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







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  • $begingroup$
    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

170k23132238




170k23132238










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










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$begingroup$

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.






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    1 Answer
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    1 Answer
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    $begingroup$

    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

        170k23132238




        170k23132238






























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