Questions tagged [branch-cuts]
A branch cut is curve in the complex extending from a branch point of the function.
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Is it possible to define a branch cut of $\log$ such that $\log(-1 + i)^{100} = 100\log(-1+i)$?
We saw a similar problem in class, but instead of raising to $100$ we just had to square it, the idea was to find a real number $\alpha$ such that if we define:
$$\log(z) = \ln|z| + i\arg(z), \alpha &...
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Is it consistent to talk about residue when evaluating indentation around simple poles lying on a branch cut?
I am trying to evaluate the following integral:
$$\mathcal{P}\int_0^\infty \frac{\sqrt{x}}{x^2-1}dx$$
using contour integration.
On the complex plane, using standard conventions, a branch cut arises ...
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Can a branch's domain include the branch cut
The Logarithmic Function chapter of Complex analysis by jcponce has the following definition of a branch,
A branch of a multiple-valued function $f$ is any single-valued function $F$ that is analytic ...
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"Portals" in $\mathbb{C}$: branch cuts that translate function arguments
First some background and motivation.
I recently came across a YouTube video where the creator tried to calculate the effects of portals on the gravitational field (although electrostatics should work ...
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On the branch selection for evaluating a Gaussian integral with complex exponent
Consider the following integral:
$$I=\int_0^\infty dx\,e^{-x^2\frac{1+j}{\sqrt{2}}}.$$
where j is the imaginary unit.
We get:
$$I^2=\int_0^\infty \int_0^\infty dx dy e^{-(x^2+y^2)\frac{1+j}{\sqrt{2}}}....
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Analytic continuation with branch cut along the whole real axis
Similar questions have been asked 1,2. Not sure I should ask my question below their.
I am also in a situation where I have such a function
$$
F(u)=\int_0^{\infty} \frac{f(k) d^3k}{k-u}+\int_0^{\infty}...
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Continuous logarithm along closed path exists if and only if winding number along path is zero
Let $f(z)$ be a meromorphic function on $\mathbb{C}$. Denote by $S$ the set of its zero's and poles. Let $\gamma$ be a (sufficiently smooth) closed curve in $\mathbb{C} \setminus S$. Is the following ...
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Contour integral with four branch points on the unit circle
I want to compute the contour integral
$$
\oint_{|z|=2} z \sqrt{z^4-1}\text{d}z,
$$
where the path is positively oriented (it is the blue one below).
It is non-zero thanks to the four branch-points $\...
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Problem in choosing value of phases when evaluating a complex integral with branch cut
I'm trying to evaluate the following integral using the residue theorem:
$$ \int_0^1 \frac{\sqrt{1-x^2}}{x^2 - 2} dx $$
First of all, we need to identify the singularities of the function, once ...
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Rigorous relation between pointing angle $\theta$ and arc length $s$ in total curvature in $\mathbb R^2$
I want to prove that
$$\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.$$
(Motivation for this is my previous question,...
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Argument of the theta function
I am looking at the Jacobi Theta function $\theta_1(z)$ with $\tau\in i\mathbb R$
$$\theta_1(z+1)=-\theta_1(z)\quad \theta_(z+\tau)=-e^{-2\pi iz}e^{-\pi i \tau}\theta_1(z)$$
I take the principal ...
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Value of a function when approaching a branch cut from different directions
I'm trying to understand how to compute real integrals using residue theorem when dealing with complex multi-valued functions. For example. Let's consider the following real function, extended to the ...
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Confusion regarding choosing multiple $\arg(z-z_0)$ restrictions for functions with several branch cuts
I thought of a very simple example to explain my confusion.
Let's say I want to describe the function $y=\sqrt{z+1}\sqrt{z-2}$. I want to make sure it is single valued, so I take $-\pi < \arg(z+1)&...
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Help in *methodical* understanding of the full Riemann surface and all possible local monodromies of $y(x)=\sqrt{x\sqrt{5} +\sqrt{2-x^2}}$
I think I understand the structure of this Riemann surface, but a) I'd like to be sure, and b) I am trying to rephrase it in a more systematic way, which will in principle allow me to solve any ...
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Tracking the 'phase' of complex contour integrals
Consider the integral $$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 ...
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