Questions tagged [branch-cuts]
A branch cut is curve in the complex extending from a branch point of the function.
359
questions
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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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0answers
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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1answer
22 views
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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0answers
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 ...
2
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1answer
66 views
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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0answers
19 views
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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0answers
38 views
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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0answers
29 views
contour integral with infinite branch cuts
I am trying to do the integral
$$ \int_{3/2-i\infty}^{3/2+i \infty} \frac{dy}{2\pi i}\frac{\Gamma(y/2)}{\Gamma((3-y)/2)}\sqrt{\frac{\Gamma(3-y)}{\Gamma(y)}}z^{3-y}\int_{0}^\infty dx \frac{e^{-mx}-1}{x^...
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0answers
39 views
$arctan(z)$ and Riemann Surfaces
How do I link the complex trigonometric function arctan(z) and it's branch cuts and branch points with Riemann surfaces? I have seen the picture of the function's Riemann surface but I don't ...
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1answer
54 views
Solving the Integral: $f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt$ when $L$ is odd
I want to solve the following integral when $L$ is odd:
$$
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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0answers
20 views
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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0answers
57 views
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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0answers
31 views
Fourier Transform Involving Logarithm in Denominator
In a theoretical physics paper, I came across the following Green's function
$$
G(\vec{r}, t)=\int \frac{d^4 p}{(2 \pi)^4} e^{i (p_0 t - \mathbf{p} \cdot \vec{r})} \frac{1}{p^2 \text{ln}(\frac{p}{k})},...
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0answers
13 views
branch cut of composite even function
Let us assume that is an analytic function in all plane, except some branch points. Assume that is an even analytic function in the entire plane. Is analytic function through the branch cut as well ...
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30 views
Branch cuts for log in a way to make $\sqrt{z^2-1}$ holomorphic
Show that a branch of the log $w \mapsto \sqrt(w)$ can be defined in such a way that $z\mapsto \sqrt{z^2-1}$ is holomorphic on $\mathbb{C}\backslash ((-\infty, -1)\cup (1,\infty))$
So, I know that ...
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0answers
38 views
Evaluation of $\int_{y-\delta}^{x}dz \frac{1}{z^2\sqrt{a-b(z-y)^2}}$
My problem consists in evaluating
$$\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$$
and both $y$ and $\delta$ are ...
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0answers
22 views
Show arctan(z) map one to one from a split plane to a strip.
How to show the Principle Branch $$\arctan(z)=\frac1{2i}\log\left(\frac{1+iz}{1-iz}\right)$$ maps the split plane $C$ \ $S$ where S is two slits along the imaginary axis, one from $i$ to $i\infty$ and ...
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0answers
55 views
How to find the single valued branch for a complex multi-valued function : $f(z)=\sqrt{z^4-z^3}$?
I need to show that such function $f(z)=\sqrt{z^4-z^3}$ admits a single-valued branch on $C-[0,1]$. So how should I start with this kind of question, separate it into $\sqrt{z}*\sqrt{z}*\sqrt{z}*\sqrt{...
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0answers
98 views
Need Help: Integrals Involving Multi-Valued Functions
I am trying to verify that $$ \int_0^\infty \frac{\log x}{(x^2+1)^2}dx = -\frac{\pi}{4}$$
using the above contour. I would ...
1
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0answers
41 views
Odd number of branch points possible for a multivalued complex function?
Is it possible for a multivalued complex function $f:\mathbb{C}\to\mathbb{C}$ to have an odd number of branch points?
I'm asking, as I've only seen examples of multivalued functions with an even ...
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2answers
143 views
contour integration with dogbone, branch cut
Compute the following integral
$\frac {1}{2\pi i} \int_{|z|=2} \frac{\sqrt{z^2-1}}{z-3}dz$
Taking a branch of $\sqrt{z^2-1}$, satisfying $\sqrt{z^2-1}>0$ for $z>0$
I tried this problem with a '...
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0answers
45 views
Show that $\int_0^\infty \frac{\log x}{\prod_{u=0}^n (1+\alpha_u\,x)}\,dx = …$
Prove that, by complex analysis,
$$\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).$$ The real ...
4
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1answer
82 views
Laurent expansion of square root
I have the following two part problem:
(a) Prove that $(z^2 - 1)^{-1}$ has an analytic square root in $\mathbb{C} - [-1,1]$
(b) Find the Laurent expansion of an analytic square root from part (a) on ...
2
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2answers
68 views
Branch cut of square root
I don't really understand how branch cuts work. Let's take the complex function $f(z) = \sqrt{z}$. Apparently this function is not defined for $\mathbb{R}^{-}$. But why? We defined $i$ to be $\sqrt{-1}...
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1answer
27 views
If $f(0) = 0$, $f$ injective, then there is single-valued branch of $\sqrt{f(z^2)}$
I need to show that if $f$ is analytic and injective in a neighborhood of $0$ and $f(0) = 0$, then there is a single-valued branch of $\sqrt{f(z^2)}$ in a possible smaller neighborhood of $0$.
The ...
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0answers
39 views
Inverse Fourier involving branch cut
I am looking for the closed form of the following integral involving inverse Fourier,
$$\int_{-\infty}^\infty \frac{e^{-i\tau\omega}}{(A-\cosh(2\tau))^{\frac{\nu}{2}-1}}\,\mathrm{d}\tau$$ where $\nu&...
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0answers
38 views
Choice of branch cuts for integral of form $\log[a + bz + bz^{-1}]$
I'm having some difficulty to understand the evaluation the following integral:
$$
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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1answer
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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0answers
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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0answers
13 views
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-...
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 ...
0
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2answers
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:
$$\frac{1}{z^4-2\cos(2\theta)z^2 ...
1
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1answer
69 views
What happens when integrating a function of which poles appear on the branch cut
I have a complicated function to integrate from $-\infty$ to $\infty$.
$$ 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{...
0
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0answers
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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1answer
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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0answers
36 views
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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0answers
30 views
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 $(...
1
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0answers
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,...
2
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1answer
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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0answers
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 ...
1
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1answer
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 ...
0
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0answers
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 ...
0
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0answers
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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0answers
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 ...
0
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0answers
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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0answers
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 ...
1
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0answers
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 ...
0
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0answers
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, ...
0
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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^...
0
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0answers
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\...
