Is it possible to prove Cauchy integral formula by parametrizing the circle?












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Theorem. Let $f$ be analytic everywhere inside and on a simple closed contour $C$, taken in the positive sense. If $z_0$ is any point interior to $C$, then $$f(z_0) = frac{1}{2pi i} int_C frac{f(z)dz}{z-z_0}.$$



The proof starts with letting $C_r$ denote a positively oriented circle $|z-z_0|=r$, where $r$ is small enough that $C_r$ is interior to $C$. Since the function $f(z)/(z-z_0)$ is analytic between and on the contours $C$ and $C_r$, we have $$ int_C frac{f(z)dz}{z-z_0} = int_{C_r} frac{f(z)dz}{z-z_0}.$$



Can we then parametrize the contour $C_r$ and calculate $int_{C_r} frac{f(z)dz}{z-z_0}$? How to proceed?










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    Theorem. Let $f$ be analytic everywhere inside and on a simple closed contour $C$, taken in the positive sense. If $z_0$ is any point interior to $C$, then $$f(z_0) = frac{1}{2pi i} int_C frac{f(z)dz}{z-z_0}.$$



    The proof starts with letting $C_r$ denote a positively oriented circle $|z-z_0|=r$, where $r$ is small enough that $C_r$ is interior to $C$. Since the function $f(z)/(z-z_0)$ is analytic between and on the contours $C$ and $C_r$, we have $$ int_C frac{f(z)dz}{z-z_0} = int_{C_r} frac{f(z)dz}{z-z_0}.$$



    Can we then parametrize the contour $C_r$ and calculate $int_{C_r} frac{f(z)dz}{z-z_0}$? How to proceed?










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      $begingroup$


      Theorem. Let $f$ be analytic everywhere inside and on a simple closed contour $C$, taken in the positive sense. If $z_0$ is any point interior to $C$, then $$f(z_0) = frac{1}{2pi i} int_C frac{f(z)dz}{z-z_0}.$$



      The proof starts with letting $C_r$ denote a positively oriented circle $|z-z_0|=r$, where $r$ is small enough that $C_r$ is interior to $C$. Since the function $f(z)/(z-z_0)$ is analytic between and on the contours $C$ and $C_r$, we have $$ int_C frac{f(z)dz}{z-z_0} = int_{C_r} frac{f(z)dz}{z-z_0}.$$



      Can we then parametrize the contour $C_r$ and calculate $int_{C_r} frac{f(z)dz}{z-z_0}$? How to proceed?










      share|cite|improve this question











      $endgroup$




      Theorem. Let $f$ be analytic everywhere inside and on a simple closed contour $C$, taken in the positive sense. If $z_0$ is any point interior to $C$, then $$f(z_0) = frac{1}{2pi i} int_C frac{f(z)dz}{z-z_0}.$$



      The proof starts with letting $C_r$ denote a positively oriented circle $|z-z_0|=r$, where $r$ is small enough that $C_r$ is interior to $C$. Since the function $f(z)/(z-z_0)$ is analytic between and on the contours $C$ and $C_r$, we have $$ int_C frac{f(z)dz}{z-z_0} = int_{C_r} frac{f(z)dz}{z-z_0}.$$



      Can we then parametrize the contour $C_r$ and calculate $int_{C_r} frac{f(z)dz}{z-z_0}$? How to proceed?







      calculus complex-numbers






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      edited Dec 14 '18 at 22:35







      user398843

















      asked Dec 14 '18 at 22:28









      user398843user398843

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      675216






















          2 Answers
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          One can proceed using mean value theorem and $rto 0$:
          begin{align}
          int_{C_r}frac{f(z)}{z-z_0},dz&=int_0^{2pi}frac{f(z_0+re^{itheta})}{re^{itheta}}re^{itheta}i,dtheta=iint_0^{2pi}f(z_0+re^{itheta}),dtheta=\
          &=[text{Re},f(z_0+re^{ic_1})+itext{Im},f(z_0+re^{ic_2})]2pi ito f(z_0)2pi i.
          end{align}






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            $f(z)=sum a_n (z-z_0)^{n}$ gives $int_0^{2pi} f(z+re^{itheta}) dtheta=sum a_n int_0^{2pi}r^{n} e^{intheta} dtheta=2 pi a_0=2 pi f(z_0)$ (since all but the first term vanish). Now $int_{C_r} frac {f(z)} {z-z_0} dz =i int_0^{2pi} f(z+re^{itheta}) dtheta=2pi i f(z_0)$.






