Possible values of winding numbers
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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 $0$:
$$N:={ninmathbb{Z}mid textrm{there is a closed, piecewise smooth curve }gamma textrm{ in } D textrm{ such that } n(gamma ,0)=n}.$$
(Here winding number $n(gamma ,0)$ of a closed, piecewise smooth ($C^1)$ curve $gamma:[a,b]rightarrow mathbb{C}$ is defined by $n(gamma,0):=int_gamma frac{1}{z}dz.$)
I was thinking about what the set $N$ look like in general. First I noticed that $N$ must be the set of the form $kmathbb{Z}$, where $k$ is a nonnegative integer. After some drawing, I started to suspect that $N$ is either ${0}$ or $mathbb{Z}.$ However, I can neither prove nor disprove it. Is it true that $N={0}$ or $mathbb{Z}?$ If so, why?
Any help is appreciated. Thanks in advance!
complex-analysis complex-integration winding-number
asked Dec 4 at 0:43
user544921
206
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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 $0$:
$$N:={ninmathbb{Z}mid textrm{there is a closed, piecewise smooth curve }gamma textrm{ in } D textrm{ such that } n(gamma ,0)=n}.$$
(Here winding number $n(gamma ,0)$ of a closed, piecewise smooth ($C^1)$ curve $gamma:[a,b]rightarrow mathbb{C}$ is defined by $n(gamma,0):=int_gamma frac{1}{z}dz.$)
I was thinking about what the set $N$ look like in general. First I noticed that $N$ must be the set of the form $kmathbb{Z}$, where $k$ is a nonnegative integer. After some drawing, I started to suspect that $N$ is either ${0}$ or $mathbb{Z}.$ However, I can neither prove nor disprove it. Is it true that $N={0}$ or $mathbb{Z}?$ If so, why?
Any help is appreciated. Thanks in advance!
complex-analysis complex-integration winding-number
asked Dec 4 at 0:43
user544921
206
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 $0$:
$$N:={ninmathbb{Z}mid textrm{there is a closed, piecewise smooth curve }gamma textrm{ in } D textrm{ such that } n(gamma ,0)=n}.$$
(Here winding number $n(gamma ,0)$ of a closed, piecewise smooth ($C^1)$ curve $gamma:[a,b]rightarrow mathbb{C}$ is defined by $n(gamma,0):=int_gamma frac{1}{z}dz.$)
I was thinking about what the set $N$ look like in general. First I noticed that $N$ must be the set of the form $kmathbb{Z}$, where $k$ is a nonnegative integer. After some drawing, I started to suspect that $N$ is either ${0}$ or $mathbb{Z}.$ However, I can neither prove nor disprove it. Is it true that $N={0}$ or $mathbb{Z}?$ If so, why?
Any help is appreciated. Thanks in advance!
complex-analysis complex-integration winding-number
asked Dec 4 at 0:43
user544921
206
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 $0$:
$$N:={ninmathbb{Z}mid textrm{there is a closed, piecewise smooth curve }gamma textrm{ in } D textrm{ such that } n(gamma ,0)=n}.$$
(Here winding number $n(gamma ,0)$ of a closed, piecewise smooth ($C^1)$ curve $gamma:[a,b]rightarrow mathbb{C}$ is defined by $n(gamma,0):=int_gamma frac{1}{z}dz.$)
I was thinking about what the set $N$ look like in general. First I noticed that $N$ must be the set of the form $kmathbb{Z}$, where $k$ is a nonnegative integer. After some drawing, I started to suspect that $N$ is either ${0}$ or $mathbb{Z}.$ However, I can neither prove nor disprove it. Is it true that $N={0}$ or $mathbb{Z}?$ If so, why?
Any help is appreciated. Thanks in advance!
complex-analysis complex-integration winding-number
complex-analysis complex-integration winding-number
asked Dec 4 at 0:43
user544921
206
asked Dec 4 at 0:43
user544921
206
asked Dec 4 at 0:43
user544921
206
asked Dec 4 at 0:43
user544921
206
asked Dec 4 at 0:43
user544921
206
206
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add a comment |
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