Questions tagged [branch-cuts]

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

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$\log(z_1z_2) = \log(z_1) +\log(z_2) + i2n\pi $ where log is the analytic branch of complex log.

In a problem provided in my course (the assignment was due a while ago, this isn't a homework help request), we needed to show : $$\log(z_1z_2) = \log(z_1) + \log(z_2) + i2n\pi $$ I seem to understand ...
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27 views

Limit of Complex logarithm towards branch cut

The textbook asks me to verify whether or not this statement is true: The limit of $Log(-1 + \frac{i}{n}) + Log(-1 - \frac{i}{n})$ as $n \rightarrow \infty$ is $0$, given that $Log(z)$ is the main ...
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Can a function have a branch cut along the real and also imaginary axis?

Is it possible for a complex function to have a branch cut along the real axis and also the imaginary axis, that cross over like a + sign?
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35 views

Traveling Around a Complex Number

In functions with complex arguments it's been said that to determine whether a point is a branch point or not we have to travel around that point and see if we get the same answer. How do I travel ...
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Branch cuts for $\ln (z^2-1)$

Consider the function $$f(z) = \ln (z^2-1),$$ which as we know is a multi-valued function. Now doing $$f(z)=\ln[(z-1)(z+1)]=\ln(r_1r_2 e^{i(\theta_1+\theta_2)}),$$ where I let $z-1=r_1e^{\theta_1}$, $...
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What is the analytic structure of this function wih a branch cut and a pole?

Consider the function $$ f(z) = \frac{1}{1+\sqrt{z}} $$ over the complex plane. Clearly it has a branch cut running along from $0$ to $\infty$ along some direction. Moreover, it seems to have a pole ...
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Interpretation of branch cut

In wikipedia, I found that a branch cut can be thought of as a sum of infinitely many poles arranged along a line in the complex plane with infinitesimal residues. And wiki takes $ln(\frac{z+1}{z-1})$ ...
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contour integral with infinite branch cuts

I am trying to do the integral $$ \int_{3/2-i\infty}^{3/2+i \infty} \frac{dy}{2\pi i}\frac{\Gamma(y/2)}{\Gamma((3-y)/2)}\sqrt{\frac{\Gamma(3-y)}{\Gamma(y)}}z^{3-y}\int_{0}^\infty dx \frac{e^{-mx}-1}{x^...
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39 views

$arctan(z)$ and Riemann Surfaces

How do I link the complex trigonometric function arctan(z) and it's branch cuts and branch points with Riemann surfaces? I have seen the picture of the function's Riemann surface but I don't ...
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Solving the Integral: $f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt$ when $L$ is odd

I want to solve the following integral when $L$ is odd: $$ f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt $$ which can be simplified to: $$ f(...
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How to realize ILT by residue theorem?

$$F(s)=\frac{e^{-x \sqrt{s}/\sqrt{D}} (e^{2C \sqrt{s}/\sqrt{D}} + e^{2x \sqrt{s}/\sqrt{D}})}{s^{3/2} (e^{2c \sqrt{s}/\sqrt{D}} -1)}$$ By the residue theorem. I tried, but didn't realize. The ...
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How can I solve this branch cut problem?

The integration is $$\int_{c-i\infty}^{c+i\infty} dz\,\frac{e^{a\sqrt{z+m^2}+bz}}{\sqrt{z+m^2}-m}$$ I understand there is a branch cut at $z = -m^2$. But a little confused about the behavior at $z = 0$...
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Fourier Transform Involving Logarithm in Denominator

In a theoretical physics paper, I came across the following Green's function $$ G(\vec{r}, t)=\int \frac{d^4 p}{(2 \pi)^4} e^{i (p_0 t - \mathbf{p} \cdot \vec{r})} \frac{1}{p^2 \text{ln}(\frac{p}{k})},...
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branch cut of composite even function

Let us assume that is an analytic function in all plane, except some branch points. Assume that is an even analytic function in the entire plane. Is analytic function through the branch cut as well ...
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Branch cuts for log in a way to make $\sqrt{z^2-1}$ holomorphic

Show that a branch of the log $w \mapsto \sqrt(w)$ can be defined in such a way that $z\mapsto \sqrt{z^2-1}$ is holomorphic on $\mathbb{C}\backslash ((-\infty, -1)\cup (1,\infty))$ So, I know that ...
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38 views

Evaluation of $\int_{y-\delta}^{x}dz \frac{1}{z^2\sqrt{a-b(z-y)^2}}$

My problem consists in evaluating $$\int_{y-\delta}^{x}dz \frac{1}{z^2\sqrt{a-b(z-y)^2}}$$ where $a,b>0$ and $y-\delta$ is one of the roots of $$a-b(z-y)^2=0$$ and both $y$ and $\delta$ are ...
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Show arctan(z) map one to one from a split plane to a strip.

