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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.

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Branch point of $\log(z)+arcsin(z)$

i try to do it by substituting $z$ into $\frac{1}{t}$. Then, it would become this function $-iln[i+t\sqrt{(1-\frac{1}{t^2})}]$. Teacher says that when $z=z_\infty$ is a branch point, only could be ...
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Branch cuts for $(z^2+1)^{1/3}$

I'm just learning about branch cuts so I'm hoping to get some clarification on this. As in the title, I'm looking at $f(z)=(z^2+1)^{1/3}$. The obvious way to write this is $f(z)=\exp(\frac{1}{3}\ln(z^...
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26 views

Countour integral with branching point and pole behaviour

I want to compute this contour integral: \begin{equation} \int\limits_C \! \mathrm{d}z \; \log(\frac{z+1}{z-1}) \frac{e^{tz}}{z-1} \end{equation} Where $C$ is a path going around the branch cut. My ...
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Does there exist a branch of $(a^2 - z^2)^{1/2}$ holomorphic on $\mathbb{C}\setminus [-a,a]$?

To compute the integral \begin{align*} \int_{-a}^a (a^2-x^2)^{1/2} \, dx \end{align*} (and practice contour integration) I am trying to define a branch of the integrand with branch cut $[-a,a]$, and ...
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Branch points on Riemann surfaces

I have a question on the exact definition of branch points (on Riemann surfaces) I have the following definition:: Let $f:X \rightarrow Y$ be a holomorphic function between Riemann surfaces. $x \in X$...
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Why is $z=1$ not a branch point of the function $w=f(z)=z^{1/2}$?

Consider the function $w=f(z)=z^{1/2}$ and the point $z=1$ on the $z$-plane. Next consider a closed circular loop of radius $2$ about the point $z=1$ so that $w=1$. As we go around $z=1$ along this ...
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Is a limit point of branch points a branch point?

I have come into a discussion with my friends over a complex analysis question: Is $\infty$ a branch point of $\log(\cos z)$? I can't get a clear answer to this from the definition of branch points. ...
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31 views

Choosing multiple branch cuts

We have $z^{\frac{1}{3}}$. I need to find three branch cuts of this function. I know branch cuts are made such that the function becomes single valued. However, I'm really uncertain on how to find the ...
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190 views

Elementary questions about branch points and branch cuts

Consider the following complex function: $$f(z) = \sqrt{z-1} \cdot \sqrt{z+1}. $$ It is posible to write it as: $$f(z) = e^{\frac{1}{2}(\ln(z-1)+\ln(z+1))}. $$ (1) From $\ln(z-1)$ I get branching ...
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101 views

Contour integral of inverse square root.

I want to calculate the below complex integral on the upper-semi-circle which its radius goes to infinity: $$ \oint \frac{dz}{\sqrt{1+z^2}}$$ I tried with the substitution $z = R e^{i \theta}$ where ...
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71 views

Branch Points and Branch Cuts for cube root

Having trouble with finding the branch points/branch cuts of this function: f(z)= $3\sqrt\frac{(z-2)(z+1)}{(z+2)}$ ive tried using the equation $z=re^{i\theta +2\pi n}$ but then I don't really know ...
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33 views

How to avoid Branch Cuts in a Function Plot that involves ArcTan

I have been looking for answers in the web and I still cannot solve it, so here it goes. I have this function $$d(z)=\arctan\frac{f_2(z)\cos(zr)-f_1(z)\sin(zr)}{ f_1(z)\cos(zr)+f_2(z)\sin(zr)}$$ ...
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Deck transformation of a branched covering

Is that true that a threefold covering of Riemann surfaces $f: Y \to X$ with a ramification point of order $2$ has a trivial group of deck transformations, provided $Y$ is connected? As an example, ...
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Branch Points of the Polylog function

The polylogarithm $$ {\rm Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s} $$ has obvious branch points at $z=1$. For integers $s\leq 0$ it is a rational function with a pole of order $1-s$ at $z=1$. If $...
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Argument range for branch cut of $[-1,1]$?

