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

learn more… | top users | synonyms

0
votes
0answers
6 views

Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
0
votes
0answers
23 views

Branch points of angular function

I am having difficulties finding out the branch points of the funtion $F(b^2)$ defined by $$F(b^2) = \int^{2M}_{r'} \frac{dr}{r^2\left[1-\frac{b^2}{r^2}\left(1-\frac{2M}{r}\right)\right]^{1/2}}$$ ...
0
votes
0answers
8 views

Find branches F, G of log(z), both analytic at 1 and i, such that: Im F(1) > Im F(i) Im G(1) > Im G(i)

Find branches F, G of log(z), both analytic at 1 and i, such that: Im F(1) > Im F(i) Im G(1) > Im G(i) I think I know how to do it, I'm just little confused about what "branches" mean. When it ...
0
votes
0answers
38 views

Contour integral with a different contour

This question is from the post: Contour integral with branch cut. My question is: if we choose the key hole contour with branch cut on the positive x axis, it seems that we have an addtional term: ...
1
vote
1answer
48 views

Why can't branch cut pass through poles?

In the wiki article, Example (IV) – branch cuts. Why can't we can't we choose the contour so that the branch cut is on the negative x axis. If we choose this, the two residual is out of the contour, ...
1
vote
0answers
34 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
3
votes
1answer
66 views

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 ...
4
votes
0answers
69 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 ...
1
vote
0answers
28 views

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 ...
1
vote
0answers
42 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 ...
0
votes
0answers
27 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 ...
0
votes
0answers
46 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 ...
3
votes
1answer
82 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 ...
2
votes
0answers
41 views

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: ...
2
votes
1answer
31 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 ...
2
votes
2answers
104 views

Integration: Branch cuts

Can someone show me how to calculate this integral using branch cuts ? $$\int_0^{\infty}\Big(\frac{x}{1-x}\Big)^{\frac{1}{3}}\frac{1}{1+x^2}dx$$
1
vote
1answer
45 views

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 ...
3
votes
1answer
79 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 ...
2
votes
0answers
41 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 ...
2
votes
2answers
60 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 ...
6
votes
2answers
274 views

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 ...
0
votes
0answers
39 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 ...
0
votes
0answers
54 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 ...
1
vote
1answer
51 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 ...
1
vote
1answer
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 ...
1
vote
1answer
44 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 ...
1
vote
1answer
113 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 ...
2
votes
2answers
53 views

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 ...
5
votes
1answer
78 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 ...
1
vote
1answer
40 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 ...
0
votes
0answers
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 ...
2
votes
2answers
88 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 ...
1
vote
1answer
99 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} + ...
1
vote
2answers
90 views

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 ...
2
votes
0answers
37 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), ...
7
votes
3answers
165 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 ...
2
votes
1answer
68 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 ...
3
votes
1answer
99 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 ...
1
vote
1answer
46 views

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 ...
0
votes
0answers
71 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. ...
0
votes
1answer
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 ...
2
votes
0answers
50 views

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 $m$ ...
0
votes
1answer
39 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 ...
1
vote
0answers
25 views

$z_0 = 0$ 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 ...
1
vote
0answers
115 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 ...
2
votes
3answers
242 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 $\frac{-1}{2\sqrt{2}i}$, ...
5
votes
2answers
191 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 ...
1
vote
1answer
91 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 ...
3
votes
1answer
105 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 ...
3
votes
1answer
129 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 ...