Entire function either constant or has a zero












1












$begingroup$


Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero?



I was thinking about either: if f is everywhere nonzero then form $g = 1/f $ and then use $ |fg| = 1 $ to show that $ f $ is bounded, hence constant, or suppose that $ f $ is not constant, then show that $ f $ has a zero somehow. But I don't know how to do either. How do I prove this?






complex-analysis entire-functions







share|cite|improve this question











$endgroup$












  • $begingroup$
    You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
    $endgroup$
    – 0x539
    Dec 30 '18 at 14:37










  • $begingroup$
    @0x539 You can't drop it:for example the exponential function doesn't have a zero.
    $endgroup$
    – ploosu2
    Dec 30 '18 at 15:04










  • $begingroup$
    @ploosu2 You're right, my bad.
    $endgroup$
    – 0x539
    Dec 30 '18 at 15:09
















1












$begingroup$


Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero?



I was thinking about either: if f is everywhere nonzero then form $g = 1/f $ and then use $ |fg| = 1 $ to show that $ f $ is bounded, hence constant, or suppose that $ f $ is not constant, then show that $ f $ has a zero somehow. But I don't know how to do either. How do I prove this?






complex-analysis entire-functions







share|cite|improve this question











$endgroup$












  • $begingroup$
    You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
    $endgroup$
    – 0x539
    Dec 30 '18 at 14:37










  • $begingroup$
    @0x539 You can't drop it:for example the exponential function doesn't have a zero.
    $endgroup$
    – ploosu2
    Dec 30 '18 at 15:04










  • $begingroup$
    @ploosu2 You're right, my bad.
    $endgroup$
    – 0x539
    Dec 30 '18 at 15:09














1












1








1





$begingroup$


Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero?



I was thinking about either: if f is everywhere nonzero then form $g = 1/f $ and then use $ |fg| = 1 $ to show that $ f $ is bounded, hence constant, or suppose that $ f $ is not constant, then show that $ f $ has a zero somehow. But I don't know how to do either. How do I prove this?






complex-analysis entire-functions







share|cite|improve this question











$endgroup$




Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero?



I was thinking about either: if f is everywhere nonzero then form $g = 1/f $ and then use $ |fg| = 1 $ to show that $ f $ is bounded, hence constant, or suppose that $ f $ is not constant, then show that $ f $ has a zero somehow. But I don't know how to do either. How do I prove this?






complex-analysis entire-functions




complex-analysis entire-functions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 30 '18 at 15:05









José Carlos Santos

175k24134243




175k24134243










asked Dec 30 '18 at 14:31









calmcalm

1387




1387












  • $begingroup$
    You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
    $endgroup$
    – 0x539
    Dec 30 '18 at 14:37










  • $begingroup$
    @0x539 You can't drop it:for example the exponential function doesn't have a zero.
    $endgroup$
    – ploosu2
    Dec 30 '18 at 15:04










  • $begingroup$
    @ploosu2 You're right, my bad.
    $endgroup$
    – 0x539
    Dec 30 '18 at 15:09


















  • $begingroup$
    You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
    $endgroup$
    – 0x539
    Dec 30 '18 at 14:37










  • $begingroup$
    @0x539 You can't drop it:for example the exponential function doesn't have a zero.
    $endgroup$
    – ploosu2
    Dec 30 '18 at 15:04










  • $begingroup$
    @ploosu2 You're right, my bad.
    $endgroup$
    – 0x539
    Dec 30 '18 at 15:09
















$begingroup$
You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
$endgroup$
– 0x539
Dec 30 '18 at 14:37




$begingroup$
You might want to drop the condition that the Taylor series converges uniformly on $mathbb{C}$, I think this is only satisfied by Polynomials.
$endgroup$
– 0x539
Dec 30 '18 at 14:37












$begingroup$
@0x539 You can't drop it:for example the exponential function doesn't have a zero.
$endgroup$
– ploosu2
Dec 30 '18 at 15:04




$begingroup$
@0x539 You can't drop it:for example the exponential function doesn't have a zero.
$endgroup$
– ploosu2
Dec 30 '18 at 15:04












$begingroup$
@ploosu2 You're right, my bad.
$endgroup$
– 0x539
Dec 30 '18 at 15:09




$begingroup$
@ploosu2 You're right, my bad.
$endgroup$
– 0x539
Dec 30 '18 at 15:09










1 Answer
1






active

oldest

votes


















2












$begingroup$

Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.






share|cite|improve this answer









$endgroup$














    Your Answer








    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%2f3056874%2fentire-function-either-constant-or-has-a-zero%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












    $begingroup$

    Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.






        share|cite|improve this answer









        $endgroup$



        Since the Taylor series of $f$ converges uniformly to $f$, $f$ is a polynomial function. Therefore, if $f$ is not constant, then it has at least a zero, by the Fundamental Theorem of Algebra.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 30 '18 at 14:37









        José Carlos SantosJosé Carlos Santos

        175k24134243




        175k24134243






























            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%2f3056874%2fentire-function-either-constant-or-has-a-zero%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

            How to send String Array data to Server using php in android

            Image
            0 I have a String array which contains value string array = {6,37,38,837,7,...} and I have also used some string values. I want to send that all string values and string array to my database which is located at server side. When I am click on submit button the data will stored in table. On the PHP side I want to read this array as it is,other thing i will do at background means PHP side. See my below code then you will get it what I am says. Here is my code: public class Hotels extends Activity { // Declare Variables JSONArray jsonarray = null; private static final String TAG_ID = "id"; public static final String TAG_NAME = "name"; public static final String TAG_LOCATION = "location"; public static final String TAG_DESC = "description"; String f_date, l_date;...
            Read more

            Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

            Image
            up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx...
            Read more

            Is anime1.com a legal site for watching anime?

            Image
            up vote 4 down vote favorite 1 Other anime streaming sites are blocking my IP because I'm Iranian while anime1.com isn't. I wanted to know if it's legal to watch/stream anime from this site. Thanks. anime-production share | improve this question edited Nov 27 at 22:38 W. Are 1,599 2 23 asked Nov 27 at 20:36 mhnajafi 23 3 add a comment ...
            Read more