Let $f$ be an entire function. Assume that $\mid f(1/n)\mid\le e^{-n}$ for all $n\in \mathbb{N}$. Show that $f$ vanishes identically.
Show that $f$ vanishes identically.
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complex-analysis
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0Are you sure it's not $|f(n^{-1}|\leq e^{-n}$? Otherwise take $f(z)=e^{-1}$. Hint: you can show that $f(0)=0$, then by induction that $f^{(n)}(0)=0$. – 2012-10-08
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0Oh, yes, sorry for mistake, I have corrected it. Thanks for hint, I will try it. – 2012-10-08