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

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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 ...
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59 views

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 ...
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Using Multiple Branch Cuts in a Contour Integral

I have the integral $$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$ which I am trying to integrate using complex integration. I know that ...
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32 views

Complex Line integral of 1/z over the principle branch cut

I would appreciate it if someone checked my work to ensure that it's consistent. Compute the integral $\int_{C}\frac 1 z {dz}$ by obtaining an appropriate branch of the logarithm. There's an ...
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22 views

Integrate function with 2 branch points

Every example I see in textbooks so far has not shown me cases like this, so please help with the following question. I wish to integrate a function $f(z)$ around the contour shown below. $f(z)$ has ...
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40 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
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63 views

Regarding branch cuts and contour integration

I am trying to compute the following integral through the use contour integration. $$ \int_0^1 \frac{dx}{\sqrt{x^2-1}} $$ So, I am considering the same integrand but from $-1$ to $1$, then doing the ...
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compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
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26 views

What is the domain of the function f(z) = log(z) if it has a branch cut at angle alpha in rectangular coordinates?

What I am given: $log(z) = ln(r) + i\theta$ , where $(r>0, \alpha<\theta<\alpha+2\pi)$ $log(z)=\frac{1}{2}ln(x^2+y^2)+itan^{-1}(\frac{y}{x})$ What I am told: Use the Cauchy-Riemann ...
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Integration: Branch cuts

Can someone show me how to calculate this integral using branch cuts ? $$\int_0^{\infty}\Big(\frac{x}{1-x}\Big)^{\frac{1}{3}}\frac{1}{1+x^2}dx$$
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What is a simple way of describing branch cuts?

Branch cuts have been asked about and discussed on MSE extensively. That is, every answer to something along the lines of "What is a branch cut?" is... extensive. I'm looking for a quick, intuitive ...
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75 views

What is a branch cut? [duplicate]

This may be a strange question; but I've read and re-read the chapter in my textbook on what exactly a branch of a logarithm is and am having trouble understanding. What is a branch of a ...
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40 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
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57 views

Calculate integrals concerning a branch

i am trying to calculate this integral: $$\int_{|z|=5}\frac{dz}{\sqrt{z^2+11}}$$ Using the branch that gives : $\sqrt{36} = -6$ The function has 2 poles at $|z| < 5$, lets call them $\alpha$ and ...
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Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$$ Where $s$ is a complex parameter, and $n$ is a positive integer. The integral ...
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37 views

Branch cut for arcsin(z)

I am referring to this particular example found here: http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/arcsin.pdf On page one, I have difficulty understanding the region where $Arcsin(z)$ is ...
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45 views

Contour integral with two branch cuts

I'm trying to solve this integral: \begin{equation} \int_0^\infty d\omega \,\frac{\left(\left(\omega ^2+1\right) \cos (\delta )-2 \omega \right) \log ...
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50 views

There is no branch of $\arg z$ in $\{0 \lt |z| \lt 1 \}$ using the fact that $\int_{\gamma} \frac{dz}{z}=2\pi i$

Show that there is no branch of $\arg z$ in $\{0 \lt |z| \lt 1 \}$ using the fact that $\int_{\gamma} \frac{dz}{z}=2\pi i$ , where $\gamma(t)=Re^{it}, o \le t\le 2\pi$, $R\gt 0$. Suppose there is a ...
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30 views

Evaluate the integral $\int_{\gamma} \frac{dz}{z+\frac{1}{2}-\frac{i}{3}}$

Let $\gamma(t)=i+e^{it}$, $t \in [0,2\pi]$. Then Evaluate the integral: $$ \int_{\gamma}\frac{dz}{z+\frac{1}{2}-\frac{i}{3}}$$ Now $\gamma(t)$ is a closed curve . Since ...
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40 views

Complex Analysis Branch cuts

Take the branch of $\log(z)$ to lie in $(-\pi, \pi]$. With complex numbers when does $\sqrt{z^2} = z$ hold and when doesn't it? If we take $z=-1$, this equality holds but then $1=-1$. I have a feeling ...
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1answer
112 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
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Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
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How to perform this contour integration with $\log$ in the denominator?

Let $k > 0$ and $ a>1$ be constants. As far as I can tell, the integral $$ J = \int_{-\infty}^\infty dx\frac{e^{i k x}}{1+x^2}\frac{1}{\log(a - ix)} $$ converges, since the argument of the ...
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1answer
36 views

Residue of function with different branch points

I'm wondering what would one do when one wishes to find the residue of a function $$\text{Res}_{z\to z_0} f(z)$$ where $f(z)$ has multiple branch points, for instance $f(z)$ may be a function such as ...
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17 views

Polar coordinates used to evaluate a function containing a branch cut

I'm having a lot of trouble understanding how to approach these kinds of problems, if anyone could explain the approach, it would be really helpful. The problem is as follows: The function $f(z)$ is ...
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86 views

Branch cuts of $\frac{1}{\sqrt{z^2 + m^2}}$

I am reading up about Quantum Field theory and the integral of the following function pops up: $$\frac{1}{\sqrt{z^2 + m^2}}$$The details are actually explained in this question. The book says that ...
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Branch cut of $\sqrt{z}$ along the *positive* real axis

Consider the function $\sqrt{z}$, with $z\in\mathbb{C}$. Writing $z = re^{i\theta}$, the imaginary part of $\sqrt{z}$ can be expressed as: $$Im({\sqrt{z}) = r^{\frac{1}{2}}\sin({\frac{\theta}{2} + ...
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Argument at branch cut

I try to use residue to calculate this integral $$\int_1^2 \frac{\sqrt {(x-1)(2-x)}} {x}\ dx$$ I let $$f(z)=\frac{\sqrt {(z-1)(2-z)}} {z}$$ and evaluate the integral $$\int_{(\Gamma)} f(z)dz$$ along ...
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Finite integral involving branch cut. Basic Argument Question

I am reading this Wikipedia article on examples of contour integrals using complex analysis (http://en.wikipedia.org/wiki/Methods_of_contour_integration). In particular, I am looking at Example (VI), ...
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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 ...
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64 views

Branch cut and principal value

I do not understand the principal value and it is relation to branch cut. Please tell me about principal value with some examples, then explain the branch cut concept. For instance, what is the ...
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Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
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How do I solve this integral with a branch point at z =0?

