# 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.

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### Simple Connectivity on the Complex Plane

There is a ton of equivalent definitions of a simply connected set on the plane, reading Ullrich's Complex Made Simple I have some of these equivalences, my question is about (iii) and (iv) ...
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### Question about the logarithm function in the Argument Principle

I have found the proofs of the argument principle from Ponnusamy and Silverman's Complex variables with applications and Brown and Churchill's Complex variables. But I am not sure how to make sense of ...
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### Winding numbers in QCD and winding numbers in complex analysis. Is there a relation through a differential geometric generalization?

I have a background in theoretical physics and the first time I came across winding numbers was in the context of the vacuum of QCD. By the way physicists treat this topic, I thought it had little to ...
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### Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms?

The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through ...
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### Two non closed paths homotopic relative to the endpoints have the same winding number?

Let $\gamma:[0,1]\rightarrow {C}$ be a path. The winding number $\omega(\gamma)$ is defined as $\omega(\gamma)=\frac{\theta(1)-\theta(0)}{2\pi}$. I have read that two paths homotopic to the endpoints ...
1 vote
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### Let $f(z)=\exp(z)\cos(z)$ prove $f(z)=a$ has infinite solutions outside any circunference.

I've been trying to solve this problem for a couple of days but i can´t find a convincent solution. The problem says: Let $f(z)=\exp(z)\cos(z)$. Prove for all $a \in \mathbb{C}$ except maybe one only ...
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• 2,108
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### How do I compute the local degree of the following map?

I have the following porblem. Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear mapping, represented by a $2\times 2$ matrix. Moreover let the determinant be nonzero. Show that the local ...
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### How do I show that the winding number of this closed continuous path is zero?

I have the following problem: Show that if $\gamma:[a,b]\rightarrow U$ is a closed continuous path, $p\in \mathbb{R}^2$ and $U\subset \mathbb{R}^2\setminus \{p\}$ is an open subset on which there is ...
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### If $|\phi_1| <| \phi_2|$ and $\phi = \phi_1 + \phi_2$ then $\mathrm {Ind} _{\phi} (0) = \mathrm {Ind} _{\phi_2} (0)$

Let $\phi1, \phi2: [0,1] \to \mathbb {C}$ be Paths, such that $|\phi_1 (t) | <| \phi_2 (t) |$ for all $t \in [0,1]$. If $0 \leq t \leq 1$, we set $\phi (t) = \phi_1 (t) + \phi_2 (t)$. Prove ...
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### Small perturbations of a loop in $\mathbb C$ does not change its winding number

From Visual Complex Fuctions by Elias Wegert, Lemma 2.7.19, it says that Let $\gamma_0 : [0,1] \to \mathbb C - \{0\}$ be a loop, i.e. a continous function with same endpoints. Denote by $d$ the ...
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### How can I prove that this defintion of a winding number is valid?

I have read a definition of a winding number on wikipedia and it involves finding the continuous polar parametrization of the curve, but then the question arises, why does such a parametriation always ...
• 2,104
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### Does injective continuous / holomorphic functions preserve winding number?

I am trying to find if the winding number is invariant to some transformations. I already know that it is invariant respect to translation, rotation and scaling. Also, it is not invariant respect to ...
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### Can I compute contour orientation without using polygon area sign?

Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
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### How do I prove this statement about winding numbers and continuous maps?

I have the following problem: Let $D$ be a disk with boundary circle $C$ and let $f:D\rightarrow \mathbb{R}^2$ be a continuous map. Suppose $P\in \mathbb{R}^2\setminus f(C)$ and the winding number of ...
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