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I have come across the following questions on a past exam paper, but I'm not sure how to go about answering the,. Any help would be greatly appreciated....

(i) Does there exist a function $f$ differentiable on the open unit disc $\{z\in \mathbb{C}: |z|<1\}$ such that $f(1/n)=f(-1/n)=1/n^2$ for all $n=1,2,3,\ldots$?

(ii) Does there exist a function $g$ differentiable on the open unit disc $\{z\in \mathbb{C}: |z|<1\}$ such that $g(1/n)=g(-1/n)=1/n^3$ for all $n=1,2,3,\ldots$?

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1 Answer

By differentiable, I assume you mean complex differentiable, i.e. holomorphic.

(a) How about $f(z) = z^2$?

(b) Compare the presumptive $g$ to $z^3$ and $-z^3$ and use the identity theorem for holomorphic functions.

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