I read somewhere on this site that for a branch cut between $(-\infty, 0]$, the range of values for $\arg(z)$ is $[-\pi,\pi]$, while for a branch cut between $[0,\infty)$, the range of values is $[0,2\...
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When/where to account for arguments/phases in branch cut problems?

Here I present two cases. 1) I first consider the keyhole contour. For instance, if I want to find $\int_{0}^{\infty} \frac{{\text{d}}x}{(x+a)^2 \sqrt{x}}$, when I consider the piece of the contour ...
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190 views

Branch Points of $\sqrt{z - 1/z}$

I'm working through Gamelin's Complex Analysis and am confused by the following Example: Consider $\sqrt{z - 1/z}$. We rewrite this as $\sqrt{z-1}\sqrt{z+1}/\sqrt{z}$. The function has three finite ...
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66 views

Smooth branch divisor implies smooth covering space

Let $c: S'\to S$ be a branched cover of a smooth projective surface $S$ that is branched over the smooth divisor $B\subset S$. Why does smoothness of $S$ and $B$ imply smoothness of $S'$?
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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) ...
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Contour integration on branch cuts

Hopefully this is a very easy question for someone out there to answer: Is it possible to have a branch cut that could contain one or more singular points (I am assuming it is). If that is the case ...
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87 views

Lagrange interpolation with multiplicities

I was wondering if it were possible to do Lagrange interpolation with multiplicities. Lagrange interpolation gives us a polynomial that obtains certain values on a given set of points. More precisely, ...
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Approaching a branch point along different paths

There's a very nice characterization of the three main types of isolated singularities of an analytic function $f(z)$ that takes oriented curves $\gamma$ that terminate at the singularity and ...
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Image of an analytic function near a branch point

Picard's great theorem says that any analytic function of one complex variable defined on a punctured neighborhood of an essential singularity takes on every complex value (with at most one possible ...
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How to find the values of a complex function on either side of a branch cut?

I have the complex function $f(z)=(z^2+1)^{1/2}$ which had branch point singularities at z=i, z=-i. To try to find the value of $f$ on either side of the cut I considered $z=i+\epsilon e^{i\theta}$, ...
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branch cut and branch points for $f(z)=[(z-1)(z-2)]^{1/3}$

I am trying to identify a possible branch cut and branch points for $f(z)=[(z-1)(z-2)]^{1/3}$. I am confused why this would even need a branch cut. My problem says it is multivalued, but it seems ...
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414 views

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 ...
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373 views

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}\...
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122 views

How to choose a branch when there are multiple branch points?

$\sqrt[n]{1-z^2}$ =$\sqrt[n]{(1-z)(1+z)}$ two branch points at -1,and 1. $\sqrt[n]{z}$ hast n branches and only one branch point at z=0. My understanding: When we want to integrate on different ...
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Branch point of $\frac{\sin\sqrt{z}}{\sqrt{z}}$

Show that $z=0$ is not a branch point for the function $f(z)= \frac{\sin\sqrt{z}}{\sqrt{z}}$. Is it a removable singularity? Definition- a branch point of a multi-valued function is a point such ...
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97 views

What did I miss using the residue theorem?

Using the residue theorem, I want to evaluate the integral along the entire real line, \begin{equation} \int_{-\infty}^{\infty} \dfrac{1}{\sqrt{1-q^2}\cos(\sqrt{1-q^2})}\ \mathrm{d}q, \end{equation} ...
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Choice of coordinates to analyze branch points of $f(z) = (z^2+1)^{1/2}$

The parametrizations $$z-i = r_1\exp(i\theta_1) \quad \text{and}\quad z+i = r_2\exp(i\theta_2)$$ are used to show how $f(z) = (z^2+1)^{1/2}$ changes when we make a complete loop around the branch ...
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Riemann surface with infinite sheets but algebraic branch points

I recently read that all infinitely sheeted Riemann surfaces can be related to exponentials, which I took to mean that they occur when you have logarithmic branch points. Is it possible to have an ...
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Justifying exchanging limit and integral arising in a contour integration with a branch point

