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







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    0












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






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      asked Dec 4 '18 at 0:43









      user544921user544921

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