# 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.

115 questions
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### Winding number of outside point [closed]

If $\Gamma$ is a contour completely contained in the smallest possible G (open subset of the complex plane), then is the winding number of $\Gamma$ about x = $0$ for all x in C\G? If not, could you ...
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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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### 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 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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### Clarification over Ahlfors page 116, 2.1 about winding numbers

Everything on this question is in complex plane. As the book describes a property of a winding number, it says that: Outside of the [line segment from $a$ to $b$] the function $(z-a) / (z-b)$ is ...
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### when are the winding number of homotopic curves not equal?

For what cases do we have it that even though two closed rectifiable curves are homotopic, that they do not share the same winding number ?
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