I'd like to solve $\displaystyle \int_0^\infty \frac{x^\alpha}{x(x+1)}dx,$ where $\alpha \in (0,1).$ The answer is $\frac{\pi}{\sin (\alpha \pi)}.$ Typically, to solve it, we use contour integral ...
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Contour integral for integrand with branch point

I'd like to solve $\displaystyle \int_0^\infty \frac{x^\alpha}{x(x+1)}dx,$ where $\alpha \in (0,1).$ The answer is $\frac{\pi}{\sin (\alpha \pi)}.$ Typically, to solve it, we use contour integral ...
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How to approach branch cut question that involves infinitesmal stuff?

anybody have any idea how to approach part b) of the branch cut question like the one below???
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370 views

How to find branch point of z^(-1/2)

I know branch points are the point where $z^{-1/2} =0$ or $z^{-1/2} =\infty$, so for this I think $z$ should be $\infty$ for $z^{-1/2}$ to go $\infty$?? However for $z^{-1/2}$ to be equal $0$, I ...
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what are the branch points of this function $w = \ln(1+\sin(z))?$

Book says there are infinitely many branch points, but i don't know how to find one. It really confused me.
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118 views

Taylor series around a branch point

The function $w:=f(z)$ defined implicitly by $\Phi(w,z)=w^2-z^2-z^3=0 $ has two critical points, $z=0$ and $z=-1$. I thought both of them were branch points (and hence singularities) but I realized it'...
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363 views

Branch Cuts of $1/\sqrt{1-z^4}$

I'm having a rather difficult time wrapping my head around branch points and branch cuts. Specifically I'm looking at $f : \mathbb C \to \mathbb C$ where $$ z \mapsto \frac{1}{\sqrt{1 - z^4}} $$ I ...
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156 views

Integral of Coulomb potential in cylindrical coordinates

When calculating the screened Coulomb potential energy between electrons in cylindrical coordinates (need those for my particular geometry) the following triple integral always appears, $$\int_0^\...
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98 views

Complex roots and singularities

Maybe the question is trivial, but I cannot find an answer according to the standard books of Complex Analysis. We have three kinds of singularities: isolated, pole, and essential. What kind of ...
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How to find the branch points of the following curve?

For a function such as $f(z)=z^{1/2}$ it is easy and intuitive to me how to find its branch points. Now, I have to deal with a complex curve with the defining equation $$ z + \frac{1}{z} = \frac{y^2-u}...
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Contour Integral of $1/(x^2+1)^{3/2}$

If I want to find the contour integral of $$ \oint_{|x-i|=\rho} \frac{1}{(x^2+1) \sqrt{ x^2+1}} dx $$ where $$ \rho \rightarrow 0 $$ How can I do that? Is there any useful approximation I can used ...
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200 views

Abel-Plana formula for specific function

Suppose we have the function $f(n) = \sqrt{n^{2}+a^{2}}$, where $a$ is real. I need to calculate the finite expression $$ \sum_{n = 0}^{\infty}f(n) - \int \limits_{0}^{\infty}f(x)dx $$ By using the ...
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257 views

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-...
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Integral: $\int_{|z|\mathop=2}\sqrt{z^4-z}\,dz$ - finding Residues / Laurent Series

I wish to calculate the integral $$\int_{|z|\mathop=2}\sqrt{z^4-z}\,dz$$ $$z^4-z=z(z-1)(z-e^{2i\pi/3})(z-e^{-2i\pi/3})$$ We must make branch cuts which go through the branch points. The branch ...
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Concluding Even-ness From a Simple Functional Identity

Let $f\left(z\right)$ be an analytic function on an open disk $D$ (of possibly infinite radius) centered at zero satisfying the functional equation: $$f\left(z^{3}\right)=f\left(-z^{3}\right),\textrm{...
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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 (\...
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Check if branches exist or not

We have four multivalued functions: $f\left( z\right) =\left( \dfrac {z-1}{z+1}\right) ^{\dfrac {1}{2}}$ $g\left( z\right) =\left( \dfrac {z-1}{z+1}\right) ^{\dfrac {1}{3}}$ $h\left( z\right) =\left( ...