how can i prove that an even, entire function of exponential type, which is bounded on the real axis has an infinite number of complex roots (using Hadamard's theorem); it must be straightforward but i am mission something.
thanx
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what about $\sin(z)$ ? – reuns Sep 20 '16 at 09:12
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you are right. obviously $sin(z)$ satisfies the conditions and has only real roots. my function also satisfies that $f(k)<0,\ \forall k\in \mathbb{R}-{0}$. So a better question is: any even, entire function of exponential type, bounded on the real line and strictly negative has infinite complex roots; – nikosp Sep 20 '16 at 10:13