How to show the Principle Branch $$\arctan(z)=\frac1{2i}\log\left(\frac{1+iz}{1-iz}\right)$$ maps the split plane $C$ \ $S$ where S is two slits along the imaginary axis, one from $i$ to $i\infty$ and ...
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How to find the single valued branch for a complex multi-valued function : $f(z)=\sqrt{z^4-z^3}$?

I need to show that such function $f(z)=\sqrt{z^4-z^3}$ admits a single-valued branch on $C-[0,1]$. So how should I start with this kind of question, separate it into $\sqrt{z}*\sqrt{z}*\sqrt{z}*\sqrt{...
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Need Help: Integrals Involving Multi-Valued Functions

I am trying to verify that $$ \int_0^\infty \frac{\log x}{(x^2+1)^2}dx = -\frac{\pi}{4}$$                                                                              using the above contour. I would ...
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Odd number of branch points possible for a multivalued complex function?

Is it possible for a multivalued complex function $f:\mathbb{C}\to\mathbb{C}$ to have an odd number of branch points? I'm asking, as I've only seen examples of multivalued functions with an even ...
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143 views

contour integration with dogbone, branch cut

Compute the following integral $\frac {1}{2\pi i} \int_{|z|=2} \frac{\sqrt{z^2-1}}{z-3}dz$ Taking a branch of $\sqrt{z^2-1}$, satisfying $\sqrt{z^2-1}>0$ for $z>0$ I tried this problem with a '...
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Show that $\int_0^\infty \frac{\log x}{\prod_{u=0}^n (1+\alpha_u\,x)}\,dx = …$

Prove that, by complex analysis, $$\int_0^\infty \frac{\log x}{\prod_{u=0}^n (1+\alpha_u\,x)}\,dx = \frac{1}{n! \,2^{n+1}}\sum_{m=0}^n (-1)^{m+1}\binom{n}{m}\alpha_m^{n-1}\log^2(\alpha_m).$$ The real ...
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82 views

Laurent expansion of square root

I have the following two part problem: (a) Prove that $(z^2 - 1)^{-1}$ has an analytic square root in $\mathbb{C} - [-1,1]$ (b) Find the Laurent expansion of an analytic square root from part (a) on ...
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Branch cut of square root

I don't really understand how branch cuts work. Let's take the complex function $f(z) = \sqrt{z}$. Apparently this function is not defined for $\mathbb{R}^{-}$. But why? We defined $i$ to be $\sqrt{-1}...
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If $f(0) = 0$, $f$ injective, then there is single-valued branch of $\sqrt{f(z^2)}$

I need to show that if $f$ is analytic and injective in a neighborhood of $0$ and $f(0) = 0$, then there is a single-valued branch of $\sqrt{f(z^2)}$ in a possible smaller neighborhood of $0$. The ...
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39 views

Inverse Fourier involving branch cut

I am looking for the closed form of the following integral involving inverse Fourier, $$\int_{-\infty}^\infty \frac{e^{-i\tau\omega}}{(A-\cosh(2\tau))^{\frac{\nu}{2}-1}}\,\mathrm{d}\tau$$ where $\nu&...
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38 views

Choice of branch cuts for integral of form $\log[a + bz + bz^{-1}]$

I'm having some difficulty to understand the evaluation the following integral: $$ I = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{d} \vartheta_2 \ln \left[ \underbrace{\cosh(2 \beta E_1) \cosh(2 \beta E_2) ...
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82 views

Let $a\in\mathbb{C}, |a|=1$ and $c$ an irrational real number. Prove: $a^c$ is dense in the unit circle.

Let $a\in\mathbb{C}$ a complex number such that $|a|=1$ and $c$ an irrational real number. Prove: The set $a^c$ is dense in the unit circle. The problem is taken from Notes on Complex Function ...
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23 views

Simplification of an integral involving complex square root

Consider the integral $$f(a) = \text{Im} \int_0^a \text{d}z \frac{1-z/a}{z \sqrt{1-z}}$$ The integral only picks up a non zero contribution for $a > 1$ since the square root function $\sqrt{1-z}$ ...
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Discontinuity of generalized hypergeometric function

The discontinuity of hypergeometric function ${}_2F_1(a_1,a_2;b_1;z)$ is $ \frac{2\pi i \Gamma(b_1)z^{1-b_1}(z-1)^{b_1-a_1-a_2}}{\Gamma(a_1)\Gamma(a_2)\Gamma(b_1-a_1-a_2+1)}{}_2F_1(1-a_1,1-a_2;b_1-...
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1answer
56 views

How to evaluate this contour integral with branches?

