## Weierstrass Theorem [closed]

Hi--

Where can I find a proof of this theorem: For each $r \in \mathbb{Z}_{+}$, there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but $f(r+1)=f(r+2)=\cdots =0$, i.e. $f(z) \in I_{r+1}$ but $f(z) \neq I_{r}$, where $I_{r}= \{ f \in R \ | f(r)=f(r+1)= \cdots =0\}$ where $R$ is the ring of complex entire functions

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## closed as too localized by Robin Chapman, Felipe Voloch, S. Carnahan♦Jul 31 2010 at 19:21

Why don't you start with a function with zeros at the integers, for instance $\sin\pi z$, and then somehow eliminate the zero at $r$?
There is a direct answer to your question using the reciprocal of the $\Gamma$-function with a linear change of variables. You really should look at the Weierstrass product of the $\Gamma$-function to understand what is going on here so that you see what its Weierstrass product achieves in terms of zeros/poles.