Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
241 views

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 $\gamma$, ...
0
votes
0answers
18 views

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 ...
1
vote
0answers
21 views

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 ...
1
vote
1answer
21 views

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. ...
0
votes
1answer
42 views

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 ...
5
votes
1answer
218 views

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 [...
1
vote
1answer
63 views

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 ...
3
votes
2answers
112 views

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 ...
0
votes
0answers
45 views

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 ...
1
vote
1answer
74 views

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 $\rho(0)=\rho(1)$. ...
2
votes
1answer
61 views

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 ...
0
votes
0answers
15 views

Alternate Representation for the Winding Number

Let $c: I \rightarrow \mathbb{R}^2$ be a closed curve which is parameterized by arc length. Further, let $P$ be the period of $c$ and $Q \not\in \mathtt{trace}(c)$. The winding number of $c$ w.r.t. $...
3
votes
3answers
89 views

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 ...
1
vote
1answer
57 views

Generalization of winding number for surfaces of the form $\mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$

Idea I have an idea, that it is possible to generalize the winding number for surfaces of the form $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$ The winding number for $n=1$ is $w_{\gamma}(x) = \...
0
votes
0answers
14 views

Possible values of winding numbers

Suppose that $D$ is a plane domain. Let $N$ be the set of all integers $n$ such that there is a closed, piecewise smooth curve $\gamma$ in $D$ whose winding number about the origin equals $0$: $$N:=\{...
0
votes
0answers
26 views

winding number of a triangle

I'm reading Complex Function Theory by Palka. Given a closed, piecewise smooth curve $\gamma:[a,b] \to \mathbb{C}$, its winding number about $z_0 \in \mathbb{C}$ (which doesn't intersect the curve) is ...
3
votes
0answers
74 views

Rudin's Proof about Winding Numbers

This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $\Gamma_1\sim \Gamma_2$ are two homotopic closed paths in a region $\Omega$, and if $\alpha\...
2
votes
0answers
44 views

How to calculate the index number for a curve around a linear system's fixed point without integrals?

We know $$ \phi = \tan^{-1} \frac{\dot{y}}{\dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral $$ \frac{1}{2\pi} \oint_C \frac{\dot{y}\ddot{x} - \dot{x}\...
0
votes
0answers
21 views

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 ...
0
votes
0answers
11 views

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 ...
1
vote
0answers
27 views

When is the winding number undefined

Let $\phi_R(t)=R(\cos 2t + i\sin 2t)$ be the closed circle of radius $R \geq 0$ going twice around the origin. Consider the closed curve $P_R(t)=p(\phi_R(t))$, where $p(z)$ is the polynomial $z^3+z^2-...
0
votes
0answers
30 views

Topology associated to winding number

Let $X$ be the space of continuous functions $\mathbb{S}^1\to\mathbb{C}\setminus\{0\}$. We define a map $w:X\to\mathbb{Z}$ as the winding number, or degree of such a map: $w(f) := \frac{1}{2\pi i}\...
0
votes
0answers
65 views

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 ?
1
vote
2answers
60 views

Am I correct in my understanding of the importance of homotopy in my proof?

I want to prove that given that $\gamma_0,\gamma_1$ are closed rectifiable curves in a region G with $\gamma_0 \sim \gamma_1\ $(i.e. homotopic) in G and $w \in \Bbb C- \{G\}$. Then $n(\gamma_0,w)=n(\...
0
votes
2answers
57 views

Is this an appropriate proof to show that $n(\gamma,a)=-n(-\gamma,a)$

Given the closed rectifiable curve $\gamma:[0,1]\rightarrow \Bbb C$. we define $-\gamma(t)=\gamma(1-t)$. I want to prove that $n(\gamma,a)=-n(-\gamma,a).$, where n is the winding number. Is the ...
1
vote
0answers
45 views

Curl of a vector field with respect to a closed curve

In the paper by Fedotov and Dragomir the formula to calculate the Index of a vector field $\mathbf{F} = (f(x,y), g(x,y))^T$ is given as $$ I \equiv \frac{1}{2 \pi} \oint_C \frac{f(x,y) \, \mathrm{d} g(...
1
vote
1answer
34 views

Showing existence of a root for a holomorphic function near $\cos z$

I have a holomorphic function $f(z)$ which is "near" $\cos z$ in the sense that as $z\to\pm i\infty$, $f(z)$ is dominated by the exponential behavior of $\cos z$. For example $f(z)=\cos z+a$ or $f(z)=\...
1
vote
1answer
46 views

Clarification regarding Winding Number Theorem

In the course notes for my class, we have an example of calculating a complex integral about a path. The specific example uses the Winding number but I am having some difficulty understanding. For ...
1
vote
1answer
75 views

Direction of path depends on sign of determinant of Jacobian

Let $f: K_1(0) \rightarrow \mathbb{R}^2$ be continuously differentiable, $\{z_1,..,z_m\}=f^{-1}(a)$ with a regular $a \in \mathbb{R}^2$. We choose $\epsilon$ small enough so that $f\vert_{\overline{U}...
2
votes
0answers
124 views

connection between winding number and topological degree

I'm writing a thesis on the topological degree (mapping or Brower degree), and I'm having trouble with the equation, that links the mapping degree to the winding number in $\mathbb{R}^2$(marked below ...
1
vote
1answer
52 views

Winding number of a radius dependent function?

