1
$\begingroup$

I have two questions regarding analyticity. They are pretty easy, and I think I have them correct, but I just want to make sure.

First, regarding the question in the title, I think that it is necessary. By the definition of analyticity, we must have the power series in some radius of the function. So, if it is analytic on all of $\mathbb{R}$, it must be within some radius (and thus convergent) for all of $\mathbb{R}$.

Second, if $f$ is analytic, is it necessary for $\text{exp}(f)$ to be analytic? Pretty sure that this again is necessary since $e^{x}$ is analytic, and the composition of analytic functions is analytic.

$\endgroup$
  • $\begingroup$ This is correct. $\endgroup$ – Zanzi Nov 16 '18 at 8:13
  • $\begingroup$ Only an accumuled point is required. $\endgroup$ – Zanzi Nov 16 '18 at 8:14
  • 1
    $\begingroup$ Counterexample to question 1: $f(x)=\frac1{1+x^2}$. $\endgroup$ – Kemono Chen Nov 16 '18 at 8:14
  • $\begingroup$ Yes. This is correct. $\endgroup$ – Andrej Nov 16 '18 at 8:16
  • $\begingroup$ @KemonoChen Why not an official answer even it is short? $\endgroup$ – Paul Frost Nov 16 '18 at 8:21
2
$\begingroup$

Counterexample to question 1: $f(x)=\frac1{1+x^2}$. The Maclaurin series of it does not converge everywhere is because $f$ is a meromorphic function in $\mathbb{C}$.
If $f(z)$ is analytic on $\mathbb{C}$, the series you gave is convergent everywhere.
Your explanation of question 2 is correct.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.