Analytic functions with uncountable zeros [closed]

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Suppose for an analytic function $f$ its zero set $Z_f$ is uncountable. Show that $f ≡ 0$.
Can we show that the set has a limit point? Hence the result.




complex-analysis




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closed as off-topic by Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy Aug 18 at 17:08


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • This is false without further hypotheses. Start by telling us what the domain of $f$ is.
    – zhw.
    Aug 17 at 6:05














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Suppose for an analytic function $f$ its zero set $Z_f$ is uncountable. Show that $f ≡ 0$.
Can we show that the set has a limit point? Hence the result.




complex-analysis




share|cite|improve this question












closed as off-topic by Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy Aug 18 at 17:08


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • This is false without further hypotheses. Start by telling us what the domain of $f$ is.
    – zhw.
    Aug 17 at 6:05












up vote
0
down vote

favorite









up vote
0
down vote

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Suppose for an analytic function $f$ its zero set $Z_f$ is uncountable. Show that $f ≡ 0$.
Can we show that the set has a limit point? Hence the result.




complex-analysis




share|cite|improve this question












Suppose for an analytic function $f$ its zero set $Z_f$ is uncountable. Show that $f ≡ 0$.
Can we show that the set has a limit point? Hence the result.





complex-analysis





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share|cite|improve this question




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asked Feb 20 '16 at 11:48









jaggu

736




736




closed as off-topic by Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy Aug 18 at 17:08


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy Aug 18 at 17:08


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Leucippus, Strants, Brahadeesh, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.











  • This is false without further hypotheses. Start by telling us what the domain of $f$ is.
    – zhw.
    Aug 17 at 6:05
















  • This is false without further hypotheses. Start by telling us what the domain of $f$ is.
    – zhw.
    Aug 17 at 6:05















This is false without further hypotheses. Start by telling us what the domain of $f$ is.
– zhw.
Aug 17 at 6:05




This is false without further hypotheses. Start by telling us what the domain of $f$ is.
– zhw.
Aug 17 at 6:05










3 Answers
3






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The complex plane is the union of countably many compact sets, and some of the compact sets must contain infinitely many zeros.






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    Yes every uncountable set has a limit point. More than that it has a condensation point.






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    • I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
      – Upstart
      Feb 20 '16 at 12:07











    • Can you help me in doing that?
      – jaggu
      Feb 20 '16 at 12:34










    • The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
      – Upstart
      Feb 20 '16 at 12:39

















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    0
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    The zeros of an analytic function that is not the zero function are isolated.



    This means that for each zero there is a disk that contains it and no other zero. In each such disk there is a point with rational coordinates. The rational point for each disk is different because the disks are disjoint.



    This gives an injection from the set of zeros into $mathbb Q times mathbb Q$, which is countable. Therefore, the set of zeros is countable.






    share|cite|improve this answer



























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote













      The complex plane is the union of countably many compact sets, and some of the compact sets must contain infinitely many zeros.






      share|cite|improve this answer
























        up vote
        4
        down vote













        The complex plane is the union of countably many compact sets, and some of the compact sets must contain infinitely many zeros.






        share|cite|improve this answer






















          up vote
          4
          down vote










          up vote
          4
          down vote









          The complex plane is the union of countably many compact sets, and some of the compact sets must contain infinitely many zeros.






          share|cite|improve this answer












          The complex plane is the union of countably many compact sets, and some of the compact sets must contain infinitely many zeros.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Feb 20 '16 at 11:51









          Qiyu Wen

          1,360415




          1,360415




















              up vote
              1
              down vote













              Yes every uncountable set has a limit point. More than that it has a condensation point.






              share|cite|improve this answer






















              • I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
                – Upstart
                Feb 20 '16 at 12:07











              • Can you help me in doing that?
                – jaggu
                Feb 20 '16 at 12:34










              • The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
                – Upstart
                Feb 20 '16 at 12:39














              up vote
              1
              down vote













              Yes every uncountable set has a limit point. More than that it has a condensation point.






              share|cite|improve this answer






















              • I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
                – Upstart
                Feb 20 '16 at 12:07











              • Can you help me in doing that?
                – jaggu
                Feb 20 '16 at 12:34










              • The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
                – Upstart
                Feb 20 '16 at 12:39












              up vote
              1
              down vote










              up vote
              1
              down vote









              Yes every uncountable set has a limit point. More than that it has a condensation point.






              share|cite|improve this answer














              Yes every uncountable set has a limit point. More than that it has a condensation point.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Aug 17 at 5:02









              Cornman

              2,60721128




              2,60721128










              answered Feb 20 '16 at 11:55









              Upstart

              1,680517




              1,680517











              • I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
                – Upstart
                Feb 20 '16 at 12:07











              • Can you help me in doing that?
                – jaggu
                Feb 20 '16 at 12:34










              • The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
                – Upstart
                Feb 20 '16 at 12:39
















              • I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
                – Upstart
                Feb 20 '16 at 12:07











              • Can you help me in doing that?
                – jaggu
                Feb 20 '16 at 12:34










              • The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
                – Upstart
                Feb 20 '16 at 12:39















              I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
              – Upstart
              Feb 20 '16 at 12:07





              I guess jaggu u dont have any of your own ideas. As we can see from your questions..please try a bit more than that i guess you're a nbhm aspirant and csir aspirant....
              – Upstart
              Feb 20 '16 at 12:07













              Can you help me in doing that?
              – jaggu
              Feb 20 '16 at 12:34




              Can you help me in doing that?
              – jaggu
              Feb 20 '16 at 12:34












              The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
              – Upstart
              Feb 20 '16 at 12:39




              The help,if any has to come from your end only..just try to give your thoughts on the questions maybe wrong but still..
              – Upstart
              Feb 20 '16 at 12:39










              up vote
              0
              down vote













              The zeros of an analytic function that is not the zero function are isolated.



              This means that for each zero there is a disk that contains it and no other zero. In each such disk there is a point with rational coordinates. The rational point for each disk is different because the disks are disjoint.



              This gives an injection from the set of zeros into $mathbb Q times mathbb Q$, which is countable. Therefore, the set of zeros is countable.






              share|cite|improve this answer
























                up vote
                0
                down vote













                The zeros of an analytic function that is not the zero function are isolated.



                This means that for each zero there is a disk that contains it and no other zero. In each such disk there is a point with rational coordinates. The rational point for each disk is different because the disks are disjoint.



                This gives an injection from the set of zeros into $mathbb Q times mathbb Q$, which is countable. Therefore, the set of zeros is countable.






                share|cite|improve this answer






















                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  The zeros of an analytic function that is not the zero function are isolated.



                  This means that for each zero there is a disk that contains it and no other zero. In each such disk there is a point with rational coordinates. The rational point for each disk is different because the disks are disjoint.



                  This gives an injection from the set of zeros into $mathbb Q times mathbb Q$, which is countable. Therefore, the set of zeros is countable.






                  share|cite|improve this answer












                  The zeros of an analytic function that is not the zero function are isolated.



                  This means that for each zero there is a disk that contains it and no other zero. In each such disk there is a point with rational coordinates. The rational point for each disk is different because the disks are disjoint.



                  This gives an injection from the set of zeros into $mathbb Q times mathbb Q$, which is countable. Therefore, the set of zeros is countable.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 20 '16 at 12:42









                  lhf

                  156k9161367




                  156k9161367












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