Holomorphic function $F:Bbb Hto Bbb C$ having the bounded sequence ${ir_n}$ as zeros.












2












$begingroup$


Stein and Shakarchi, Complex Analysis, Chapter 8 Problem 5.





  1. Suppose that $F:mathbb{H}tomathbb{C}$ is holomorphic and bounded. Also, suppose
    $F(z)$ vanishes when $z=ir_n$, $n=1,2,3,ldots,$ where ${r_n}$ is
    a bounded sequence of positive numbers. Prove that if $sum r_n=infty$ then $F=0$.

  2. If $sum r_n<infty$, it is possible to construct a bounded function
    on the upper half-plane with zeros precisely at the points $ir_n$.




There is something weird about the first part. If the sequence ${ir_n}$ is infinite then it has a convergent subsequence since it's bounded. Hence the zeros of $F$ accumulate in $Bbb H$ and $F$ is zero. Am I missing something here?






complex-analysis







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Stein and Shakarchi, Complex Analysis, Chapter 8 Problem 5.





    1. Suppose that $F:mathbb{H}tomathbb{C}$ is holomorphic and bounded. Also, suppose
      $F(z)$ vanishes when $z=ir_n$, $n=1,2,3,ldots,$ where ${r_n}$ is
      a bounded sequence of positive numbers. Prove that if $sum r_n=infty$ then $F=0$.

    2. If $sum r_n<infty$, it is possible to construct a bounded function
      on the upper half-plane with zeros precisely at the points $ir_n$.




    There is something weird about the first part. If the sequence ${ir_n}$ is infinite then it has a convergent subsequence since it's bounded. Hence the zeros of $F$ accumulate in $Bbb H$ and $F$ is zero. Am I missing something here?






    complex-analysis







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      1



      $begingroup$


      Stein and Shakarchi, Complex Analysis, Chapter 8 Problem 5.





      1. Suppose that $F:mathbb{H}tomathbb{C}$ is holomorphic and bounded. Also, suppose
        $F(z)$ vanishes when $z=ir_n$, $n=1,2,3,ldots,$ where ${r_n}$ is
        a bounded sequence of positive numbers. Prove that if $sum r_n=infty$ then $F=0$.

      2. If $sum r_n<infty$, it is possible to construct a bounded function
        on the upper half-plane with zeros precisely at the points $ir_n$.




      There is something weird about the first part. If the sequence ${ir_n}$ is infinite then it has a convergent subsequence since it's bounded. Hence the zeros of $F$ accumulate in $Bbb H$ and $F$ is zero. Am I missing something here?






      complex-analysis







      share|cite|improve this question











      $endgroup$




      Stein and Shakarchi, Complex Analysis, Chapter 8 Problem 5.





      1. Suppose that $F:mathbb{H}tomathbb{C}$ is holomorphic and bounded. Also, suppose
        $F(z)$ vanishes when $z=ir_n$, $n=1,2,3,ldots,$ where ${r_n}$ is
        a bounded sequence of positive numbers. Prove that if $sum r_n=infty$ then $F=0$.

      2. If $sum r_n<infty$, it is possible to construct a bounded function
        on the upper half-plane with zeros precisely at the points $ir_n$.




      There is something weird about the first part. If the sequence ${ir_n}$ is infinite then it has a convergent subsequence since it's bounded. Hence the zeros of $F$ accumulate in $Bbb H$ and $F$ is zero. Am I missing something here?






      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 Dec 16 '18 at 20:18







      UserA

















      asked Dec 16 '18 at 20:08









      UserAUserA

      542216




      542216






















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          If ${ir_n}$ accumulates at a point other than $0$, then $F=0$ is trivial as you said. But the problem is requiring you to show that if ${ir_n}$ accumulates at $0$ and the convergence $r_nto 0$ is "slow" enough to make $sum_n r_n =infty$, then $F$ must be $0$. Put differently, if $Fneq 0$ is bounded on the upper half plane and ${z_n}$ are zeros of $F$, then it must be that $Im(z_n)to 0 $ fast enough to make $$
          sum_n Im (z_n)<infty.$$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:36










          • $begingroup$
            That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
            $endgroup$
            – Song
            Dec 16 '18 at 20:38












          • $begingroup$
            Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:40












          • $begingroup$
            I guess that will work :)
            $endgroup$
            – Song
            Dec 16 '18 at 20:43










          • $begingroup$
            what about the second part?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:46



















          1












          $begingroup$

          The sequence $(ir_n)$ accumulates at a point of $mathbb{C}$, but not necessarily at a point of $mathbb{H}$. Indeed, if $r_nto 0$ then they accumulate only at $0$, which is not in $mathbb{H}$.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3043091%2fholomorphic-function-f-bbb-h-to-bbb-c-having-the-bounded-sequence-ir-n%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            If ${ir_n}$ accumulates at a point other than $0$, then $F=0$ is trivial as you said. But the problem is requiring you to show that if ${ir_n}$ accumulates at $0$ and the convergence $r_nto 0$ is "slow" enough to make $sum_n r_n =infty$, then $F$ must be $0$. Put differently, if $Fneq 0$ is bounded on the upper half plane and ${z_n}$ are zeros of $F$, then it must be that $Im(z_n)to 0 $ fast enough to make $$
            sum_n Im (z_n)<infty.$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:36










            • $begingroup$
              That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
              $endgroup$
              – Song
              Dec 16 '18 at 20:38












            • $begingroup$
              Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:40












            • $begingroup$
              I guess that will work :)
              $endgroup$
              – Song
              Dec 16 '18 at 20:43