The question: Let $\gamma:[0,2\pi] \rightarrow \mathbb{C}, t \mapsto e^{int} + re^{imt}$, with $n,m \in \mathbb{Z}$ and $0<r \neq 1$. Calculate the winding number. My attempt: My first try was to ...
0
votes
0answers
141 views

Defining the winding number on a sphere

This is for “point-in-polygon” testing on the sphere. I’ve defined a spherical polygon as a list of points on the sphere, coupled with whether the corresponding great circle arcs on the sphere are ...
1
vote
1answer
205 views

Showing Equality of Winding Numbers

Let $ w \in \Bbb C $, and let $ \gamma, \delta : [0,1] \rightarrow \Bbb C $ be closed curves such that for all $ t \in [0,1], |\gamma(t) - \delta(t)| < |\gamma(t) - w| $. By computing the winding ...
0
votes
0answers
38 views

Show $\int_{\Gamma} \frac{f'(w)dw}{f(w)}\in 2\pi i \mathbb{Z}$ for every cicle $\Gamma$ in $D$. [duplicate]

Let $D$ be a domain, $c:[0,1]\rightarrow D$ a continuous curve and $f\in \mathcal{O}(D)$ has no zero of a function. Show $$f(c(1))=f(c(0))exp\int_{c} \frac{f'(w)dw}{f(w)}$$ and conclude $\int_{\...
4
votes
1answer
103 views

Existence of complex branch for real exponents

I've recently encountered a problem regarding complex branches that made me feel like there is something fundamental about branches I do not understand: The problem Let $G$ be an open subset of $\...
0
votes
1answer
53 views

Winding number of two closed courves in C with the constraint ∀t:|c1(t)−c2(t)|≠|c1(t)|+|c2(t)|.

Show that the curves $c_1,c_2$ are running in $\mathbb C^{\times}$ with $win(c_1,0)=win(c_2,0)$. Hint: How does the curve $c=\frac{c_1}{c_2}$ look like? First I tried to find out how $c_1,c_2$ ...
1
vote
1answer
57 views

Hanging picture on the wall with two nails

There's picture which we have to hang on the wall with two nails. The two ends of a string are attached to both of the upper edges of the picture. a) How do we have to hang the picture if it falls (...
4
votes
0answers
61 views

Consider homotopy of closed curves. Show equality of winding numbers.

a) Let $H: [0,K] \times [0,1] \rightarrow \mathbb{R}^2 $ be a homotopy of closed curves, so $H$ is continuous and for every $\sigma \in [0,1]$ it holds that $ c_{\sigma}:[0,K] \rightarrow \mathbb{R}...
1
vote
2answers
84 views

Identifying the winding number of $C$ around points in the regions shown

I am trying to understand the concept of the winding number of a curve. As the title suggests, I would like to figure out the winding number of C around points in the regions 1, 2, 3 and 4 (It ...
0
votes
1answer
733 views

Total curvature/Turning Number/Winding Number of simple curve

I am totally confused by the various notions and formulas for Total curvature/Turning Number/Winding Number of plane curves. I did some calculations for the curve $\alpha(\phi)=(r(\phi),\phi)$ in ...
1
vote
1answer
130 views

Proof about sum of indices of singular points

In Lee's Topological Manifolds, problem 8-10 discusses vector fields in the plane, and defines them as continuous maps $v:\mathbb{R}^2\rightarrow\mathbb{R}^2$. An isolated singular point is a point $p$...
1
vote
0answers
208 views

Residue Theorem: Inside vs. Outside

My lecturer told me that when I use the residue theorem, it does not matter if I sum over the residues of the poles on the inside or the outside of a positively oriented simple closed curve $\gamma$. ...
0
votes
2answers
81 views

how to prove this homotopic problem

For maps $f,g: S^1\rightarrow S^1$, show that $f \circ g$ is always homtopic $f \circ g$ my friends asked me , i have no idea to solve it. could anyone help me?
1
vote
2answers
44 views

how to prove the image of a loop is the entire space

Show that there exists a loop $\gamma:[0,1]\rightarrow S^2$ such that the image of $\gamma$ is the entire $S^2$ i cannot prove it in a exactly correctly way, could anyone help me?
1
vote
0answers
115 views

Winding number in physics and mathematics

I'm learning about winding numbers in the subject of topological insulators and I'm having problem proving something that will sort of, connect the winding number as it is defined in physics with how ...
5
votes
1answer
147 views

How to show that the integral $\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z - a}$ is integer-valued when the curve $\gamma$ is not piecewise smooth?

In Conway's Functions of One Complex Variable, there is a proposition which is as follows: 5.1 Proposition. If $\gamma\colon [0,1] \to \mathbb{C}$ is a closed rectifiable curve and $a \notin \{\...
2
votes
1answer
242 views

Calculate the winding number of a triangle

My question is about calculating the winding number of a triangle, i.e. $$ Ind_\gamma(z) = \frac{1}{2i\pi} \int_\gamma\frac{1}{w-z}dw, $$ where $\gamma$ is the path given by the border of a triangle $...
2
votes
1answer
52 views

$f(x)=\frac{x-x_0}{||x-x_0||}$ is a zero degree map

Let $S\subset\mathbb{R}^n$ be a compact, connected, embedded hypersurface. For $x_0\notin S$, define: \begin{align*} f_{x_0}:S &\to \mathbb{S}^{n-1}\subset \mathbb{R}^n\\ x&\mapsto \frac{x-...
0
votes
1answer
52 views

Show winding number is non-zero over conformal equivalence

Suppose $D = \{ z: r< |z| < 1\}$ and $D' = \{z: s < |z| < 1\}$. Then let $f:D\to D'$ be a conformal bijection. Now suppose $\gamma$ is a closed piecewise smooth path in $D$ with winding ...
0
votes
1answer
78 views

Why does the integral over the closed contour $\gamma$ be zero?

I am going through the property (ii) of winding number from L.V.Ahlfors' book where I have failed to understand the reasoning "and if $\gamma$ does not meet the segment we must have $$\int_{\gamma} \...