If f has power series expansion around each point then does it have a single power series expansion on following sets

Question

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!!

Answer 1

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 $\sqrt{a^2+1}<\infty$.