Does there exist an analytic function $f:D\to\mathbb{C}$ such that $f(1/n)=f(-1/n)=1/n^3$

Question

"Does there exist an analytic function $f:D\to\mathbb{C}$ such that $f(1/n)=f(-1/n)=1/n^3$?"

This is one of the past qualifying exam problems that I am working on and I found that $f(0)=0$, $f^{(n)}(0)=0,n=1,2$, $f^{(3)}(0)=1$ using the definition of derivative of a function. I am trying to use a Taylor expansion at z=0 since f is analytic in $D=\{z\in \mathbb{C}||z|=1\}$. However I do not know how to use $f(1/n)=f(-1/n)=1/n^3$ to prove or disprove the existence of such function $f$.

Any help would be appreciated.

Thank you in advance.

http://www.proofwiki.org/wiki/Zeroes_of_Analytic_Function_are_Isolated — Andrés E. Caicedo, Dec 30 '12 at 07:54
@Andres Caicedo: We know that $f$ is not a constant function but I still do not understand how to use the fact that zero being isolated to disprove the existence of such function $f$. Would you give me more clues? — Yeonjoo Yoo, Dec 30 '12 at 21:06

P.. · Accepted Answer · 2012-12-30T08:23:57.420

3

No. Use identity theorem to show that if $g(z)=z^3$ and $h(z)=-z^3$ then $f=g=h↯.$

edited Dec 30 '12 at 08:23

answered Dec 30 '12 at 08:14

P..

14,929

1

@ Pambos: I understood your idea that if you can find a nbhd of $\frac{1}{n}$ s.t. $f=g$ and a nbhd of $\frac{-1}{n}$ s.t. $f=h$ then by the identity theorem that $f=g=h$ in $D$ which is a contradiction. However, how do you guarantee that such neighborhoods exist? – Yeonjoo Yoo Dec 30 '12 at 21:01
I supposed that $D={z\in\mathbb C: |Z|<1}$. What is $D$? – P.. Dec 31 '12 at 08:43
D is D={z∈C:|Z|<1}. – Yeonjoo Yoo Jan 01 '13 at 15:54
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Read the section "An improvement" here. – P.. Jan 01 '13 at 18:56

Does there exist an analytic function $f:D\to\mathbb{C}$ such that $f(1/n)=f(-1/n)=1/n^3$

1 Answers1