I know that there are three types of isolated singularities. Removable singularities , poles and essential singularity . Consider tan(z) function on $ \mathbb C $ . Then it has poles and an essential singularity at $ \infty$. But why $ \infty $ is isolated singularity here ? I'm confused because there are infinitely many poles near $ \infty $ . Can anyone help me to understand this?
isolated singularity
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complex-analysis
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0$\infty$ isn't an isolated singularity. Often, the term essential singularity denotes all singularities that aren't removable or poles, so in particular it includes all non-isolated singularities. – 2017-02-01
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0That means isolated singularity are classified by removable singularities and poles only? – 2017-02-01
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0No, there are also essential isolated singularities. But an essential singularity is not necessarily isolated. (Some people use the term "essential singularity" only for essential isolated singularities, others use it for essential isolated singularities and also for all non-isolated singularities.) – 2017-02-01