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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35 views

Choice of branch cuts for integral of form $\log[a + bz + bz^{-1}]$

I'm having some difficulty to understand the evaluation the following integral: $$ I = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{d} \vartheta_2 \ln \left[ \underbrace{\cosh(2 \beta E_1) \cosh(2 \beta E_2) ...
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Simplification of an integral involving complex square root

Consider the integral $$f(a) = \text{Im} \int_0^a \text{d}z \frac{1-z/a}{z \sqrt{1-z}}$$ The integral only picks up a non zero contribution for $a > 1$ since the square root function $\sqrt{1-z}$ ...
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Evaluating the following integral $\int_{0}^{\infty} \frac{\ln(x^{2}+1)} {(x(x^{2}+1))} dx$

I have tried many different methods on this particular integral, none of them yielding any fruitful results. Here was an attempt I(t) = $\int_{0}^{\infty} \frac{ln(tx^{2}+1)} {(x(x^{2}+1))}dx$, I(1) ...
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Contour integration with fractional powers in denominator

I wish to integrate the following $$\int_0^{\infty} \frac{\sin(k r)}{ r^{1/2}(r-2m)^{1/2}} dr$$ I believe contour integration is the best way to proceed. I see that there are branch points at $0$ and $...
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26 views

Branch points but no branch cut. Is it possible?

I've stumbled upon a certain problem regarding the function $f(z)=log((z-2i)(z+3i))$ Going for the long path and taking the real and imaginary parts of $g(z)=(z-2i)(z+3i)$, then checking what points $(...
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25 views

Weierstrass-type factorization (reconstruction) of a function with branch cuts

Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,...
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Integrating a function with two branch point $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $

I was studying a book about residue intigrating then I get into trouble solving this : $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $ I know that I have to choose a contour like this: but I don't ...
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Proof Check: Complex function and Branch points

requesting a proof (rationale) check as i'm unsure whether or not my rationale is legitamate or not. Thanks in advance Consider the function $$f(z) = \frac{\omega}{(z^2 + \omega^2)e^{\alpha\sqrt{z/d}}...
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Branch Points and Cuts

I am having a lot of problems understanding what exactly branch points are and how they are computed for a function. There is this one problem that I just can't seem to get around to understanding, ...
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Branch cuts of complex logarithm for Friedel oscillation

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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Finding Branch Cuts and Branch Points of $f(z)=\left(\frac{z-a}{z-b}\right)^\frac{1}{2}$

I want to find the branch points and branch cut structure of $f(z)=(\frac{z-a}{z-b})^{1/2}$, but I'm stuck on the process. I tried to rewrite it in bipolar coordinates first using $z-a = r_1e^{i\...
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Why is this not a logarithmic branch point?

Good afternoon! I wanted to ask a quick question, as a beginner in complex analysis, I am trying to get my head around branch points. I came upon some lecture notes, but then did not understand ...
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85 views

Finding the branch points of $\log(z^2-1)$

This example is presented in the following paper (http://math.mit.edu/classes/18.305/Notes/n00Branch_Points_B_Cuts.pdf, example 2.2, page 11). The author uses the identity $\log(z^2-1)=\log(z+1)+\log(...
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171 views

How to solve this integral with contour integration $\int ^{\infty }_{0}\frac{\ln( x)}{( x+1)^{\alpha}}\,dx$

I know how to solve this integral without contour integration. The answer to the integral is $$\int^{\infty }_{0}\frac{\ln( z)}{( z+1)^{\alpha}} \,dx=\frac{H_{\alpha-2}}{1-\alpha} ,\; \alpha>1,$$ ...
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39 views

Why is it the case that infinity is also a branch point for log

I thought about considering $L(z)=log|\frac{1}{z}|+i\theta$ and consider the behaviour as $z$ tends to $0$? All I am observing now is that the function tends to infinity with it. I am really stuck on ...
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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} \...
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Branch cuts of $f(z)=\frac{1}{\sqrt{1+z^4}}$

Let $f(z)=\frac{1}{\sqrt{1+z^4}}$. The branch points are $e^{\frac{2k-1}{4}\pi i}$. I am going to find ALL possible branch cuts. When $z$ traces a closed curve (anticlockwise) around any of the above ...
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Prove $[z^{1/2}]$ has no holomorphic branch on any open ball $B(0,R)$

Prove the multifunction $[z^{1/2}]$ has no holomorphic branch on $B(0,r)$. Here is my attempt: Let $f: B(0,r) /(-\infty,0] \to \mathbb{C}$ be a holomorphic branch of $[z^{1/2}]$. Note $-f$ defines ...
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Help on finding the branch of a multi-valued function

I'm trying to answer the following question: Find a branch of the following multiple-valued function that is analytic in the given domain: $(z^2-1)^{1/2}$ in the unit disk |z|<1 I tried to answer ...
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83 views

Integrand with branch point

I am trying to learn how to work with branch points. I decided to change the sign in the next classical example: $$f(x)=\int_{-1}^{+1} \frac{1}{\sqrt{1-x^2}}$$ In this example, simply select the ...
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How to integrate $\frac{1}{z^{1/2}}$ with respect to complex variable $z$?

