Questions tagged [branch-cuts]

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

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29 views

How would yo make a branch cut to define a single-valued branch of the function log(z - 1 + i)?

The question says it all. I understand the concept of branch cuts, but I have not quite yet figured out how to find branch cuts of functions. If I am not mistaken, the branch cut of logz is πi + 2πz ...
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65 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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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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47 views

Show this is a branch for $z^{ab}$.

Let $f: G \to \mathbb{C}$ and $g: G \to \mathbb{C}$ branches of $z^a$ resp. $z^b$. Suppose that $g(G) \subseteq G$ and $f(G) \subseteq G$. Show that $$f \circ g, g \circ f$$ are branches of $z^{ab}$...
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55 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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How to construct a function whose inverse branches are related by a radical?

Consider a function $f(x)$ that has a minimum $c$ and whose inverse around that minimum has two branches, $x_1(f)$ and $x_ {2}(f)$. I look for functions $f$ such that $\frac{1}{x_1(f)}-\frac{1}{x_2(f)...
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60 views

$f(z)=\sqrt{1-z^2}$ pole at infinity

Consider integrating $f(z)=\sqrt{1-z^2}$ with a branch cut of $[-1,1]$ around the following contour. $\gamma_1:[-1,1]\to\mathbb{C}, t\mapsto t+\epsilon i$ $\gamma_2:[-\pi/2,\pi/2]\to\mathbb{C}, t \...
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51 views

Contour integral around a branch cut

I'm working on computing a contour integral (Actually it's a part of a inverse Laplace transform problem, see my question) $$ \oint_{\Gamma}g(s)ds $$ $$ g(s)=e^{-\tau s\sqrt{\frac{s+q}{s+p}}}e^{ts} $$ ...
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simple properties of branches of complex square root and integrating a function with a branch of square root

I'm trying to grasp the difference between branches for the complex square root and I'm having difficulty with some very basic examples. First example, if I choose $\sqrt{\,}$ to denote the branch ...
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37 views

How to find branch cut of $\ln(\sin(z))$?

I know that $\ln(z)$ is undefined when $z=0$ or $z=\inf$ , so I manged to find the critical points to be $z=\{k*\pi,\inf*i, -\inf*i\}$ , for every integer k, but I don't know how to proceed from here ...
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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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1answer
38 views

Defining a holomorphic branch of $f(z)=\log(z^2-1)$

I am revising Complex Analysis and am confused about how to approach this question. I want to define a holomorphic branch of $f(z)=\log(z^2-1)$ on the cut plane $$\Bbb{C}\backslash\{(-\infty,-1]\cup[...
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Contour Integration of product of cube roots

I have been trying to evaluate the residue at infinity of the following function: $\exp(-\frac{1}{3}Log(1-z))\exp(-\frac{2}{3}Log(1+z))$ The first Log (multiplied by $-\frac{1}{3}$) has branch cut $[...
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Mapping a 2-sheeted Riemann surface with 2 branch cuts to a Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
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Find the branch of each complex function making it holomorphic

There are 3 questions about branch cuts in my textbook, and I cannot wrap my head around the idea. For the first 2, I think I have an answer but it's a branch cut of a related function, not the ...
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49 views

Why is it that $ \ln(1) = 0 $ but $ e^{(i2\pi)}=1$?

Why is it that $ e^{(i2\pi)} = 1 $ but $ \ln(1) = 0 $? In other words, is $2\pi i = 0$? I know that $e^0 = 1$, but should it also equal $e^{2\pi i}$?
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Find a branch cut where $f(z)=\log_{\tau}(z^3 - 2)$ is holomorphic at $z=0$

There is a question in my complex analysis textbook whose answer I don't understand, and cannot find a clear explanation online: Find a branch cut $C_\tau = \{z=re^{i\tau} : r>0,\tau\in\mathbb{R}...
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$\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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Fourier transform with branch cut 2

How to evaluate the following integral by using residue theorem and how we choose the contour. $$W=\int_{-\infty}^{\infty}\frac{d\lambda}{\sqrt{i\lambda}} \frac{e^{-ib\lambda}}{\lambda^2-a}$$, where ...
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1answer
49 views

Discontinuity of Hypergeometric function along the branch cut

I am trying to evaluate an expression involving the hypergeometric function evaluated near its (principal) branch cut discontinuity, which is placed on the real line from $1$ to infinity. For $x>1$...
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102 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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33 views

Branch cut of $\int_0^{\infty} \frac{x^{s-1}}{1+x}$

Why do I need to use a branch cut from $0$ to $\infty$ when evaluating $\int_0^{\infty} \frac{x^{s-1}}{1+x}$? I know there is a pole at $x=-1$ I can't seem to understand the purpose of the branch ...
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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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Is a multivalued function analytic over its branch cut?

