Let \(f(z)\) be a non-constant analytic function in \(|z|>0\) such that \(f(z_n) = 0\) for infinite many points \(z_n\) with \(\lim_{n \rightarrow \infty} z_n =0\).
Show that \(z=0\) is an essential singularity for \(f(z)\).
Hint: an example of such a function is \(f(z) = \sin (1/z)\).
Solution
The point \(z=0\) is not a removable singularity: otherwise \(f\) would extend to a holomorphic function on \(\abs z<\infty\) with \(f(0) = \lim f(z_n) = 0\), so the zeros \(z_n\) would accumulate at the point \(0\) of the domain, and the identity theorem would force \(f\equiv 0\), contradicting that \(f\) is nonconstant.
The point \(z=0\) is not a pole: otherwise \(\abs{f(z)}\to \infty\) as \(z\to0\), whereas \(f(z_n) = 0\) with \(z_n\to0\).
Hence \(z=0\) is an essential singularity.