Let \(f\) be holomorphic on a punctured disc \(D_r(z_0)\setminus\{z_0\}\). The isolated singularity \(z_0\) is an essential singularity of \(f\) if it is neither a removable singularity nor a pole.
The isolated singularity \(z_0\) is essential if and only if \(\lim_{z\to z_0}f(z)\) does not exist in \(\CC\) and \(\abs{f(z)}\not\to\infty\) as \(z\to z_0\).
If \(z_0\) is removable, \(f\) agrees near \(z_0\) with a holomorphic \(g\), so \(f(z)\to g(z_0)\in\CC\); if \(z_0\) is a pole, \(\abs{f(z)}\to\infty\) by the proposition on D-AUD6K. Conversely, if \(f(z)\) has a limit in \(\CC\), then \(f\) is bounded near \(z_0\) and \(z_0\) is removable by Riemann's removable singularity theorem; if \(\abs{f(z)}\to\infty\), then \(z_0\) is a pole by the proposition on D-AUD6K.
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FD-ZPKLQEssential singularityvariant-of