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It follows from Weierstrass theorem that there exists an entire function with zeros only at prescribed points.

But let's say we have a particular sequence of prescribed points, for example $\{a_n\} = \mathbb{N}$.

How do we built the entire function with zeros only at those points constructively?

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  • $\begingroup$ You can choose $\exp (2\pi z) - 1$; this also has the negative integers as zeros. $\endgroup$ – user258700 Jul 31 '16 at 17:25
  • $\begingroup$ @AhmedHussein My bad, I meant "with zeros only at prescribed points" $\endgroup$ – G. Schiele Jul 31 '16 at 17:27
  • $\begingroup$ maybe try $f(z)=\prod_{i=1}^{\infty} (1-\frac{z}{n})$ and hope it converges and is analytic on $\mathbb{C}$? $\endgroup$ – M. Van Jul 31 '16 at 17:30
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    $\begingroup$ See this. $\endgroup$ – Seewoo Lee Jul 31 '16 at 17:30
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    $\begingroup$ The proof of the Weierstrass theorem is constructive, so if you want to know what to do with a general sequence you should probably just learn it. If you're interested in finding an elegant-looking function for your specific sequence, that's another story. Something along the lines of $\frac{1}{\Gamma(-z)}$ will probably do the trick... $\endgroup$ – Micah Jul 31 '16 at 17:31

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