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2
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0answers
37 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 ...
1
vote
1answer
53 views

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 ...
0
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0answers
18 views

Resolving branch cut away from branch point

The standard example $\sqrt{z}$ where z is complex has a branch cut which we take to be the negative real axis. Plotting a circle around zero the branch cut is resolved by a Riemann sheet. This boils ...
2
votes
1answer
30 views

$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ 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 ...
2
votes
1answer
20 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?
0
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0answers
30 views

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 ...
4
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0answers
70 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 ...
1
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0answers
51 views

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 $a$ ...
1
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0answers
27 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, ...
0
votes
1answer
28 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 ...
1
vote
1answer
35 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 ...
0
votes
2answers
115 views

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 ...
0
votes
2answers
37 views

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 ...
0
votes
3answers
61 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 ...
0
votes
1answer
36 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 $O(R^{-2})$? $$\int f(z) dz = \int \frac{1}{z(1-z^2)^{\frac{1}{2}}} dz $$
0
votes
1answer
36 views

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 ...
1
vote
1answer
21 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
2
votes
1answer
89 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 ...
0
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0answers
18 views

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, ...
0
votes
0answers
45 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 ...
0
votes
1answer
152 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? ...
0
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0answers
23 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
votes
1answer
165 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 ...
0
votes
0answers
23 views

How to deal with the poles on the branch cut

(1) I have a function $F(s)=\dfrac{1}{ (s+p/2)\sqrt{s(s+2p)} }$. The branch cut runs from $s=0$ to $s=-2p$. My questions is if I can consider $s=-p/2$ as a pole. If somebody can provide me with some ...
0
votes
1answer
93 views

Branch points and Branch cut of $f(z) = \log(z^2)$

I am trying to find the branch points and choose a branch cut for the function $f(z) = \log(z^2)$. I know that both $z = 0$ and $z = \infty$ are branch points, so it seems reasonable to just choose ...
0
votes
1answer
59 views

Finding Branch Values

I have a function $f(z) = (\frac{8}{7}z^3-\frac{64}{7})^\frac{1}{3}$ which I have found to have branch points at $z_1 = 2$, $z_2 = 2\exp(\frac{2i\pi}{3})$, and $z_3 = 2\exp(\frac{-2i\pi}{3})$. The ...
2
votes
0answers
30 views

Complex integral with branch cuts.

The problem is the following, $$ \ \int_{-\infty}^{\infty} du e^{-iu w }\bigg( \cos (\theta u) - i \frac{\xi }{\theta} \sin (\theta u)\bigg)^{-1/2} $$ When we go to complex $u$ plane there are branch ...
5
votes
4answers
324 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
3
votes
0answers
42 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
0
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0answers
30 views

Suitable branch for Arg

I have the following function: $ \ f(z)=\log(z^3-2) $ and I am asked to find a branch where $f(z) $ is defined. I understand that since $ f(0)=\log(-2)$ the main branch of Arg is not defined and we ...
2
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0answers
215 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
1
vote
2answers
70 views

Defining a branch of $(1-\zeta^2)^{-1/2}$

In this question I brought up a passage from Stein/Shakarchi's Complex Analysis page 232: ...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z \frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the ...
2
votes
3answers
92 views

power series expansion of $z^a$ at $z = 1$

I'm working through some problems in a complex analysis book, and one asks to compute the power series expansion of $f(z) = z^a$ at $z = 1$, where $a$ is a positive real number. The series should ...
0
votes
2answers
106 views

Choose the branch for $(1-\zeta^2)^{1/2}$ that makes it holomorphic in the upper half-plane and positive when $-1<\zeta<1$

From Stein/Shakarchi's Complex Analysis page 232: ...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z \frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the integral is taken from $0$ to $z$ along ...
5
votes
2answers
125 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
3
votes
1answer
115 views

Branch cut problem

I am looking at the text by G. K. Batchelor, An Introduction to Fluid Dynamics, pg. 428-9. I am looking at the inverse mapping of $z = \zeta + \frac{\lambda^2}{\zeta}$ given by ...
6
votes
1answer
96 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
1
vote
1answer
42 views

Holomorphic problem

I have a function $f(z)$ holomorphic in $\mathbb{C}\setminus\mathbb{R}^-$. I have these information: $f(x+i\epsilon) = f(x-i\epsilon)$ on $\mathbb{R}^+$ (the $\epsilon$ is indented as a shorthand ...
0
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0answers
148 views

Difficult Fourier transform

While looking at non-local modifications to wave propagation in 2d I have run into the following integral $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega dk \ln(k^2-\omega^2)e^{-i\omega ...
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0answers
51 views

Why is the Chern class of a line bundle well defined?

Let $M$ be a compact Riemann surface with a finite open covering $$ M = \bigcup_{i=1}^{n} U_i $$ which has the property that every intersection is contractible (i.e. it is a good cover). To each two ...
0
votes
1answer
136 views

Branch cut for $\log (iz)$ in the region $\{z:\mathrm{Im}(z)>0\}$

If someone could explain branch cuts and branch points to me that would be fantastic. I understand that a branch cut is a curve that we remove from the domain to make a function (usually a logarithm) ...
0
votes
1answer
95 views

Branch cut of $w=\left(\dfrac{z+1}{z-1}\right)^{1/3}$

I'd like todetermine a branch cut for the function $w=\left(\dfrac{z+1}{z-1}\right)^{1/3}$ that allows to construct analytic branches defined on $|z|>1 \;,\; \forall z\in \mathbb C$. How can I do ...
1
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0answers
115 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
0
votes
0answers
81 views

Branch cuts and multidimensional residues

When dealing with a multidimensional contour integral, how do you treat multivalued functions with branch cuts? In simple domains with singe valued functions it's not too hard to calculate these ...
0
votes
2answers
1k views

Finding branch points and branch cuts of arctan

I am studying complex analysis and I do not yet fully understand branch points and branch cuts. I am trying to figure out how it works by looking at the following: $z \rightarrow \frac{1}{2i} ...
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0answers
25 views

possible results of integral along closed path after defining branch of sqrt

Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ ...
0
votes
1answer
67 views

edge disjoin Cut Set

prove that a graph G=(V,E) where | v | =n there are at most n-1 edge disjoint cut sets. I was thinking that for tree it is true since each edge is cut set. but i have no idea how to prove above ...
0
votes
1answer
105 views

handwaving substitution in integral involving branch cut and derivative of sqrt + generalization

Want to compute $$ I = \int_0^i \mathrm{d}z \frac{z}{\sqrt{z^2-1}}$$ on the complex plane using complex methods. QUESTION: is the result $i \left( \sqrt{2}-1 \right)$ which one gets imposing ...
1
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0answers
270 views

Branch Cut for $\ln(1 - z^2)$

"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 $g(i) = \ln(2)$" I ...
1
vote
2answers
112 views

Is discontinuity along a line equivalent to branch cut?

Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this? Is it sufficient to prove that $f(z)$ is discontinuous across this line? ...