6 true or false questions in complex analysis












0












$begingroup$


Let $f:mathbb Ctomathbb C$ be an entire function. Which of the following statements are true and which are false?




  1. If $f(z)inmathbb R$ for all $zinmathbb C$, then $f$ is constant.

  2. If $f(frac{1}{n})=frac{i}{n}$ for all $ninmathbb N$, then $f(z)=iz$ for all $zinmathbb C$.

  3. If $f$ is a non-constant polynomial, then there is a continuously differentiable path $gamma: [0,1]tomathbb C$ such that $int_gamma f(z)dz=2pi i$.

  4. The function $1/f$ has no pole in $0$.

  5. The function $rmapstoint_{|z|=r}f(z)dz$ is constant on $(0,infty)$.

  6. The series $sum_{n=0}^inftyfrac{1}{n!}f^{(n)}(1)(z-1)^n$ converges for all $zinmathbb C$



Here is what I got:




  1. True. If $f$ is not constant, then $f(mathbb C)$ must be a domain which conflicts with $f(mathbb C)subseteqmathbb R$.

  2. True. On the set ${1/n|ninmathbb N$} the function $f$ coincides with $g(z)=iz$. Since the set does have an accumulation point in the domain of $f$, the identity theorem yields $f=g$.

  3. I have no idea. How do I start here?

  4. False. $f(z)=z$.

  5. True. Since $f$ is holomorphic, the integral is always $0$.

  6. True. This is just the taylor series of $f$ in $z_0=1$.






complex-analysis proof-verification entire-functions







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    Let $f:mathbb Ctomathbb C$ be an entire function. Which of the following statements are true and which are false?




    1. If $f(z)inmathbb R$ for all $zinmathbb C$, then $f$ is constant.

    2. If $f(frac{1}{n})=frac{i}{n}$ for all $ninmathbb N$, then $f(z)=iz$ for all $zinmathbb C$.

    3. If $f$ is a non-constant polynomial, then there is a continuously differentiable path $gamma: [0,1]tomathbb C$ such that $int_gamma f(z)dz=2pi i$.

    4. The function $1/f$ has no pole in $0$.

    5. The function $rmapstoint_{|z|=r}f(z)dz$ is constant on $(0,infty)$.

    6. The series $sum_{n=0}^inftyfrac{1}{n!}f^{(n)}(1)(z-1)^n$ converges for all $zinmathbb C$



    Here is what I got:




    1. True. If $f$ is not constant, then $f(mathbb C)$ must be a domain which conflicts with $f(mathbb C)subseteqmathbb R$.

    2. True. On the set ${1/n|ninmathbb N$} the function $f$ coincides with $g(z)=iz$. Since the set does have an accumulation point in the domain of $f$, the identity theorem yields $f=g$.

    3. I have no idea. How do I start here?

    4. False. $f(z)=z$.

    5. True. Since $f$ is holomorphic, the integral is always $0$.

    6. True. This is just the taylor series of $f$ in $z_0=1$.






    complex-analysis proof-verification entire-functions







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Let $f:mathbb Ctomathbb C$ be an entire function. Which of the following statements are true and which are false?




      1. If $f(z)inmathbb R$ for all $zinmathbb C$, then $f$ is constant.

      2. If $f(frac{1}{n})=frac{i}{n}$ for all $ninmathbb N$, then $f(z)=iz$ for all $zinmathbb C$.

      3. If $f$ is a non-constant polynomial, then there is a continuously differentiable path $gamma: [0,1]tomathbb C$ such that $int_gamma f(z)dz=2pi i$.

      4. The function $1/f$ has no pole in $0$.

      5. The function $rmapstoint_{|z|=r}f(z)dz$ is constant on $(0,infty)$.

      6. The series $sum_{n=0}^inftyfrac{1}{n!}f^{(n)}(1)(z-1)^n$ converges for all $zinmathbb C$



      Here is what I got:




      1. True. If $f$ is not constant, then $f(mathbb C)$ must be a domain which conflicts with $f(mathbb C)subseteqmathbb R$.

      2. True. On the set ${1/n|ninmathbb N$} the function $f$ coincides with $g(z)=iz$. Since the set does have an accumulation point in the domain of $f$, the identity theorem yields $f=g$.

      3. I have no idea. How do I start here?

      4. False. $f(z)=z$.

      5. True. Since $f$ is holomorphic, the integral is always $0$.

      6. True. This is just the taylor series of $f$ in $z_0=1$.






      complex-analysis proof-verification entire-functions







      share|cite|improve this question











      $endgroup$




      Let $f:mathbb Ctomathbb C$ be an entire function. Which of the following statements are true and which are false?




      1. If $f(z)inmathbb R$ for all $zinmathbb C$, then $f$ is constant.

      2. If $f(frac{1}{n})=frac{i}{n}$ for all $ninmathbb N$, then $f(z)=iz$ for all $zinmathbb C$.

      3. If $f$ is a non-constant polynomial, then there is a continuously differentiable path $gamma: [0,1]tomathbb C$ such that $int_gamma f(z)dz=2pi i$.

      4. The function $1/f$ has no pole in $0$.

      5. The function $rmapstoint_{|z|=r}f(z)dz$ is constant on $(0,infty)$.

      6. The series $sum_{n=0}^inftyfrac{1}{n!}f^{(n)}(1)(z-1)^n$ converges for all $zinmathbb C$



      Here is what I got:




      1. True. If $f$ is not constant, then $f(mathbb C)$ must be a domain which conflicts with $f(mathbb C)subseteqmathbb R$.

      2. True. On the set ${1/n|ninmathbb N$} the function $f$ coincides with $g(z)=iz$. Since the set does have an accumulation point in the domain of $f$, the identity theorem yields $f=g$.

