Questions tagged [branch-points]
A branch point is a point in the complex that can map from a single point to multiple points in the range.
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Branch Points and Branch Line of the Function $f(z) = \cos^{-1}z$
I know the branch line and branch points of common function like $\log{z}$ and aware of the definitions. But what's the general approach here to find branch line and branch points of any multivalued ...
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Integral involving a simple pole and 4 branch points, pairwise in-and-out of the contour
For some time lately, I am facing difficulty in evaluating the following integral coming from a physics context:
$$I = \frac{1}{2 \pi}\int^{\pi}_{-\pi} \frac{h-\cos k}{\sqrt{(h-\cos k)^2+\gamma^2 \sin^...
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Complex Integral and Residue involving multiple Branch Points inside the contour
I encountered the integral:
$$\oint_{|z|=1} \frac{f(z) \ dz}{\sqrt{(z-a)(z-b)}} \ \ \ \ \text{with} \ \ \ |a|,|b| < 1$$
So that the branch points are inside the contour. I am not adding the ...
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Branch points of $\sqrt{z-1}$
Let, $f(z)=\sqrt{z-1}$
I am reading the book "Visual complex analysis" by Tristan Needham and after reading the section of branch points and branch cuts, I gave myself this definition of a ...
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totally ramified covering of $\mathbb{P}^1$ with 3 branch points.
I'm trying to construct a totally ramified covering (of order $d$) of the complex projective line $\mathbb{P}^1$ with exactly 3 branch points $0,1,\infty$.
Here is what I'm trying: consider the smooth ...
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Question About Existence Of Branch For $(z^2 -1)^{ \frac12}$.
Consider the function $f :z\mapsto (z^2 -1)^{
\frac12}$.
Now Here proved that it has branch such that $f$ is analytic in $|z|>1$.Where branch cut is $[-1,1]$.
Now f is composition of two function ...
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Rigorous proof that $\sqrt{z}$ has a branch point at zero.
Here is the definitions I am working with
Define the map $$ (z-1)^{\frac{1}{2}}$$ defined as the inverse of $$\begin{align} f: \mathbb{C} & \rightarrow \mathbb{C} \\ z &\mapsto z^{2}+1 \end{...
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When working with multiple branch cuts, is there a way to chose the arguments so that log of a product can be opened as sum of individual logarithms?
Suppose a function $\eta (z)=log(\psi (z))$ where $$\psi (z)=\prod_{k=1}^{n} \left(z-z_k\right)$$ We know that $log(z)=log|z|+i(argz)$, this implies that $$log(\prod_{k=1}^{n} \left(z-z_k\right))=\...
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An improper integral of an inverse of a square root of a higher degree polynomial.
Let $1 \le n_1 < n $ and $ n\ge 3$ be integers and let ${\bf \lambda}= \left( \lambda_j \right)_{j=1}^n \in {\mathbb R}$ such that $\lambda_j > 0 $ for $j=1,\cdots, n_1$ and $\lambda_j <0 $...
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What is the branch cut of composite of multivalued complex function
I have the following function where I want to identify the Riemann surface.
$$
f(z)=\log\left(\sqrt{z^2+1}\right). \quad\quad\quad (1)
$$
The square root function has a Riemann surface $R_{SR}$ with ...
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Radius of convergence at $2+i$ of $f_i(z)=\frac{1}{\phi_i-2^{1/4}}$ with $2^{1/4}$ being the positive real root
Let $\phi_k(z), k=0,1,2,3$ the branch cuts of $z^{1/4}$. Consider
$$f_k(z)=\frac{1}{\phi_k-2^{1/4}}, \quad 2^{1/4}=|2|^{1/4}e^{i0}>0$$
Find the radius of convergence of the series expansion at $2+i$...
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Branch cut in integral function
I'm not very versed in complex analysis and I'm trying to understand some concepts on branch cuts and contour integration. Consider a function
$$
I(s)=\int_0^1 d\alpha\ \frac{1}{f(s,\alpha)},
$$
such ...
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Riemann surface of the hypergeometric function
The hypergeometric function $_2F_1(a,b,c,z)$ has a branch cut extending from $z=1$ to $z=\infty$. Does this define an infinite-sheeted Riemann surface (like that for $\log{z}$) or one with a finite ...
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Branch cut of $\log(z^2-1)$ if $\arg(z^2-1)\in\left(-\frac{7\pi}{4}, \frac{\pi}{4}\right]$
Using the principal branch $\arg(\xi(z))\in\left(-\dfrac{7\pi}{4}, \dfrac{\pi}{4}\right]$ for
$$f(z)=\log(z^2-1)=\log(\xi(z)),$$
what's its branch cut's equation/how does it look like?
I was doing the ...
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Branch cuts of $\log(z^2-1)$ for unusual principal values
Say we got a multivalued function
$$\log(z^2-1)\equiv\log(\xi(z)).$$
Usually, we would choose principal values such as $\arg\xi(z)\in[0,2\pi)$ or $\arg\xi(z)\in[-\pi,\pi)$. In order to find the branch ...
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