# Entire function with zeros only at the natural numbers

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?

• You can choose $\exp (2\pi z) - 1$; this also has the negative integers as zeros. – user258700 Jul 31 '16 at 17:25
• @AhmedHussein My bad, I meant "with zeros only at prescribed points" – G. Schiele Jul 31 '16 at 17:27
• maybe try $f(z)=\prod_{i=1}^{\infty} (1-\frac{z}{n})$ and hope it converges and is analytic on $\mathbb{C}$? – M. Van Jul 31 '16 at 17:30
• See this. – Seewoo Lee Jul 31 '16 at 17:30
• 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... – Micah Jul 31 '16 at 17:31