Mathematics Stack Exchange
Mathematics Stack Exchange

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)

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Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want to clarify my understanding of multivalued…
AmadeusDrZaius
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Contour Integration and Branch Cuts: $\oint_{|\omega|=x}d\omega\:(1-\omega)^{-3/4}(x-\omega)^{-3/4}\omega^{-3/4}$

I have a embarrassingly simple question. For some reason I haven't really studied complex before and I'm suffering under this now. I need to evaluate contour integrals of multi-valued functions and I'm confused about a few details (the standard…
Heidar
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What are the branches of the square root function?

I am studying branches of logarithm. I came to know that there are infinitely many branches of logarithm where $\log z = \log |z| + i (\arg z +2k\pi)$, $k \in \mathbb Z$ and $z \neq 0$. Now for each $\alpha \in [0,2\pi)$ if we restrict $\arg z$ to…
Arnab Chattopadhyay.
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Branch cut for $\sqrt{1-z^2}$ - Can I use the branch cut of $\sqrt{z}$?

I was trying to clarify some questions I had about elliptic integrals using http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf There they define the map $$\phi\colon w\mapsto \int_0^w\frac{\mathrm{d}z}{\sqrt{1-z^2}}$$ on…
Gregor Botero
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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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Why would a branch cut not end at a branch point?

Both Wikipedia and MathWorld (here and here) seem to place some imporantance on saying, but not elaborating on It should be noted that the endpoints of branch cuts are not necessarily branch points. When would it make sense to have a branch cut…
hmakholm left over Monica
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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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Using complex analysis, prove $\int\limits_0^\infty \frac{\log(x) \arctan(x)}{1 + x^2} \, dx = \frac78 \zeta(3)$

Expanding on an earlier question, I am trying to prove that $$I = \int_0^\infty \frac{\log(x) \arctan(x)}{1 + x^2} \, dx = \frac78 \zeta(3)$$ using complex analysis but am running into some issues. My attempt: Let $f(z) = \dfrac{\log^2(z)…
user170231
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How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form $f(x) = ab^x$ where $a \neq 0, b > 0$). I've taught…
Nick H
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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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Smoothness of $\frac12[W_0(x)+W_{-1}(x)]$ for real $x<0$

The Lambert W-function, i.e. the multivalued inverse of $z=we^w$, has countably many complex-valued branches $W_k(z)$. The relations between the branches are a bit involved and are summarized here. We will consider the behavior of the $k=0,-1$…
Semiclassical
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Branch of $\sqrt{1-z^2}$

Show that a branch of $\sqrt{1-z^2}$ can be defined in any region $\Omega$ where the points $1,-1$ are in the same component of its complement. This is a question in Ahlfors' Complex Analysis (P.148 Q5) that I came across while trying to self-study…
Chi Cheuk Tsang
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An arctan series with a parameter $\sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right)$

I'm trying to evaluate $$\sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) \ , \ a >0. $$ The answer I get only seems to be correct for small values of $a$. What accounts for this? Using the principal branch of the logarithm, I…
Random Variable
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