I'm looking at example 7.54 in the book by Ponnusamy, available publicly here, on page 240 (249 in the pdf viewer). Example 7.54. Consider $$\int_{|z|=1} f(z)\ dz,\quad f(z)=1/z^{1/2}.$$ Then $z=...
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Branch point versus points along branch cut

I am trying to get better intuition for branch points and a branch cut. I understand the textbook definitions: Branch point (of an analytic function $f$): Point in the complex plane whose value in ...
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What are the branch points of the function w(z)=arctan(z)

I have the next question: What are the branch points of the function $w(z)=\arctan(z)$. I don't have any idea of how to start to solve this question. Can someone please guide me on how to approach ...
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About ramification points of a map between Riemann surfaces

I have trouble formulating rigorously the concept of ramification points and their multiplicity (=ramification degree). I am in the case of maps from the Riemann sphere $\Bbb{CP^1}$ to itself. I ...
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24 views

Integral over circle around branch point

I am having some problems with the integral $$\int_\gamma \frac{(-z)^{s-1}}{e^z-1}\;dz$$ where $s>1$ and $\gamma$ is the cirle with radius $\delta$ around $0$. This integral should approach $0$ for ...
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Is this an admissible branch cut for $f(z)=\sqrt{z}+\sqrt{z-1}$?

For the function $f(z)=\sqrt{z}+\sqrt{z-1}$, on the one-hand I feel like since we have two branch points at $z=0$ and $z=1$, we would be able to define an admissible branch cut simply going from $z=0$ ...
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103 views

How to find the branch cuts of $\sqrt{g(z)}$ and the contour integral $\int_{z_1}^{z_2}d z\sqrt{g(z)}$

I need to evaluate the following integral: \begin{equation} \int_{z_1}^{z_2} d z \sqrt{g(z)}, \end{equation} where the function $g(z)$ is given by \begin{equation} g(z)=-\left(\alpha-\frac{\beta}{...
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84 views

Finding the branch of a function

I am having a really hard time understanding exactly how to determine the Riemann surface of a complex function f(z). I understand the concept: images of these complex functions are periodic, and so ...
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(Branch cut of z^2) Can someone explain this picture?

I am trying to understand exactly what is going on in the picture below: From what I understand so far, these are two complex planes. The left one is z, and the right one is the image of z under $f(z)...
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108 views

Residue at $\infty$ for $1/\sqrt{z^2-1}$?

This feels rather silly to ask, but this has been confusing me as of late. One exam question I was attempting recently was to find the contour integral of $1/\sqrt{z^2-1}$ over the contour $\Gamma$ ...
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86 views

Understanding the function $\frac{1}{\sqrt{z^2+11}}$

I want to understand the function: $f(z) = \frac{1}{\sqrt{z^2+11}} dz$. This functions seems to have two branch points $\sqrt{11}i$ and $-\sqrt{11}i$. Does it also have a brunch at $\infty$? Usually ...
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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 ...
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60 views

Meromorphic function at a point.

i'm study meromorphic function at the complex plane extended, but i have a trouble. I know if $ f \colon D \subseteq \mathbb{C} \to \mathbb{C} $ $f$ is meromorphic on D if the singularities of $f$ ...
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126 views

Questions about branch point of holomorphic map

In order to calculate genus of compact Riemann surface using Riemann-Hurwitz theorem, we have to determine the branch points first. Question: For holomorphic maps between $\Bbb{CP^1}$, is there a ...
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$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$ difficult integral with two branch cuts

$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$ I defined two branch cuts along the real axis: $[-\infty ,-\frac{1}{a}]$ & $[0,\infty]$ with the following contour: I defined the $arg{(z)} =0$ ...
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63 views

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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324 views

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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34 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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83 views

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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106 views

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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54 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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593 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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188 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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258 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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221 views

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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201 views

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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88 views

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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468 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 ...