      3. I have no idea. How do I start here?

      4. False. $f(z)=z$.

      5. True. Since $f$ is holomorphic, the integral is always $0$.

      6. True. This is just the taylor series of $f$ in $z_0=1$.






      complex-analysis proof-verification entire-functions




      complex-analysis proof-verification entire-functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 19 '18 at 12:20









      José Carlos Santos

      172k23132240




      172k23132240










      asked Dec 19 '18 at 12:14









      RedLanternRedLantern

      517




      517






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          What you did is correct.



          The assertion 3. is true. For each $winmathbb C$, let $gamma_wcolon[0,1]longrightarrowmathbb C$ be the path $gamma_w(t)=tw$. Then the map $wmapstoint_{gamma_w}f(z),mathrm dz$ is just the antiderivative $F$ of $f$ such that $F(0)=0$. But then $F$ is a non constant polynomial function and therefore it is surjective. So, there is so $w$ such that $F(w)=2pi i$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
            $endgroup$
            – RedLantern
            Dec 19 '18 at 12:35






          • 1




            $begingroup$
            By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
            $endgroup$
            – Fred
            Dec 19 '18 at 12:54












          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%2f3046319%2f6-true-or-false-questions-in-complex-analysis%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









          1












          $begingroup$

          What you did is correct.



          The assertion 3. is true. For each $winmathbb C$, let $gamma_wcolon[0,1]longrightarrowmathbb C$ be the path $gamma_w(t)=tw$. Then the map $wmapstoint_{gamma_w}f(z),mathrm dz$ is just the antiderivative $F$ of $f$ such that $F(0)=0$. But then $F$ is a non constant polynomial function and therefore it is surjective. So, there is so $w$ such that $F(w)=2pi i$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
            $endgroup$
            – RedLantern
            Dec 19 '18 at 12:35






          • 1




            $begingroup$
            By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
            $endgroup$
            – Fred
            Dec 19 '18 at 12:54
















          1












          $begingroup$

          What you did is correct.



          The assertion 3. is true. For each $winmathbb C$, let $gamma_wcolon[0,1]longrightarrowmathbb C$ be the path $gamma_w(t)=tw$. Then the map $wmapstoint_{gamma_w}f(z),mathrm dz$ is just the antiderivative $F$ of $f$ such that $F(0)=0$. But then $F$ is a non constant polynomial function and therefore it is surjective. So, there is so $w$ such that $F(w)=2pi i$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
            $endgroup$
            – RedLantern
            Dec 19 '18 at 12:35






          • 1




            $begingroup$
            By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
            $endgroup$
            – Fred
            Dec 19 '18 at 12:54














          1












          1








          1





          $begingroup$

          What you did is correct.



          The assertion 3. is true. For each $winmathbb C$, let $gamma_wcolon[0,1]longrightarrowmathbb C$ be the path $gamma_w(t)=tw$. Then the map $wmapstoint_{gamma_w}f(z),mathrm dz$ is just the antiderivative $F$ of $f$ such that $F(0)=0$. But then $F$ is a non constant polynomial function and therefore it is surjective. So, there is so $w$ such that $F(w)=2pi i$.






          share|cite|improve this answer









          $endgroup$



          What you did is correct.



          The assertion 3. is true. For each $winmathbb C$, let $gamma_wcolon[0,1]longrightarrowmathbb C$ be the path $gamma_w(t)=tw$. Then the map $wmapstoint_{gamma_w}f(z),mathrm dz$ is just the antiderivative $F$ of $f$ such that $F(0)=0$. But then $F$ is a non constant polynomial function and therefore it is surjective. So, there is so $w$ such that $F(w)=2pi i$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 19 '18 at 12:19









          José Carlos SantosJosé Carlos Santos

          172k23132240




          172k23132240












          • $begingroup$
            So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
            $endgroup$
            – RedLantern
            Dec 19 '18 at 12:35






          • 1




            $begingroup$
            By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
            $endgroup$
            – Fred
            Dec 19 '18 at 12:54


















          • $begingroup$
            So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
            $endgroup$
            – RedLantern
            Dec 19 '18 at 12:35






          • 1




            $begingroup$
            By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
            $endgroup$
            – Fred
            Dec 19 '18 at 12:54
















          $begingroup$
          So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
          $endgroup$
          – RedLantern
          Dec 19 '18 at 12:35




          $begingroup$
          So all non constant polynomials are surjective? I didn't know that. And what about injective? Is the proof rather easy or complicated?
          $endgroup$
          – RedLantern
          Dec 19 '18 at 12:35




          1




          1




          $begingroup$
          By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
          $endgroup$
          – Fred
          Dec 19 '18 at 12:54




          $begingroup$
          By the fundamental theorem of algebra, a non constant polynomial is surjective. In general, a polynomial is not injective. Example: $p(z)=z^2$.
          $endgroup$
          – Fred
          Dec 19 '18 at 12:54


















          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%2f3046319%2f6-true-or-false-questions-in-complex-analysis%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

          Plaza Victoria

          Image
          Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
          Read more

          Puebla de Zaragoza

          Image
          Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
          Read more

          Musa

          Image
          Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
          Read more