All the complex polynomials $p$ and $q$ for which $p(z)sin^2(z)+q(z)cos^2(z)=1$ holds.












0












$begingroup$


Find all the complex polynomials $p$ and $q$ for which $$p(z)sin^2(z)+q(z)cos^2(z)=1$$ holds.

We know that the function $sin^2(z)+cos^2(z)-1$ is analytic on $mathbb{C}$ and bounded, so it is constant. And since this function in $0$ on real line, $$sin^2(z)+cos^2(z)-1=0 text{ on } mathbb{C}$$ But I have no idea for the polynomials $p$ and $q$ other than $1$. Please help me to find all such complex polynomials $p$ and $q$ such that the above holds.






complex-analysis







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    Find all the complex polynomials $p$ and $q$ for which $$p(z)sin^2(z)+q(z)cos^2(z)=1$$ holds.

    We know that the function $sin^2(z)+cos^2(z)-1$ is analytic on $mathbb{C}$ and bounded, so it is constant. And since this function in $0$ on real line, $$sin^2(z)+cos^2(z)-1=0 text{ on } mathbb{C}$$ But I have no idea for the polynomials $p$ and $q$ other than $1$. Please help me to find all such complex polynomials $p$ and $q$ such that the above holds.






    complex-analysis







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Find all the complex polynomials $p$ and $q$ for which $$p(z)sin^2(z)+q(z)cos^2(z)=1$$ holds.

      We know that the function $sin^2(z)+cos^2(z)-1$ is analytic on $mathbb{C}$ and bounded, so it is constant. And since this function in $0$ on real line, $$sin^2(z)+cos^2(z)-1=0 text{ on } mathbb{C}$$ But I have no idea for the polynomials $p$ and $q$ other than $1$. Please help me to find all such complex polynomials $p$ and $q$ such that the above holds.






      complex-analysis







      share|cite|improve this question











      $endgroup$




      Find all the complex polynomials $p$ and $q$ for which $$p(z)sin^2(z)+q(z)cos^2(z)=1$$ holds.

      We know that the function $sin^2(z)+cos^2(z)-1$ is analytic on $mathbb{C}$ and bounded, so it is constant. And since this function in $0$ on real line, $$sin^2(z)+cos^2(z)-1=0 text{ on } mathbb{C}$$ But I have no idea for the polynomials $p$ and $q$ other than $1$. Please help me to find all such complex polynomials $p$ and $q$ such that the above holds.






      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 6 at 20:35









      rtybase

      11.5k31534




      11.5k31534










      asked Jan 6 at 17:29









      abcdmathabcdmath

      337110




      337110






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          Let $p_1=p-1$, $q_1=q-1$, then $q_1(z)cos^2(z)+p_1(z)sin^2(z)=0$.



          Now take $z=kpi$, $k$ an integer, then $q_1(z)=0$. Thus $q_1$ has infinitely many roots so is zero. Similarly, $p_1=0$, hence $p=q=1$.






          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%2f3064148%2fall-the-complex-polynomials-p-and-q-for-which-pz-sin2zqz-cos2z-1%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            Let $p_1=p-1$, $q_1=q-1$, then $q_1(z)cos^2(z)+p_1(z)sin^2(z)=0$.



            Now take $z=kpi$, $k$ an integer, then $q_1(z)=0$. Thus $q_1$ has infinitely many roots so is zero. Similarly, $p_1=0$, hence $p=q=1$.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Let $p_1=p-1$, $q_1=q-1$, then $q_1(z)cos^2(z)+p_1(z)sin^2(z)=0$.



              Now take $z=kpi$, $k$ an integer, then $q_1(z)=0$. Thus $q_1$ has infinitely many roots so is zero. Similarly, $p_1=0$, hence $p=q=1$.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Let $p_1=p-1$, $q_1=q-1$, then $q_1(z)cos^2(z)+p_1(z)sin^2(z)=0$.



                Now take $z=kpi$, $k$ an integer, then $q_1(z)=0$. Thus $q_1$ has infinitely many roots so is zero. Similarly, $p_1=0$, hence $p=q=1$.






                share|cite|improve this answer









                $endgroup$



                Let $p_1=p-1$, $q_1=q-1$, then $q_1(z)cos^2(z)+p_1(z)sin^2(z)=0$.



                Now take $z=kpi$, $k$ an integer, then $q_1(z)=0$. Thus $q_1$ has infinitely many roots so is zero. Similarly, $p_1=0$, hence $p=q=1$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 6 at 17:38









                MindlackMindlack

                4,900211




                4,900211






























                    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%2f3064148%2fall-the-complex-polynomials-p-and-q-for-which-pz-sin2zqz-cos2z-1%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