Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [branch-points]

A branch point is a point in the complex that can map from a single point to multiple points in the range.

A branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point (Ablowitz & Fokas 2003, p. 46). Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. (Wikipedia)

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Branch cuts for $\sqrt{z^2+1}$

Consider the complex function $$f(z) = \sqrt{z^2+1}.$$ Obviously, $f(z)$ has branch points at $z = \pm i$. One way of defining a branch cut would be to exclude the points on the imaginary axis with $|z| \geq 1$. Another way of defining a branch cut…
Étienne Bézout
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What is a branch point?

I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a multivalued function. However, surely this…
user122316
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how to find the branch points and cut

For $f(z) = \sqrt{z^2+1}$, how can I find the branch points and cuts? I took $z=re^{i\theta+2n\pi}$ and substitute in $f(z)$ $$\sqrt{r^2 e^{i(2\theta +4n\pi)}+e^{i 2k\pi}}=$$ then, I don't know how to deal with this any more and by guessing, I…
leave2014
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Integrating $1/\sqrt{z^{2}-1}$ on some contour

If I wanted to integrate $$\oint \frac{1}{\sqrt{z^{2}-1}}$$ Say around a circular contour radius $2$ centre $0$, how would I do that? Does the function have poles at $\pm 1$ or are they just "branch points" without residue? Would the definition of…
John Fernley
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Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} \frac{dk}{\sqrt{k^2+m^2}}…
bnado
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How to properly understand branches of complex functions

$\DeclareMathOperator{\Log}{Log}$ I have several problems to understand the concept of branches and how to find analytic branches. From what I learned, for example for the complex logarithm, it is a multi valued function, and if we want it to be…
Gabi G
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Understanding the branch cut and discontinuity of the hypergeometric function

I am going through this paper, and I am having trouble understanding page 20. I am still learning my way around managing multi valued complex functions, so I'd like your help in understanding what's happening there. I have the definition of the…
Salvatore Baldino
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Defining the square root of $z$ squared and determining the location of branch cuts

I am asked the following: For $\epsilon > 0$, we define $$ \sqrt{z^2} = \lim_{\epsilon \to 0} \sqrt{z^2 + \epsilon^2}\,, $$ where the principle value square root is used on the right-hand side. Determine the location of the branch cuts…
Dennis van den Berg
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Keyhole Contour with Square Root Branch Cut on Imaginary Axis

Consider integrating the following function of a complex variable, $z$, $$ f(z)=\frac{z e^{irz}}{\sqrt{z^2+m^2}}, $$ around the contour It seems straightforward to show that the infinite radius arc segments, $C_1$ and $C_2$, vanish and we are left…
user143410
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Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From what I can tell there are three singularities: one…
William J. Cunningham
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Does $\sqrt{z^2}$ have any branch cuts?

I'm confused about the Riemann surface needed to properly describe the function $f(z)=\sqrt{z^2}$ on the complex plane. In other words, I am not interested in the definition on the real axis, $\sqrt{x^2}=|x|$ . It seems to me that this function is…
sillyQsman
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Branch points of $f(z)= \frac{\sqrt{z} \log(z)}{(1+z)^2}$

How does one go about finding the branch points/holomorphic branches of a multi-function composed of several other multi-functions? Here is an example of what I mean: Let $f(z)= [\frac{\sqrt{z} \log(z)}{(1+z)^2}]$ be a multifunction. Identify the…
user489116
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Classifying singularities of $\frac {z^{1/2}-1}{\sin{\pi z}}$

I am trying to classify the singularities of $$\frac {z^{1/2}-1}{\sin{\pi z}}$$ where $-\pi<\arg z<\pi$. I am confused by this because of the branch cut of $\sqrt z$ but here is my (bad) attempt: The singularities are at $z = n$ for $n \in \mathbb…
user85798
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Branch cut of $\sqrt{z^2 + 1}$

In my studies I've come across the concept of branch cuts, and I'm having a little bit of trouble digesting the topic. As I understand it, in this example: $$f(z) = \sqrt{z^2 + 1} = \sqrt{z + i}\sqrt{z - i}$$ With the substitution $ z + i =…
Vpen
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Calculate $\int_{-\infty}^\infty \frac{\cos{(kx)}}{\sqrt{x^2+a^2}} \,dx$

I want to evaluate the integral $\int_{-\infty}^\infty \frac{\cos{(kx)}}{\sqrt{x^2+a^2}}dx$, where $k$ and $a$ are constants. Clearly the integrand has two branch points at $\pm ia$. Let $f(z)=Re (\frac{e^{ikz}}{\sqrt{z^2+a^2}})$. Considering the…
Anindita Bera
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