branch cut of $z^{-s}$

I need to perform the following integration: $I(s) = \int_{\gamma}\text{d}z\ z^{-s} \frac{\text{d}\ln\mathcal{F(z)}}{\text{d}z}$, where $\mathcal{F(z)}$ is analytic everywhere on the complex plane. For the purpose of integration, the branch cut…
nightmarish
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Can someone please explain why $\log\left(\frac{z-1}{z+1}\right)$ has branch points at 1 and -1

Can someone please explain a) why $\log\left(\frac{z-1}{z+1}\right)$ has branch points at 1 and -1. I know what a branch point is as it is defined in my text. I've seen that sometimes we have a branch cut along the negative real axis for $log(z)$.…
Elli
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Branch cut of $\log(z)$ in comparison with branch cut of a function like $\sqrt{z}$

I am looking at the explanation of the branch cut for $\log(z)$ defined at $ -\pi$ $< \theta \leq \pi$ and the explanation seems to be that the limit does not exist on the negative real axis because there is a jump of $2\pi$ as we cross the negative…
Elli
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