I want to evaluate the integral $\int_Cf(z)dz$, where C is the unit circle centered at the origin, and $f(z)$ = log(z+2). I know that the integral of a function over a simple, closed contour is $0$ ...
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32 views

Distinct minimum spanning tree

For a connected, weighed, undirected graph G: G has a unique MST, if for every cut of G there is a unique minimum weight edge crossing the cut. Is this statement true? I think false because for the ...
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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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101 views

Finding a continuous branch of $F(z)=\sqrt{\frac{(z^2-1)(z-2)}{z}}$

I am having problems understanding branch cuts. For instance, I am given the following function, for $z \in \mathbb{C},$ let $$F(z)=\sqrt{\frac{(z^2-1)(z-2)}{z}}=\sqrt{\frac{(z+1)(z-1)(z-2)}{z}}.$$ ...
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29 views

B&B and simplex algorithm

I'm starting studying OR, I read that when solving PLI problem it's common to use Branch and Bound techinque which "decompose" the problem and solves smaller problems. My question is the following: ...
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1answer
78 views

Finding Laurent Series using Binomial Theorem - HOW?

I'm working on a fairly simple question asking to work out the necessary branch cut(s) for the function $f(z)=(z^2+1)^{1/2}$. I am comfortable doing this and the rigour required to explained why I ...
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111 views

Evaluate $\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz$.

Evaluate $$\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz\,.$$ What is an elegant way to evaluate this integral for Im $\alpha >0$? I imagine using residue theorem will lead ...
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62 views

Inverse Laplace transform of Gamma with branch cut

In solving a particular physical problem I have had to perform inverse Laplace transforms of sum and products of Gamma functions. Since my actual problem is complicated, I will state a simple example. ...
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1answer
39 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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Is the hypergeometric function a simple resurgent function

In the study of Airy functions you obtain, by Borel transforming components of a transseries, a hypergeometric function $_2F_1(a,b,c|z)$ where the constant are positive. Now I would like to know if ...
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58 views

Branch cut for $\sqrt{z^2-a^2}$ in the lower half of the complex plane (instead of $[-a,a]$) - How does this change the function?

I have given a function $\sqrt{z^2-a^2}$ with $a>0$. At first i have chosen the branch cut on the real axis at $-a<z<a$: $$f(z)=\sqrt{z-a}\sqrt{z+a}=\sqrt{|z-a||z+a|} \exp({i\frac{\theta_1+\...
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35 views

Shifting the branch cut of the complex logarithm

I have some troubles understanding branch cuts of the complex logarithm, or to be more precise, the shift thereof. So typically, the branch cut is along the negative real axis, connecting the branch ...
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1answer
311 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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129 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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106 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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Difference Between Branches and Injective Functions in the Complex Plane

Let $z = re^{i\theta}$ denote a complex number. Consider the square root mapping $w = z^{1/2}$. We can define single valued function by $$f_1 (z) = \sqrt{r} \, \exp\Big(i \, \frac{\theta}{2}\Big), \...
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Determining $g(-i)$ for $g(z) = \ln(1-z^2)$, given that $g(i) = \ln(2)$.

Consider $g(z) = \ln(1-z^2)$ defined on $C$ \ $(-\infty, 1]$. Find $g(-i)$ given that $g(i) = \ln(2)$. I've begun by stating $\ln(1-z^2) = \ln|1-z^2| + i*arg(1-z^2)$. The real part at $g(-i) = \ln(2)...
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1answer
50 views

Relation between dilogarithm and its complex conjugate

I am looking for a relation between the dilog and its complex conjugate, that is can I simplify the following summation of terms $$f(z) = \text{Li}_2(z) + (\text{Li}_2(z))^*?$$ I have looked through ...
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1answer
145 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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83 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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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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1answer
84 views

Choosing any branch cut, or some branch cut?

In An Introduction to Harmonic Analysis by Katznelson I once stumbled upon this: with the footnote Here $D$ is the complex unit disk around $0$. $f$ is holomorphic with no zeros, and since $D$ is ...
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71 views

Complex analysis - branch cut of log

I am trying to show that there is no continuous branch of log on and open set containing the origin. wlog I am taking the contour integral around the unit circle. I know that an antiderivative of a ...
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167 views

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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194 views

Question about Hankel's contour applied for Gamma functions

I would like to have some explanation (or pehaps some intuition) on the Hankel's contour for the following contour integral: $$ F(n)= \oint_{C} \frac{e^{ -z} }{z^{n+1}} \,dz$$ ,where n is non-...
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60 views

Branch cut rotation of the logarithm function?

in my text book i have an exercise that uses a different way of expressing the logarithm: $\log z = \ln + iθ$, $\ z = e^{i\theta}$, $\ r>0$, $\ -\pi/2 < \theta < 3\pi/2 $ I wonder how the ...