Prove simple closed curves $f$'s exist, so $Gamma = C-sum_{i=1}^{k}{f_i}$ satisfies $...











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Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $kin mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, dots ,f_k$ such that the cycle $Gamma = C-sum_{i=1}^{k}{f_i}$ satisfies
$$ int_{Gamma}{frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$$



Note: $C^1$ means the curve has continuous derivatives for each $t$ within the curve's interval $[a,b]$.



What would be the simplest way to prove this?










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    Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $kin mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, dots ,f_k$ such that the cycle $Gamma = C-sum_{i=1}^{k}{f_i}$ satisfies
    $$ int_{Gamma}{frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$$



    Note: $C^1$ means the curve has continuous derivatives for each $t$ within the curve's interval $[a,b]$.



    What would be the simplest way to prove this?










    share|cite|improve this question
























      up vote
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      down vote

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      up vote
      1
      down vote

      favorite











      Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $kin mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, dots ,f_k$ such that the cycle $Gamma = C-sum_{i=1}^{k}{f_i}$ satisfies
      $$ int_{Gamma}{frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$$



      Note: $C^1$ means the curve has continuous derivatives for each $t$ within the curve's interval $[a,b]$.



      What would be the simplest way to prove this?










      share|cite|improve this question













      Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $kin mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, dots ,f_k$ such that the cycle $Gamma = C-sum_{i=1}^{k}{f_i}$ satisfies
      $$ int_{Gamma}{frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$$



      Note: $C^1$ means the curve has continuous derivatives for each $t$ within the curve's interval $[a,b]$.



      What would be the simplest way to prove this?







      complex-analysis complex-integration cauchy-integral-formula simple-functions






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      asked Nov 18 at 18:49









      wtnmath

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          The simplest way to prove it is by providing an example.



          Note that the singularities of the integrand are at $a_{1,2}=pm i,a_{3,4}=(-1pm isqrt3)/2,a_5=0$.



          Thus, let $f_n=C(a_n,epsilon)$ where $1le nle 5$ and $epsilon$ being sufficiently small.



          Then, clearly, $C-sum f_n$ encloses no singularities. Therefore, by Cauchy’s integral theorem,
          $$oint_{Gamma} frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz =0$$






          share|cite|improve this answer





















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            The simplest way to prove it is by providing an example.



            Note that the singularities of the integrand are at $a_{1,2}=pm i,a_{3,4}=(-1pm isqrt3)/2,a_5=0$.



            Thus, let $f_n=C(a_n,epsilon)$ where $1le nle 5$ and $epsilon$ being sufficiently small.



            Then, clearly, $C-sum f_n$ encloses no singularities. Therefore, by Cauchy’s integral theorem,
            $$oint_{Gamma} frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz =0$$






            share|cite|improve this answer

























              up vote
              0
              down vote













              The simplest way to prove it is by providing an example.



              Note that the singularities of the integrand are at $a_{1,2}=pm i,a_{3,4}=(-1pm isqrt3)/2,a_5=0$.



              Thus, let $f_n=C(a_n,epsilon)$ where $1le nle 5$ and $epsilon$ being sufficiently small.



              Then, clearly, $C-sum f_n$ encloses no singularities. Therefore, by Cauchy’s integral theorem,
              $$oint_{Gamma} frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz =0$$






              share|cite|improve this answer























                up vote
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                up vote
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                The simplest way to prove it is by providing an example.



                Note that the singularities of the integrand are at $a_{1,2}=pm i,a_{3,4}=(-1pm isqrt3)/2,a_5=0$.



                Thus, let $f_n=C(a_n,epsilon)$ where $1le nle 5$ and $epsilon$ being sufficiently small.



                Then, clearly, $C-sum f_n$ encloses no singularities. Therefore, by Cauchy’s integral theorem,
                $$oint_{Gamma} frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz =0$$






                share|cite|improve this answer












                The simplest way to prove it is by providing an example.



                Note that the singularities of the integrand are at $a_{1,2}=pm i,a_{3,4}=(-1pm isqrt3)/2,a_5=0$.



                Thus, let $f_n=C(a_n,epsilon)$ where $1le nle 5$ and $epsilon$ being sufficiently small.



                Then, clearly, $C-sum f_n$ encloses no singularities. Therefore, by Cauchy’s integral theorem,
                $$oint_{Gamma} frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz =0$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 19 at 23:54









                Szeto

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                6,2292726






























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