Questions tagged [branch-cuts]
A branch cut is curve in the complex extending from a branch point of the function.
535 questions
2
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answers
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Contour integral with four branch points on the unit circle
I want to compute the contour integral
$$
\oint_{|z|=2} z \sqrt{z^4-1}\text{d}z,
$$
where the path is positively oriented (it is the blue one below).
It is non-zero thanks to the four branch-points $\...
- 31
1
vote
1
answer
109
views
Problem in choosing value of phases when evaluating a complex integral with branch cut
I'm trying to evaluate the following integral using the residue theorem:
$$ \int_0^1 \frac{\sqrt{1-x^2}}{x^2 - 2} dx $$
First of all, we need to identify the singularities of the function, once ...
- 492
2
votes
1
answer
56
views
Rigorous relation between pointing angle $\theta$ and arc length $s$ in total curvature in $\mathbb R^2$
I want to prove that
$$\kappa(s)=\frac{d\theta}{ds}\tag{1}$$
to verify that
$$\int_{s_1}^{s_2}\kappa(s)\,ds=\int_{\theta_1}^{\theta_2}=\theta_2-\theta_1.$$
(Motivation for this is my previous question,...
- 860
0
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0
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44
views
Argument of the theta function
I am looking at the Jacobi Theta function $\theta_1(z)$ with $\tau\in i\mathbb R$
$$\theta_1(z+1)=-\theta_1(z)\quad \theta_(z+\tau)=-e^{-2\pi iz}e^{-\pi i \tau}\theta_1(z)$$
I take the principal ...
- 41
1
vote
1
answer
75
views
Value of a function when approaching a branch cut from different directions
I'm trying to understand how to compute real integrals using residue theorem when dealing with complex multi-valued functions. For example. Let's consider the following real function, extended to the ...
- 492
0
votes
0
answers
42
views
Confusion regarding choosing multiple $\arg(z-z_0)$ restrictions for functions with several branch cuts
I thought of a very simple example to explain my confusion.
Let's say I want to describe the function $y=\sqrt{z+1}\sqrt{z-2}$. I want to make sure it is single valued, so I take $-\pi < \arg(z+1)&...
- 109
0
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0
answers
34
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Help in *methodical* understanding of the full Riemann surface and all possible local monodromies of $y(x)=\sqrt{x\sqrt{5} +\sqrt{2-x^2}}$
I think I understand the structure of this Riemann surface, but a) I'd like to be sure, and b) I am trying to rephrase it in a more systematic way, which will in principle allow me to solve any ...
- 109
1
vote
0
answers
93
views
Tracking the 'phase' of complex contour integrals
Consider the integral $$I = \int_{-1}^{1}{(x+1)^{\frac{1}{3}}(1-x)^{\frac{2}{3}} dx}$$
A common method of evaluating the integral is to consider $f(z) = {(z+1)^{\frac{1}{3}}(z-1)^{\frac{2}{3}}}$ with ...
- 351
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views
When is $\sqrt{(z-a)(z-b)} = \sqrt{z-a}\sqrt{z-b}$ true?
This seems a quite easy concept, but I feel like I am missing something. The function $f(z) = \sqrt{(z-a)(z-b)}$, with $a, b \in \mathbb{C}$, is a multi-valued function, and as such, it's true that
$$
...
- 31
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0
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62
views
Is there a specific way to name the branches of a multivalued complex function?
A question of vocabulary I am not sure about.
Let's take the complex function (with values in $\mathbb{R} \in \mathbb{C}$ in that case...):
$$z \mapsto \arg(z)$$
Let's take $z$ and $\theta$ ...
- 713
3
votes
1
answer
214
views
On the behaviour of a function around branch cuts
I'm a physics student. I'm currently taking a complex analysis course and I'm struggling to understand how the determination of a function is given on its branch cut. Since it is hard for me to even ...
- 492
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1
answer
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Apparent paradox from deforming a Hankel contour on the Riemann sphere
An integral representation of the modified Bessel function is
$$I_\nu(t) = \frac1{2\pi i} \oint_C \exp(t (z+1/z))\frac{\mathrm dz}{z^{1+\nu}}$$
where the contour $C$ starts in $\infty-i\epsilon$ and ...
- 81
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0
answers
44
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Determining Discontinuity Across Branch Cuts for $\phi^3$ propagator
I am considering the $\phi^3$ theory in $d = 3$ dimensions. The propagator is given by the following formula
$$
G(p^2) = \frac{-i}{p^2 + m_0^2 + M^2(p^2) - i\epsilon}
$$
where
$$
M^2(p^2) = -\frac{g_0^...
- 952
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Complicated contour integral with branch cut
Let's consider the function
\begin{equation}
f(x)
=
\frac{1}{x - iy_0}
\frac{1}{x - iy_1}
\left(
\frac{ia - x}{ia + x}
\right)^{b - i\frac{c}{x}}
.
\end{equation}
Here, $\{a,c,y_0,y_1\}\in\mathbb{R}$ ...
- 145
1
vote
1
answer
85
views
Is there some lift of the complex plane on which these contours are different?
Consider the following contour in the complex plane
$$\Gamma_m = \{e^{(2m+1)\pi i} r \in \mathbb{C} : r>0\},\quad m\in \mathbb{Z}$$
For any value of $m$ this is exactly the same locus in $\mathbb{C}...
- 253