The integral $\int_{-\infty}^{\infty}e^{\iota\left(k+\iota\delta\right)x^{2}}dx$ can be written as $\int_{-\infty}^{\infty}\frac{e^{\iota\left(k+\iota\delta\right)z}}{\sqrt{z}}dz$. Here, the branch ...
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Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
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38 views

A question about the complex logarithm

So frist, define $L(z) = \log(r)+i\theta $ is the holomorphic branch of $\log(z)$ on the cut-plane $\mathbb{C} \setminus (-\infty,0]$ such that $L(1)=0$ Let$[1,i]$ denote the line segment from 1 to ...
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How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
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38 views

Showing $1$ is not a branch point for $f(z) = z^2$?

I can see geometrically why $1$ is not a branch point for $f(z) = z^\frac{1}{2}$ as if we take a a point $p$ on the Riemann surface for $z$, $\epsilon$ distance away from $1$ are able to rotate that ...
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$z_0 = 0$ a branch point for $f(z) =(z + i)^{\frac{1}{2}}$?

I seem to have a mental block regarding branch points...I thought that the singularities of a function determined its branch points but then I read that they are irrelevant when deciding if a point is ...
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98 views

Contour integration and the square root branch cut

Consider the following equation $$ \int_0^\infty f(\sqrt{x(x-a)}) dx $$ For $a>0$ real and some analytic function $f(z)$ which dies off sufficiently fast for $\Re[z]>0$ and $\Im[z]>0$ so ...
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Integral with branch cut ( Problem while calculating residue)

While calculating this integral $\int_{-1}^{1}\frac{dx}{\sqrt{1-x^2}(1+x^2)}$ , I am really struggling to calculate the residue at (-i), I am getting the value of residue as $\frac{-1}{2\sqrt{2}i}$, ...
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Why do we need a branch cut for $\int_0^{\infty} \frac{x^{\frac{1}{2}}}{{(1 + x)^2}}dx$?

What is the significance of the $x^{\frac{1}{2}}$ in the numerator of this integral. I have read this kind of integral requires taking a branch cut. Why do we need a branch cut, what does it enable us ...
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89 views

A branch of $\tanh^{-1}z$?

$\def\Log{\operatorname{Log}}$ How can I show that $$\frac{1}{2}\Log\left(\frac{1+z}{1-z}\right)$$ defines a branch of $\tanh^{-1}(z)$ on $\mathbb{C}\backslash((-\infty,-1]\cup[1,\infty))$? (where ...
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Branches of $\log(z)$ on $\mathbb{C}\backslash(-\infty,0]$?

I know this is the most typical example of branches and I think I don't get the concept... Could you help me by giving a detailed development leading to all the required branches? It'd help me ...
3
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1answer
110 views

how to find the branch points and cut

for $\sqrt{z^2+1}$, how can I find the branch points and cuts? I let $z=re^{i\theta+2n\pi}$ and substitute into $$\sqrt{r^2 e^{i(2\theta +4n\pi)}+e^{2k\pi}}=$$ then, I don't know how to deal with ...
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101 views

Derive branch cuts for $\log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

Attempt: First, we examine $\sqrt{1-z^2}$. Note that it can be written $\sqrt{1-z}\sqrt{1+z}$, so the appropriate branch cuts are $(-\infty,-1)$ and $(1,\infty)$ for the inner square root term. ...
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81 views

How do we define the branch cuts for $\sin^{-1}z = \frac{1}{i} \log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

As $\sin^{-1}z$ is a function of complex $\log$, it is multivalued. The branch cuts to make $\log$ single-valued are defined conventionally as $-\pi < Arg(z) \leq \pi$. Why wouldn't this carry over ...
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65 views

How does one define appropriate branch cuts for arcsin(z) in the complex plane?

In learning about complex numbers, I have come across the following: http://upload.wikimedia.org/wikipedia/commons/b/be/Complex_arcsin.jpg Clearly, arcsin is multi-valued because it is a function of ...
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1answer
53 views

Branch of multiple-valued function (complex analysis)

Find a branch of $(z^3 - 1)^{1/3}$ which is analytic in $|z| > 1$ So we essentially want to study $\frac 13\text{Log} (1 - \frac 1{z^3} )$, the principal branch of the logarithm where $-\pi ...
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97 views

Analyze branch cuts and discontinuities of function $f(z)=\sqrt{1-z^2}$

Analyze the function $f(z)=\sqrt{1-z^2}$, where the square root is defined by the principal branch of the log function. Where does it have discontinuities? Here's what I did: We have $I = ...
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162 views

Understanding branch cuts by manually choosing the branch cuts of a function

Below I will explain what I have done in order to illustrate my confusion with branch cuts of a typical function. If I say something wrong at any point please do not hesitate to correct me! In order ...