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I am solving previous year question paper some competitive exam. Give me some hint to solve the following problem.

Let $f$ be an entire function. Suppose for each $a \in \mathbb{R} $ there exists at least one coefficient $c_n$ in $f(z) = \sum_{n=0}^{\infty} c_n (z-a)^n$ which is zero. Then,

a) $f^{(n)}(0) = 0$ for infinitely many $n \ge 0$

b) $f^{(2n)}(0)=0$ $ \forall n \ge 0$

c) $f^{(2n+1)}(0)=0$ $ \forall n \ge 0$

d) $f^{(n)}(0) = 0$ for all sufficiently large n.

Thanks in advance.

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GA316
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1 Answers1

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$f$ is a polynomial and d) is correct.

If $f$ is not a polynomial, then $f^{(n)}$ is a non constant entire function for all $n$. The set of zeros of $f^{(n)}$ is discrete, and in particular, countable. Thus, $$ Z=\{z\in\mathbb{C}:f^{(n)}(z)=0\text{ for some }n\} $$ is countable. But $\mathbb{R}\subset Z$, a contradiction.

Julián Aguirre
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  • Thanks . nice answer. if d is correct then a is also correct no sir. so answer is a and d. – GA316 Oct 17 '14 at 16:51