            • $begingroup$
              what about the second part?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:46
















            1












            $begingroup$

            If ${ir_n}$ accumulates at a point other than $0$, then $F=0$ is trivial as you said. But the problem is requiring you to show that if ${ir_n}$ accumulates at $0$ and the convergence $r_nto 0$ is "slow" enough to make $sum_n r_n =infty$, then $F$ must be $0$. Put differently, if $Fneq 0$ is bounded on the upper half plane and ${z_n}$ are zeros of $F$, then it must be that $Im(z_n)to 0 $ fast enough to make $$
            sum_n Im (z_n)<infty.$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:36










            • $begingroup$
              That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
              $endgroup$
              – Song
              Dec 16 '18 at 20:38












            • $begingroup$
              Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:40












            • $begingroup$
              I guess that will work :)
              $endgroup$
              – Song
              Dec 16 '18 at 20:43










            • $begingroup$
              what about the second part?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:46














            1












            1








            1





            $begingroup$

            If ${ir_n}$ accumulates at a point other than $0$, then $F=0$ is trivial as you said. But the problem is requiring you to show that if ${ir_n}$ accumulates at $0$ and the convergence $r_nto 0$ is "slow" enough to make $sum_n r_n =infty$, then $F$ must be $0$. Put differently, if $Fneq 0$ is bounded on the upper half plane and ${z_n}$ are zeros of $F$, then it must be that $Im(z_n)to 0 $ fast enough to make $$
            sum_n Im (z_n)<infty.$$






            share|cite|improve this answer











            $endgroup$



            If ${ir_n}$ accumulates at a point other than $0$, then $F=0$ is trivial as you said. But the problem is requiring you to show that if ${ir_n}$ accumulates at $0$ and the convergence $r_nto 0$ is "slow" enough to make $sum_n r_n =infty$, then $F$ must be $0$. Put differently, if $Fneq 0$ is bounded on the upper half plane and ${z_n}$ are zeros of $F$, then it must be that $Im(z_n)to 0 $ fast enough to make $$
            sum_n Im (z_n)<infty.$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 16 '18 at 20:30

























            answered Dec 16 '18 at 20:22









            SongSong

            14.4k1635




            14.4k1635












            • $begingroup$
              what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:36










            • $begingroup$
              That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
              $endgroup$
              – Song
              Dec 16 '18 at 20:38












            • $begingroup$
              Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:40












            • $begingroup$
              I guess that will work :)
              $endgroup$
              – Song
              Dec 16 '18 at 20:43










            • $begingroup$
              what about the second part?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:46


















            • $begingroup$
              what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:36










            • $begingroup$
              That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
              $endgroup$
              – Song
              Dec 16 '18 at 20:38












            • $begingroup$
              Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:40












            • $begingroup$
              I guess that will work :)
              $endgroup$
              – Song
              Dec 16 '18 at 20:43










            • $begingroup$
              what about the second part?
              $endgroup$
              – UserA
              Dec 16 '18 at 20:46
















            $begingroup$
            what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:36




            $begingroup$
            what do you suggest in this case? Does the Jensen formula help? Should we work with the unit disk instead of the upper half plane?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:36












            $begingroup$
            That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
            $endgroup$
            – Song
            Dec 16 '18 at 20:38






            $begingroup$
            That's the trick of this problem. How can we transform the domain into the unit disk without losing analyticity of $F$? (Hint)Maybe you can use $z = frac{aw+b}{cw+d}$ or something like this.
            $endgroup$
            – Song
            Dec 16 '18 at 20:38














            $begingroup$
            Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:40






            $begingroup$
            Use the map $T:Bbb Dto Bbb H$ defined by $zmapsto ifrac{1-z}{1+z}$?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:40














            $begingroup$
            I guess that will work :)
            $endgroup$
            – Song
            Dec 16 '18 at 20:43




            $begingroup$
            I guess that will work :)
            $endgroup$
            – Song
            Dec 16 '18 at 20:43












            $begingroup$
            what about the second part?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:46




            $begingroup$
            what about the second part?
            $endgroup$
            – UserA
            Dec 16 '18 at 20:46











            1












            $begingroup$

            The sequence $(ir_n)$ accumulates at a point of $mathbb{C}$, but not necessarily at a point of $mathbb{H}$. Indeed, if $r_nto 0$ then they accumulate only at $0$, which is not in $mathbb{H}$.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              The sequence $(ir_n)$ accumulates at a point of $mathbb{C}$, but not necessarily at a point of $mathbb{H}$. Indeed, if $r_nto 0$ then they accumulate only at $0$, which is not in $mathbb{H}$.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                The sequence $(ir_n)$ accumulates at a point of $mathbb{C}$, but not necessarily at a point of $mathbb{H}$. Indeed, if $r_nto 0$ then they accumulate only at $0$, which is not in $mathbb{H}$.






                share|cite|improve this answer









                $endgroup$



                The sequence $(ir_n)$ accumulates at a point of $mathbb{C}$, but not necessarily at a point of $mathbb{H}$. Indeed, if $r_nto 0$ then they accumulate only at $0$, which is not in $mathbb{H}$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 16 '18 at 20:16









                Eric WofseyEric Wofsey

                187k14215344




                187k14215344






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3043091%2fholomorphic-function-f-bbb-h-to-bbb-c-having-the-bounded-sequence-ir-n%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    Probability when a professor distributes a quiz and homework assignment to a class of n students.

                    Image
                    0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
                    Read more

                    Aardman Animations

                    Image
                    Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
                    Read more

                    Are they similar matrix

                    Image
                    0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
                    Read more