Complicated Cauchy complex integral.












0














I have the function $f(z)$ that defined on the closed disc $ bar D_3(0)$
by the integeral on the boundry $C_3(0)$:



$f(z) = int_{C_3(0)} frac{3w^2+7w+1}{w-z}dw $



I need to find $f'(1+i)$



Caushy integral theorem and it`s conclusions tells me that on $C_3(0)$ boundry, my function is:



$f(w) = 2 pi i(3w^2+7w+1)$
and that the derivative of $f(z)$ is:



$f'(z)=int_{ C_3(0)}frac {(3w^2+7w+1)}{(w-z)^2}dw$
we use $gamma:[0,1] to Bbb C $ defined $ gamma (t)=3e^{2pi it}$ to be our path,
so we get:



$f'(1+i)=int_{0}^{1}frac {6 pi ie^{2pi i t}(3e^{4pi it}+7e^{2pi it}+1)}{(3e^{2pi it}-(1+i))^2}dt$ as integral over a path is defined.



Is there a way to pass over this complicated integral? or there is an algebric trick with whom I`ll solve this integral?, or maybe somehow to choose another parametrization to my path?










share|cite|improve this question





























    0














    I have the function $f(z)$ that defined on the closed disc $ bar D_3(0)$
    by the integeral on the boundry $C_3(0)$:



    $f(z) = int_{C_3(0)} frac{3w^2+7w+1}{w-z}dw $



    I need to find $f'(1+i)$



    Caushy integral theorem and it`s conclusions tells me that on $C_3(0)$ boundry, my function is:



    $f(w) = 2 pi i(3w^2+7w+1)$
    and that the derivative of $f(z)$ is:



    $f'(z)=int_{ C_3(0)}frac {(3w^2+7w+1)}{(w-z)^2}dw$
    we use $gamma:[0,1] to Bbb C $ defined $ gamma (t)=3e^{2pi it}$ to be our path,
    so we get:



    $f'(1+i)=int_{0}^{1}frac {6 pi ie^{2pi i t}(3e^{4pi it}+7e^{2pi it}+1)}{(3e^{2pi it}-(1+i))^2}dt$ as integral over a path is defined.



    Is there a way to pass over this complicated integral? or there is an algebric trick with whom I`ll solve this integral?, or maybe somehow to choose another parametrization to my path?










    share|cite|improve this question



























      0












      0








      0







      I have the function $f(z)$ that defined on the closed disc $ bar D_3(0)$
      by the integeral on the boundry $C_3(0)$:



      $f(z) = int_{C_3(0)} frac{3w^2+7w+1}{w-z}dw $



      I need to find $f'(1+i)$



      Caushy integral theorem and it`s conclusions tells me that on $C_3(0)$ boundry, my function is:



      $f(w) = 2 pi i(3w^2+7w+1)$
      and that the derivative of $f(z)$ is:



      $f'(z)=int_{ C_3(0)}frac {(3w^2+7w+1)}{(w-z)^2}dw$
      we use $gamma:[0,1] to Bbb C $ defined $ gamma (t)=3e^{2pi it}$ to be our path,
      so we get:



      $f'(1+i)=int_{0}^{1}frac {6 pi ie^{2pi i t}(3e^{4pi it}+7e^{2pi it}+1)}{(3e^{2pi it}-(1+i))^2}dt$ as integral over a path is defined.



      Is there a way to pass over this complicated integral? or there is an algebric trick with whom I`ll solve this integral?, or maybe somehow to choose another parametrization to my path?










      share|cite|improve this question















      I have the function $f(z)$ that defined on the closed disc $ bar D_3(0)$
      by the integeral on the boundry $C_3(0)$:



      $f(z) = int_{C_3(0)} frac{3w^2+7w+1}{w-z}dw $



      I need to find $f'(1+i)$



      Caushy integral theorem and it`s conclusions tells me that on $C_3(0)$ boundry, my function is:



      $f(w) = 2 pi i(3w^2+7w+1)$
      and that the derivative of $f(z)$ is:



      $f'(z)=int_{ C_3(0)}frac {(3w^2+7w+1)}{(w-z)^2}dw$
      we use $gamma:[0,1] to Bbb C $ defined $ gamma (t)=3e^{2pi it}$ to be our path,
      so we get:



      $f'(1+i)=int_{0}^{1}frac {6 pi ie^{2pi i t}(3e^{4pi it}+7e^{2pi it}+1)}{(3e^{2pi it}-(1+i))^2}dt$ as integral over a path is defined.



      Is there a way to pass over this complicated integral? or there is an algebric trick with whom I`ll solve this integral?, or maybe somehow to choose another parametrization to my path?







      complex-analysis cauchy-integral-formula






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      edited Nov 20 at 18:01









      Praneet Srivastava

      762516




      762516










      asked Nov 20 at 17:51









      Daniel Vainshtein

      18811




      18811






















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          You stated that $f(w) = 2pi i (3w^2 + 7w + 1)$. Isn't it immediate that $f'(w) = 2 pi i (6w + 7)$?






          share|cite|improve this answer





















          • but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
            – Daniel Vainshtein
            Nov 20 at 18:23













          Your Answer





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          1 Answer
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          0














          You stated that $f(w) = 2pi i (3w^2 + 7w + 1)$. Isn't it immediate that $f'(w) = 2 pi i (6w + 7)$?






          share|cite|improve this answer





















          • but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
            – Daniel Vainshtein
            Nov 20 at 18:23


















          0














          You stated that $f(w) = 2pi i (3w^2 + 7w + 1)$. Isn't it immediate that $f'(w) = 2 pi i (6w + 7)$?






          share|cite|improve this answer





















          • but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
            – Daniel Vainshtein
            Nov 20 at 18:23
















          0












          0








          0






          You stated that $f(w) = 2pi i (3w^2 + 7w + 1)$. Isn't it immediate that $f'(w) = 2 pi i (6w + 7)$?






          share|cite|improve this answer












          You stated that $f(w) = 2pi i (3w^2 + 7w + 1)$. Isn't it immediate that $f'(w) = 2 pi i (6w + 7)$?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 20 at 18:07









          Umberto P.

          38.5k13064




          38.5k13064












          • but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
            – Daniel Vainshtein
            Nov 20 at 18:23




















          • but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
            – Daniel Vainshtein
            Nov 20 at 18:23


















          but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
          – Daniel Vainshtein
          Nov 20 at 18:23






          but this is only relevant on the boundry of the disc $D_3(0)$ when definitely 1+i is on the interior of $D_3(0)$
          – Daniel Vainshtein
          Nov 20 at 18:23




















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