Rudin's Proof about Winding Numbers












2












$begingroup$


This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $Gamma_1sim Gamma_2$ are two homotopic closed paths in a region $Omega$, and if $alphanotin Omega$, then $n(Gamma_1;alpha)=n(Gamma_2;alpha).$ I've seen several proofs of this, using approximation by polygonal paths. Rudin's is (surprise!) the slickest, but of course, he leaves some of the details to the reader, and when I do the calculation, I am off by a factor of two at a certain step, which does not affect the proof (one can scale the original hypothesis), but I must be making a mistake, and it has always bugged me. So I'd like to see where my error is.



Let $H:Itimes Ito Omega$ be the homotopy. and choose an integer $n$ such that



$|s-t|+|s'-t'|<1/nRightarrow$



$ |H(s)-H(t)|+|H(s')-H(t')|<epsilon. $



Define the paths ${gamma_0,cdots ,gamma_n}$ by



$gamma_k(s)=H(i/k,k/n)(ns+1-i)+H((i-1)/n,k/n)(i-ns)$



if $i-1le nsle i.$



The claim is then that $|gamma_k(s)-H(s,k/n)|<epsilon.$



Here is what I am getting, after substituting and applying the triangle inequality:



$|H(i/n,k/n)-H((i-1)/n,k/n)|(ns-i)+|H(i/n,k/n)-H(s,k/n)|$



which is easily seen to be $<2epsilon.$ It seems like the only way to avoid the factor of two, would be to arrive at a tractable expression without using the triangle inequality. But I do not see how to do this. Unless at the outset, we should have simply required that



$|s-t|+|s'-t'|<1/nRightarrow$



$|H(s)-H(t)|+|H(s')-H(t')|<epsilon/2. $










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$endgroup$

















    2












    $begingroup$


    This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $Gamma_1sim Gamma_2$ are two homotopic closed paths in a region $Omega$, and if $alphanotin Omega$, then $n(Gamma_1;alpha)=n(Gamma_2;alpha).$ I've seen several proofs of this, using approximation by polygonal paths. Rudin's is (surprise!) the slickest, but of course, he leaves some of the details to the reader, and when I do the calculation, I am off by a factor of two at a certain step, which does not affect the proof (one can scale the original hypothesis), but I must be making a mistake, and it has always bugged me. So I'd like to see where my error is.



    Let $H:Itimes Ito Omega$ be the homotopy. and choose an integer $n$ such that



    $|s-t|+|s'-t'|<1/nRightarrow$



    $ |H(s)-H(t)|+|H(s')-H(t')|<epsilon. $



    Define the paths ${gamma_0,cdots ,gamma_n}$ by



    $gamma_k(s)=H(i/k,k/n)(ns+1-i)+H((i-1)/n,k/n)(i-ns)$



    if $i-1le nsle i.$



    The claim is then that $|gamma_k(s)-H(s,k/n)|<epsilon.$



    Here is what I am getting, after substituting and applying the triangle inequality:



    $|H(i/n,k/n)-H((i-1)/n,k/n)|(ns-i)+|H(i/n,k/n)-H(s,k/n)|$



    which is easily seen to be $<2epsilon.$ It seems like the only way to avoid the factor of two, would be to arrive at a tractable expression without using the triangle inequality. But I do not see how to do this. Unless at the outset, we should have simply required that



    $|s-t|+|s'-t'|<1/nRightarrow$



    $|H(s)-H(t)|+|H(s')-H(t')|<epsilon/2. $










    share|cite|improve this question











    $endgroup$















      2












      2








      2


      2



      $begingroup$


      This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $Gamma_1sim Gamma_2$ are two homotopic closed paths in a region $Omega$, and if $alphanotin Omega$, then $n(Gamma_1;alpha)=n(Gamma_2;alpha).$ I've seen several proofs of this, using approximation by polygonal paths. Rudin's is (surprise!) the slickest, but of course, he leaves some of the details to the reader, and when I do the calculation, I am off by a factor of two at a certain step, which does not affect the proof (one can scale the original hypothesis), but I must be making a mistake, and it has always bugged me. So I'd like to see where my error is.



      Let $H:Itimes Ito Omega$ be the homotopy. and choose an integer $n$ such that



      $|s-t|+|s'-t'|<1/nRightarrow$



      $ |H(s)-H(t)|+|H(s')-H(t')|<epsilon. $



      Define the paths ${gamma_0,cdots ,gamma_n}$ by



      $gamma_k(s)=H(i/k,k/n)(ns+1-i)+H((i-1)/n,k/n)(i-ns)$



      if $i-1le nsle i.$



      The claim is then that $|gamma_k(s)-H(s,k/n)|<epsilon.$



      Here is what I am getting, after substituting and applying the triangle inequality:



      $|H(i/n,k/n)-H((i-1)/n,k/n)|(ns-i)+|H(i/n,k/n)-H(s,k/n)|$



      which is easily seen to be $<2epsilon.$ It seems like the only way to avoid the factor of two, would be to arrive at a tractable expression without using the triangle inequality. But I do not see how to do this. Unless at the outset, we should have simply required that



      $|s-t|+|s'-t'|<1/nRightarrow$



      $|H(s)-H(t)|+|H(s')-H(t')|<epsilon/2. $










      share|cite|improve this question











      $endgroup$




      This is kind of a softball question, an untied loose end that has always bugged me. It is well-known that if $Gamma_1sim Gamma_2$ are two homotopic closed paths in a region $Omega$, and if $alphanotin Omega$, then $n(Gamma_1;alpha)=n(Gamma_2;alpha).$ I've seen several proofs of this, using approximation by polygonal paths. Rudin's is (surprise!) the slickest, but of course, he leaves some of the details to the reader, and when I do the calculation, I am off by a factor of two at a certain step, which does not affect the proof (one can scale the original hypothesis), but I must be making a mistake, and it has always bugged me. So I'd like to see where my error is.



      Let $H:Itimes Ito Omega$ be the homotopy. and choose an integer $n$ such that



      $|s-t|+|s'-t'|<1/nRightarrow$



      $ |H(s)-H(t)|+|H(s')-H(t')|<epsilon. $



      Define the paths ${gamma_0,cdots ,gamma_n}$ by



      $gamma_k(s)=H(i/k,k/n)(ns+1-i)+H((i-1)/n,k/n)(i-ns)$



      if $i-1le nsle i.$



      The claim is then that $|gamma_k(s)-H(s,k/n)|<epsilon.$



      Here is what I am getting, after substituting and applying the triangle inequality:



      $|H(i/n,k/n)-H((i-1)/n,k/n)|(ns-i)+|H(i/n,k/n)-H(s,k/n)|$



      which is easily seen to be $<2epsilon.$ It seems like the only way to avoid the factor of two, would be to arrive at a tractable expression without using the triangle inequality. But I do not see how to do this. Unless at the outset, we should have simply required that



      $|s-t|+|s'-t'|<1/nRightarrow$



      $|H(s)-H(t)|+|H(s')-H(t')|<epsilon/2. $







      complex-analysis analysis analytic-geometry winding-number






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      share|cite|improve this question













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      edited Dec 2 '18 at 5:11







      Matematleta

















      asked Dec 1 '18 at 16:56









      MatematletaMatematleta

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