holomorphy domain












0












$begingroup$


could someone please help me with this exercise?



Let $$f_n=frac{logleft(z-frac{1}{n}right)}{log^frac{3}{2}(n)left(z^2+nright)}$$
Where $log(z)$ is the natural branch of the function logarithm.

Find the holomorphy domain of $f(z)=sumlimits_{n=2}^{infty} f_n$.

I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $f$ implies the holomorphy of $f$ but i can't find a majoration.

I also would be happy to recive other approaches to this kind of exercises.






complex-analysis







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    could someone please help me with this exercise?



    Let $$f_n=frac{logleft(z-frac{1}{n}right)}{log^frac{3}{2}(n)left(z^2+nright)}$$
    Where $log(z)$ is the natural branch of the function logarithm.

    Find the holomorphy domain of $f(z)=sumlimits_{n=2}^{infty} f_n$.

    I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $f$ implies the holomorphy of $f$ but i can't find a majoration.

    I also would be happy to recive other approaches to this kind of exercises.






    complex-analysis







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      could someone please help me with this exercise?



      Let $$f_n=frac{logleft(z-frac{1}{n}right)}{log^frac{3}{2}(n)left(z^2+nright)}$$
      Where $log(z)$ is the natural branch of the function logarithm.

      Find the holomorphy domain of $f(z)=sumlimits_{n=2}^{infty} f_n$.

      I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $f$ implies the holomorphy of $f$ but i can't find a majoration.

      I also would be happy to recive other approaches to this kind of exercises.






      complex-analysis







      share|cite|improve this question











      $endgroup$




      could someone please help me with this exercise?



      Let $$f_n=frac{logleft(z-frac{1}{n}right)}{log^frac{3}{2}(n)left(z^2+nright)}$$
      Where $log(z)$ is the natural branch of the function logarithm.

      Find the holomorphy domain of $f(z)=sumlimits_{n=2}^{infty} f_n$.

      I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $f$ implies the holomorphy of $f$ but i can't find a majoration.

      I also would be happy to recive other approaches to this kind of exercises.






      complex-analysis




      complex-analysis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 7 at 21:26









      Fabio

      1079




      1079










      asked Jan 7 at 20:06









      Elia OrsiElia Orsi

      13




      13






















          1 Answer
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          $begingroup$

          Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $mathbb Csetminus E_n.$ The set where $sum f_n$ makes sense can be no larger than $mathbb Csetminus (cup ,E_n).$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
            $endgroup$
            – Elia Orsi
            Jan 7 at 22:10










          • $begingroup$
            I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
            $endgroup$
            – zhw.
            Jan 7 at 23:26













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          active

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          0












          $begingroup$

          Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $mathbb Csetminus E_n.$ The set where $sum f_n$ makes sense can be no larger than $mathbb Csetminus (cup ,E_n).$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
            $endgroup$
            – Elia Orsi
            Jan 7 at 22:10










          • $begingroup$
            I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
            $endgroup$
            – zhw.
            Jan 7 at 23:26


















          0












          $begingroup$

          Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $mathbb Csetminus E_n.$ The set where $sum f_n$ makes sense can be no larger than $mathbb Csetminus (cup ,E_n).$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
            $endgroup$
            – Elia Orsi
            Jan 7 at 22:10










          • $begingroup$
            I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
            $endgroup$
            – zhw.
            Jan 7 at 23:26
















          0












          0








          0





          $begingroup$

          Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $mathbb Csetminus E_n.$ The set where $sum f_n$ makes sense can be no larger than $mathbb Csetminus (cup ,E_n).$






          share|cite|improve this answer









          $endgroup$



          Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $mathbb Csetminus E_n.$ The set where $sum f_n$ makes sense can be no larger than $mathbb Csetminus (cup ,E_n).$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 7 at 21:07









          zhw.zhw.

          72.4k43175




          72.4k43175












          • $begingroup$
            Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
            $endgroup$
            – Elia Orsi
            Jan 7 at 22:10










          • $begingroup$
            I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
            $endgroup$
            – zhw.
            Jan 7 at 23:26




















          • $begingroup$
            Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
            $endgroup$
            – Elia Orsi
            Jan 7 at 22:10










          • $begingroup$
            I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
            $endgroup$
            – zhw.
            Jan 7 at 23:26


















          $begingroup$
          Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
          $endgroup$
          – Elia Orsi
          Jan 7 at 22:10




          $begingroup$
          Thank you, $E_n$ should be {$zin mathbb{C}:zneq isqrt{n},$ $zneq-isqrt{n}$ } $cup$ $ ${$zin mathbb{C}: Re(z)leq frac{1}{n}$ $and $ $ Img(z)=0$}.
          $endgroup$
          – Elia Orsi
          Jan 7 at 22:10












          $begingroup$
          I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
          $endgroup$
          – zhw.
          Jan 7 at 23:26






          $begingroup$
          I would write $E_n = {isqrt n,-isqrt n}cup (-infty,1/n].$
          $endgroup$
          – zhw.
          Jan 7 at 23:26




















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