# Analytic function with vanishing derivatives.

A function that is analytic in the whole plane and which vanishes along with all its derivatives at any one point in the plane is identical to $0$. Now consider a function $f(z)$, which is supposedly analytic everywhere such that $$\lim_{z\rightarrow\infty}f^{(n)}(z)=0$$ for $n=0, 1, 2...$ Is the conclusion that $f(z)$ is identical to $0$ the only possibility?

Yes. $\lim_{z\rightarrow\infty}f(z)=0$ already implies that $f \equiv 0$.
• Yes, but the answer did remind me that $f$ would be bounded. – vnd Dec 20 '15 at 22:47
• @zhw.: But the conclusion $\lim_{z\rightarrow\infty}f(z)=0 \Longrightarrow f \equiv 0$ would also follow from the maximum principle, without using Liouville, doesn't it? – Martin R Dec 20 '15 at 23:03