I want to evaluate this integral below using contour integral. $$\int_{-1}^1\frac{dx}{\sqrt{1-x^2}(1+x^2)}$$ I know this can be done transforming x to sin or cos, but I want to solve this by ...
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100 views

Evaluating the integral $\int^{\infty}_{-\infty} \frac{dx}{x^4-2\cos(2\theta)x^2 +1}$

The first part of this question required me to find out the zeroes of the denominator, and to treat the equation as that of a complex number, which allows us to write: $$\frac{1}{z^4-2\cos(2\theta)z^2 ...
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1answer
69 views

What happens when 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{...
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27 views

Contour integration with fractional powers in denominator

I wish to integrate the following $$\int_0^{\infty} \frac{\sin(k r)}{ r^{1/2}(r-2m)^{1/2}} dr$$ I believe contour integration is the best way to proceed. I see that there are branch points at $0$ and $...
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60 views

Different branchs of $w^{1/3}$

Find a branch $g(w)$ of $w^{1/3}$ which serves as the inverse function $f(z)=z^3$ with $\text{dom}(f)=\{z\in\mathbb{C}^\times\mid \text{Arg}(z)\in[0,2\pi/3)\}$. Relate $g(w)$ to the principle branch $\...
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Integral with two branch cuts

I need help calculating the following integral $$\int_0^\infty\!\mathrm{d}u\frac{u^{-s}(u+i\omega_n)^s}{u-w+i\omega_n+i\omega_m}$$ where $0<s<1$. What I have thought so far is to do contour ...
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Branch points but no branch cut. Is it possible?

I've stumbled upon a certain problem regarding the function $f(z)=log((z-2i)(z+3i))$ Going for the long path and taking the real and imaginary parts of $g(z)=(z-2i)(z+3i)$, then checking what points $(...
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28 views

Weierstrass-type factorization (reconstruction) of a function with branch cuts

Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,...
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155 views

Branch cut of $\sqrt{z^2-1}$.

I was reading something that defined the function $f(z)=\sqrt{z^2-1}$ on $\mathbb{C}\setminus [-1,1]$ where the branch cut is such that the argument of $z$ and $\sqrt{z^2-1}$ are in the same quadrant. ...
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51 views

Branch Cut/s for $f(z)=\operatorname{Log}(\frac{1}{z(z-1)})$

I've been trying to arrive at a branch cut that works for this function. Firstly, I find the branch points which are, if I'm not mistaken, $z_1=0$ and $z_2=1$. I then tried encircling them to see ...
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62 views

Integrating a function with two branch point $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $

I was studying a book about residue intigrating then I get into trouble solving this : $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $ I know that I have to choose a contour like this: but I don't ...
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39 views

Branch of incomplete gamma function

The incomplete gamma function is defined as follows: $$ \Gamma(s,z)=\int_z^{\infty}t^s e^{-t}\frac{dt}{t}, \qquad Re(s)>0, \quad z\in \mathbb C $$ Since $t^s$ is a multi-valued function, this ...
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27 views

Integral involving a Multi-Valued function

I don't really have any idea about how does one calculate integrals involving Multi-valued functions, all I know about Multi - valued functions, are the closely associated concepts of Branch Points ...
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28 views

How should I chose the branch of a multivalued intagrand?

I need to calculate the integral of $$ \oint_C \frac{log(z^2-1)}{z^2+1}$$ Where $C:2x^2+3y^2=1 $ The singularity of the denominator lies outside the contour. Consequently, I thought Cauchy ...
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37 views

How to calculate the contour integration $\int_{2}^{\infty} \dfrac{1}{x \sqrt{x^{2} - 4}}dx$?

How could I calculate the integration $\int_{2}^{\infty} \dfrac{1}{x \sqrt{x^{2} - 4}}dx$ using the following contour? I tried to use $f(z) = \int_{2}^{\infty} \dfrac{\log(z)}{z \sqrt{z^{2} - 4}}dz$ ...
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34 views

Why $\log(1/z) = -\log(z)$ is true for $- \pi < \arg(z) < \pi $ and it's false for $0 < \arg(z) < 2 \pi$?

I don't underestand why this argument is false: We take $$z = re^{i\theta}.$$ Then $$\log(z^{-1}) = \log(r^{-1}e^{-i\theta}) = -\log(r) - i\theta = -\log(re^{i\theta}) = -\log(z).$$ And why it ...
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25 views

Heterodox asymptotics for the square root at the origin

(TLDR: Need an asymptotic expansion for the square root which works at the origin, specifically. The classical Taylor series won't do: i need it to be exact at zero, outside of which it can have ...
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38 views

Branch Points and Cuts

I am having a lot of problems understanding what exactly branch points are and how they are computed for a function. There is this one problem that I just can't seem to get around to understanding, ...
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1answer
58 views

Branch cuts of complex logarithm for Friedel oscillation

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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63 views

Finding Branch Cuts and Branch Points of $f(z)=\left(\frac{z-a}{z-b}\right)^\frac{1}{2}$

I want to find the branch points and branch cut structure of $f(z)=(\frac{z-a}{z-b})^{1/2}$, but I'm stuck on the process. I tried to rewrite it in bipolar coordinates first using $z-a = r_1e^{i\...

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