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Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ? If yes, please give some example.

PS: I already thought of $f(x)=\frac{1}{\Gamma(-x+1)}$. Any other nice options?

EDIT: To avoid trivial solutions due to restriction of definition range, please consider the required function to be defined in the whole complex plane.

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    What about $\frac{\sin(\pi z)}{z}$?2012-10-12
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    @Pragabhava : your function has zeros at $z=\pm 1,~\pm 2,~\pm 3,~...$2012-10-12
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    Right, Didn't thought about the negative ones. As I did with the zero: $\frac{\sin(\pi z)}{\Pi_0^\infty (z + n)}$, but now I'm not sure about the analiticity (I think it is). If not, how about removing the negative ones using the $\Gamma$ function? They are of order one.2012-10-12
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    @Pragabhava: nice idea to use $\Gamma$ to remove zeros, but somehow it does not really enrich the pool of functions with that property...2012-10-12
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    If you already have an example, then what is it that you're looking for in additional examples that isn't already provided by the one you have?2012-10-12
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    Ok, then how about this idea: Take something like $H(1-z) + \frac{1}{2} \sin(\pi z)$, but instead of the Heaviside function, use a proper [mollifier like](http://en.wikipedia.org/wiki/Mollifier) function.2012-10-12

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