Let $f$ be a entire function. Assume that there exist a real number $a$ such that $f^{(r)}(a)=0$ for all integer $r≥0$. My question is show that the function $f$ is identically zero.
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$\begingroup$ How much do you know about holomorphic functions? For example what do you know about power series expansions (Taylor series)? $\endgroup$ – mrf Apr 11 '14 at 8:37
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$\begingroup$ @mrf: I know about them. $\endgroup$ – DER Apr 11 '14 at 8:41
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An entire function is a function that is holomorphic on the whole complex plane. Such a function can be expressed globally as its Taylor series, developed by ANY point of the complex plane. So you can take $a$. Hence you have that $f(z)=\sum_{n=0}^{+\infty}\frac{f^{(n)}(a)}{n!}(z-a)^n\;\;\forall z\in\mathbb C$. But $f^{(n)}(a)=0\;\;\forall n\geq0$ by hypotesis. Hence $f\equiv0$, as wanted.
