# Questions tagged [winding-number]

For questions about winding numbers. The winding number of a continuous curve counts how many times it "loops" around a given point.

114 questions
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### How to calculate the winding number?

I've been given the following loop $\gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $\gamma$, ...
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### Winding number and ramification index

It is known that if $f:X\rightarrow Y$ is a non-constant holomorphic map of Riemann surfaces, then there is an integer $k$ such that locally around each $x\in X$, we can choose coordinate functions ...
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### Winding number in 4D & SU(2) group

In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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### Checking homotopy of curves

Consider two curves defined as $f:S^1 \rightarrow \mathbb{R}^2-\left\lbrace x_0\right\rbrace$ and $g:S^1 \rightarrow \mathbb{R}^2-\left\lbrace x_0\right\rbrace$. You can see them in the picture below. ...
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### Evaluate $\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_1)(z-z_2)}-\frac{f(z)}{(z-z_0)^2}$

In Marsden's Complex Analysis, section 2.4, the main theorem is Cauchy's integral formula (C.I.F) and there appears this problem: Let $f$ be analytic inside and on $\gamma: |z-z_0|=R$. Prove that ...
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### Degree of maps on the 3-sphere

I am currently in the process of going through Ticciati's Quantum Field Theory for Mathematicians, which states the following (Theorem 13.7.11): "Let $g$ be a differentiable function from $S^3$ to a [...
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### Application of winding number and the roots of complex polynomial from a non simple closed cuvre

There is a formula for the simple closed curve $\gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(\gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the ...
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### What's the calculation formula of topological number for mappings of $\pi_{3}(S^2)=\mathbb{Z}$?

It is well-known that, when mapping $|\vec{n}(\vec{x})|=1$, we can use $N=\int{\mathrm{d}x_1\mathrm{d}x_2\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})}$ to calculate the topological winding ...
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### What are Winding number?

I came across the term in Tom Apostol Calculus volume 2 (page 390) as a line integral , which seemed to be equivalent to the wikipedia definition in terms of polar coordinate. The line Integral in ...
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### winding number of paths

Let $c:[0,1]\to\mathbb{R}^2\backslash\{\mathbf{0}\}$ be a closed path with winding number $k$. Let $\tilde{c}=\rho(t)c(t)$, where $\rho:[0,1]\to(0,\infty)$ is function satisfying $\rho(0)=\rho(1)$. ...
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### If two closed plane curves are outside each other, can there be a point inside both of them?

I think this recent question (also here) has a quick answer if the conjecture below is true. It looks "obviously" true, but I've learned to distrust my judgement in such matters. It also looks as if ...
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### Give an example of a bounded domain and a piecewise $C^1$ closed curve satisfy given conditions.

Give an example of a bounded domain $\Omega \subset \mathbb {C}$ and a piecewise $C^1$ closed curve $f$ in $\Omega$ such that $I(f;z)=5$ for some $z \in \mathbb {C}/\Omega$. (Here $C^1$ means the ...
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### Creating a continuous evaluation of a phase graph

I have a function $$R(\xi) := \prod_{i=0}^{n} (e^{-i\xi} - z_k)$$ where $z_{k} \in \mathbb{C}$. The $z_{k}$ are random, but clustered vaguely around the unit circle, if that is relevant. I need to ...
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