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I am trying assignment in complex analysis of an institute in which I don't study because the instructor who taught me was terrible didn't bothered to give any assignment and also was poor in teaching concepts . So , I try problems of a different institute .

Question : If $f :\mathbb{C} \to \mathbb{C}$ has a power series expansion around each point then does it have a single power series expansion valid on all of $\mathbb{C}$ ?

If $f :\mathbb{R} \to \mathbb{R}$ has a power series expansion around each point then does it have a single power series expansion valid on all of $\mathbb{R}$ ?

I have no clue on which concept should be used . Kindly tell what concepts / results should be used give a hint or two (if necessary) and rest I would like to work by myself .

Thanks!!

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The answer to the first question is affirmative. In fact a standard Complex Analysis theorem says that if $f\colon D(\subset\Bbb C)\longrightarrow\Bbb C$ is analytic and if $D_r(a)\subset D$, then the radius of convergence of the power series of $f$ centered at $a$ is at least $r$. Since, in your case, $D=\Bbb C$, then for each $a\in\Bbb C$, the radius of convergence of the power series of $f$ centered at $a$ is $\infty$. That is, it converges everywhere. And its sum is $f(z)$ (this follows from the identity theorem).

However, this is false in $\Bbb R$: if $f(x)=\dfrac1{1+x^2}$, then the radius of convergence of the Taylor series of $f$ centered at $a$ is $\sqrt{a^2+1}<\infty$.

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  • $\begingroup$ @ Jose Carlos Santos What is $D_r(A)$ in second line of your answer? $\endgroup$ – Avenger Sep 16 '20 at 11:55
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    $\begingroup$ @Ben It was a typo. I meant $D_r(a)$, which is the open disk with center $a$ and radius $r$. $\endgroup$ – José Carlos Santos Sep 16 '20 at 21:07
  • $\begingroup$ You have answered some of my questions and I am really thankful to you for that.Can you also please answer this question if you have some spare time?math.stackexchange.com/questions/3810924/… $\endgroup$ – Avenger Sep 20 '20 at 4:38
  • $\begingroup$ by using which result did you found out radius of convergence in case of reals? $\endgroup$ – Avenger Sep 20 '20 at 4:50
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    $\begingroup$ @Ben The radius of convergence of the Taylor series centered at $a$ is the distance from $a$ to the nearest pole or essential singularity. In this case, it's the distance from $a$ to $\pm i$, which is $\sqrt{a^2+1}$. $\endgroup$ – José Carlos Santos Sep 20 '20 at 7:50

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