What is Branch cut: Definition and 24 Discussions

In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.
Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w2 = z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch point typically means the former more restrictive kind: the algebraic branch points. In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.

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1. F

[Complex analysis] Contradiction in the definition of a branch

I find the following definition in my complex analysis book : Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one...
2. A Integrating a function of which poles appear on the branch cut

I have a complicated function to integrate from -\infty to \infty . I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{k^2 - \Omega^2}} \Omega d\Omega Where I0I0 and I2I2 are functions containing Hankel functions as...
3. I Drawing the contour with a branch cut

I am trying to determine the contour required in solving part b. The branch points (poles) are at s=0 and s= -a and in between these two values, there is a branch cut. I know that the branch cut cannot be included in the contour so does this mean the poles also cannot be in the contour? Would...
4. Contour integration with a branch cut

Homework Statement ∫-11 dx/(√(1-x2)(a+bx)) a>b>0 Homework Equations f(z0)=(1/2πi)∫f(z)dz/(z-z0) The Attempt at a Solution I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed...
5. A A problem about branch cut in contour integral

Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
6. I Domain of single-valued logarithm of complex number z

Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...
7. Complex integration, possibly branch cut integral

Homework Statement The integral I want to solve is $$D(x) = \frac{-i}{8\pi^2}\int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r^2\sin\theta$$ which I've reduced to $$D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}}$$ by integrating over ##\theta##. However, I...
8. Zee, Quantum Field Theory in a Nutshell, problem 1.3.1

Homework Statement I'm working through Zee for some self study and I'm trying to do all the problems, which is understandably challenging. Problem 1.3.1 is where I'm currently stuck: Verify that D(x) decays exponentially for spacelike separation. Homework Equations The propagator in question...
9. MHB Defining Branch Cuts for f(z) = z^(1/3)

Given the function $$f(z) = z^{1/3}$$. Can the branch cuts be defined by the rays from zero through the three roots of unity and to infinity?
10. MHB What is the Branch Cut for log(z)?

What is the Branch cut for the log(z) ? Correct me if I am wrong. I know that the function $f(z) = e^z$ , is periodic function with period $2 \pi$ so to define the function $\log(z)$ we have to restrict the domain of $e^z$ for example taking the points $D : z \in \mathbb{C}$ such that...
11. Solving the Branch Cut Question on f(z) = arccot(z)

Just covered branch cuts in my undergraduate course but stuck on one of the questions... Find the domain on which f(z) = arccot(z) is single valued and analytic. Now, we've looked at ln(z) in class and I understand the principal of limiting the domain but I'm not having much success and...
12. Finding g(-i) using branch cuts in ln(1-z^2)

Homework Statement Given that g(z) = ln(1-z^2), defined on \mathbb{C}\backslash \left(-\infty, 1\right], i.e. the branch cut is from -\infty to 1 along the real axis. Find g(-i) given g(i) = ln(2). Homework Equations The Attempt at a Solution I tried drawing it out but I'm having...
13. Branch Cut for (lnz)^2: Determining Branch Points and Range

Homework Statement It is simply the same as the one for lnz i.e. does it go from 0 to ∞? Also, is there any proper way to figure out branch points of a function? Homework Equations The Attempt at a Solution
14. Mellin's inversion integral, branch cut problem

I recently had to solve a problem in which i had to find the inverse laplace transform of some function with a branch cut from - ∞ to 0, so i used a contour avoiding that branch cut like this http://www.solitaryroad.com/c916/ole19.gif my problem is as follows: i know the contributions from...
15. Improper Integral With Branch Cut

I understand most of the problem, but have yet to understand where a particular term came from. The problem is as follows: Homework Statement Show that (0 to ∞)∫dx/[(x2+1)√x] = π/√2 Hint: f(z)=z−1/2/(z2+ 1) = e(−1/2) log z /(z2+ 1). The Attempt at a Solution I actually have a solutions...
16. Find the continuous branch cut of a complex logarythm

Homework Statement Find the continuous branch cut of a complex logarythm for C\[iy:y=>0] One of the complex numbers, for example, is -4i Homework Equations I don´t understand what to do with the subset. How could I find the continuous branch cut in the subset? The Attempt at a...
17. Branch cut of the principle value log

Homework Statement Find the branch points of g(z) = log(z(z+1)/(z-1)) and defining a branch of g as the principle branch of the logarithm find the location of the branch cuts. Homework Equations The Attempt at a Solution Since g(z) = log(z) + log(z+1) - log(z-1) the branch points are 0, 1...
18. What is the residue of a branch cut?

Homework Statement I'm finding the residues of the branch cut of \int^\infty_0 \frac{dx}{x^{1/4}(x^2+1)}dx Homework Equations The Attempt at a Solution I am trying to find the residue of i I am not sure how to handle lim z->i of \frac{1}{z^\frac{1}{4}(z+i)} Any nudges...
19. No Branch Cut Needed for cos(sqrt(z))

Homework Statement Branch cut for cos(sqrt(z)). Homework Equations The Attempt at a Solution Apparently there is no need for a branch cut for this function, but I am not sure why - I heard it has something to do with cos being an even function. Any clarification would...
20. Exploring the Independence of Branch Cut Placement in ln(-s^2-i\epsilon)

In the expression ln(-s^2-i\epsilon) , s^2 and \epsilon are positive (this expression can result from for example a loop diagram where s^2 is a Mandelstam variable). In mathematics, the branch cut of ln() is usually taken to be the negative real axis, so that the value above the negative axis...
21. Solving Integral: Defining Branch Cut Prior

Homework Statement Define the branch cut prior to solving the following: integrate from 0 to infinity of [log x]^4/ [1+x^2] Homework Equations The only poles inside the upper half plane is i The Attempt at a Solution How do I separate the countour?---help. Thanks.
22. Why doesn't the residue theorem work for branch cut integration?

I need help with a branch cut intgration. The problem is to show the following for 0< \alpha <1: \int_{0}^{\infty}{x^{\alpha - 1} \over x+1}={\pi \over sin\alpha\pi} I used the standard keyhole contour around the real axis (taking that as the branch cut), but using the residue theorem...
23. What is the contradiction in integrating around a contour with a branch cut?

In complex analysis we say that for fn's like lnz we apply a branch cut along positive x-axis to make sure it's single valued. i.e restrict theta s.t 0<=theta<2Pi but we never allow theta to equal 2Pi as this would make lnz take on 2nd value. Let us integrate around a contour which goes from...
24. Contour Integral with Branch Cut

Hi, I've typed up my work. Please see the attached pdf. Basically, I am trying to sovle this problem. \int_0^\infty \frac{x^\alpha}{x^2+b^2} \mathrm{d}x for 0 <\alpha < 1. I follow the procedure given in Boas pg 608 (2nd edition)...and everything seems to work. However, when I...