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

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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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65 views

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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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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62 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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51 views

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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26 views

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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144 views

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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36 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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50 views

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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60 views

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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34 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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35 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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44 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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108 views

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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158 views

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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67 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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64 views

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 ...
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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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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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48 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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37 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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71 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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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 ...
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Problem identifying branch cuts of a square root function

Just when I thought I understood the basics of branch cuts, I started to plot some standard functions to see how they were handled on a computer. I used python 2.7 I plotted the function ...
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Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
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$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$

As the title states, the definite integral in question is $$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$ where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the ...
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29 views

Non probabilistic algorithm for min-cut problem?

I know about Karger's algorithm and its variations, all of them being probabilistic. Is there non-trivial (i.e. non-brutefoce) deterministic algorithm for mincut problem?
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integration with 4 branch point

I come across a problem of contour integration with 4 branch point. The problem came down to be equivalent to integrate $\sqrt{(x^2-1)^n(x^2-2)^m}$ over the imaginary axis. So, there are two branch ...
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81 views

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 ...
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Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$ \int^1_0 dx \frac{\ln(x-a)}{x-b} $$ by turning it into something in terms of dilogarithms. But for certain values of $a$ and ...
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40 views

Constructing Branch Cuts for $\sqrt{z}$ Not on the Negative Real Axis

Can someone provide an example of different branch cut for the complex square root functioning than the classic one along the negative real axis? I'm a little hazy on the full purpose of branch cuts, ...
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39 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
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60 views

Can Gomory's cutting plane be used to solve Mixed Integer Linear Programs?

Do you know if Gomory's cut can be used to solve MILP problems ? I have read that Gomory's cut is useful when all variables are integer, but what if just some of them are integer ? Is is necessary ...
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LambertW: $ x=W(x\cdot e^{x}) $ for $ x \ge -1$ but not for $x \lt-1$. How do I express my formula/my text?

I just found by numerical heuristics for some systematic numbers $q(x)_\text{heuristical}$ depending on $x=1,2,3,4,\ldots$ using WolframAlpha the suggested interpretation in terms of the ...
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Why should $f(z)=\sqrt{z}$ be limited on $\mathbb{C}-\{z:\Re(z)\leq0\}$ to be considered as an analytic function?

A multivalued function $f(z)$ can be analytic on an open set $\Omega$ where $f(z)$ has an unique value and is differentiable on every point. If $f(z)=\sqrt{z}$, I think $\Omega$ can be defined as ...
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91 views

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 ...
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41 views

Quick question on complex integral

For $f(z) = (1-z^2)^{\frac{1}{2}}$, how do I show that the integral of $f(z)$ from $0$ to $\pi$ is $O(R^{-2})$? $$\int f(z) dz = \int \frac{1}{z(1-z^2)^{\frac{1}{2}}} dz $$
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Calculating an integral with a branch cut, using some “uniqueness property”

Consider a complex function $$\tilde{f}(z)=z\int_{M}^{\infty}ds' \frac{\rho(s')}{z-s'} \qquad (1)$$ , where $M>0$ and $$\rho(s')=\frac{1}{s'}\sqrt{1-M/s'}.$$ This function is analytic in the ...
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26 views

Principal value for this question?

I have this question in my notes: Here is the answer: How in the world do they get that starting equation for -i?? THANK YOU
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117 views

Recommend textbooks that expain branch cut, Riemann surface and contour integration with branch cut in detail

I read several textbook on complex analysis, but few of them explain the branch cut and Riemann surface in detail and treat the contour integration with branch cut. But this is very important for many ...
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Branch Cuts of $f(z) + g(z) = \sqrt{p(z)} + \sqrt{q(z)}$

How does one find the branches of $$f(z) + g(z) = \sqrt{p(z)} + \sqrt{q(z)}$$ where $p$ & $q$ are second degree polynomials? It would be very nice to see this general method applied to, say, ...
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64 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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204 views

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
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24 views

Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2]$

Problem: Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2].$ My approach to this problem was to prove that ...
3
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1answer
322 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
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85 views

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} ...