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              2 Answers
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              2 Answers
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              active

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              $begingroup$

              One can proceed using mean value theorem and $rto 0$:
              begin{align}
              int_{C_r}frac{f(z)}{z-z_0},dz&=int_0^{2pi}frac{f(z_0+re^{itheta})}{re^{itheta}}re^{itheta}i,dtheta=iint_0^{2pi}f(z_0+re^{itheta}),dtheta=\
              &=[text{Re},f(z_0+re^{ic_1})+itext{Im},f(z_0+re^{ic_2})]2pi ito f(z_0)2pi i.
              end{align}






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                One can proceed using mean value theorem and $rto 0$:
                begin{align}
                int_{C_r}frac{f(z)}{z-z_0},dz&=int_0^{2pi}frac{f(z_0+re^{itheta})}{re^{itheta}}re^{itheta}i,dtheta=iint_0^{2pi}f(z_0+re^{itheta}),dtheta=\
                &=[text{Re},f(z_0+re^{ic_1})+itext{Im},f(z_0+re^{ic_2})]2pi ito f(z_0)2pi i.
                end{align}






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  One can proceed using mean value theorem and $rto 0$:
                  begin{align}
                  int_{C_r}frac{f(z)}{z-z_0},dz&=int_0^{2pi}frac{f(z_0+re^{itheta})}{re^{itheta}}re^{itheta}i,dtheta=iint_0^{2pi}f(z_0+re^{itheta}),dtheta=\
                  &=[text{Re},f(z_0+re^{ic_1})+itext{Im},f(z_0+re^{ic_2})]2pi ito f(z_0)2pi i.
                  end{align}






                  share|cite|improve this answer











                  $endgroup$



                  One can proceed using mean value theorem and $rto 0$:
                  begin{align}
                  int_{C_r}frac{f(z)}{z-z_0},dz&=int_0^{2pi}frac{f(z_0+re^{itheta})}{re^{itheta}}re^{itheta}i,dtheta=iint_0^{2pi}f(z_0+re^{itheta}),dtheta=\
                  &=[text{Re},f(z_0+re^{ic_1})+itext{Im},f(z_0+re^{ic_2})]2pi ito f(z_0)2pi i.
                  end{align}







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 14 '18 at 23:33

























                  answered Dec 14 '18 at 23:23









                  A.Γ.A.Γ.

                  22.8k32656




                  22.8k32656























                      0












                      $begingroup$

                      $f(z)=sum a_n (z-z_0)^{n}$ gives $int_0^{2pi} f(z+re^{itheta}) dtheta=sum a_n int_0^{2pi}r^{n} e^{intheta} dtheta=2 pi a_0=2 pi f(z_0)$ (since all but the first term vanish). Now $int_{C_r} frac {f(z)} {z-z_0} dz =i int_0^{2pi} f(z+re^{itheta}) dtheta=2pi i f(z_0)$.






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        $f(z)=sum a_n (z-z_0)^{n}$ gives $int_0^{2pi} f(z+re^{itheta}) dtheta=sum a_n int_0^{2pi}r^{n} e^{intheta} dtheta=2 pi a_0=2 pi f(z_0)$ (since all but the first term vanish). Now $int_{C_r} frac {f(z)} {z-z_0} dz =i int_0^{2pi} f(z+re^{itheta}) dtheta=2pi i f(z_0)$.






                        share|cite|improve this answer









                        $endgroup$
















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                          0





                          $begingroup$

                          $f(z)=sum a_n (z-z_0)^{n}$ gives $int_0^{2pi} f(z+re^{itheta}) dtheta=sum a_n int_0^{2pi}r^{n} e^{intheta} dtheta=2 pi a_0=2 pi f(z_0)$ (since all but the first term vanish). Now $int_{C_r} frac {f(z)} {z-z_0} dz =i int_0^{2pi} f(z+re^{itheta}) dtheta=2pi i f(z_0)$.






                          share|cite|improve this answer









                          $endgroup$



                          $f(z)=sum a_n (z-z_0)^{n}$ gives $int_0^{2pi} f(z+re^{itheta}) dtheta=sum a_n int_0^{2pi}r^{n} e^{intheta} dtheta=2 pi a_0=2 pi f(z_0)$ (since all but the first term vanish). Now $int_{C_r} frac {f(z)} {z-z_0} dz =i int_0^{2pi} f(z+re^{itheta}) dtheta=2pi i f(z_0)$.







                          share|cite|improve this answer












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                          answered Dec 14 '18 at 23:23









                          Kavi Rama MurthyKavi Rama Murthy

                          65.1k42766




                          65.1k42766






























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