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Is there an example of an analytic function in the unit disc whose zeros are only the points $z_n=1-1/n$?

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    @GEdgar Thanks for pointing that out.2012-04-16

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Another one: $ f(z) = \frac{1}{\Gamma\left(\frac{1}{z-1}\right)} $ This is analytic in the plane, except one point $z=1$, and has zeros exactly $1-1/n$, $n=1,2,3\dots$. Unlike Henning's, which also has zeros ${} \gt 1$.

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    I think it's just a matter of already knowing $1/\Gamma(z)$ as a standard example of a holomorphic function whose zeroes form a singly infinite arithmetic sequence.2012-04-16
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How about $f(z) = \sin(\frac{\pi}{1-z})$?

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For the general question of functions with prescribed zeros, consider Weierstraß' factorization theorem.

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    Just use a fractional linear transformation to go from a question about a sequence of prescribed zeros with a finite limit $\alpha$ to a sequence of prescribed zeros with a limit of $\infty$: get an entire function, and transform it back (obtaining a function analytic on ${\mathbb C} \backslash \{ \alpha \}$).2012-04-16