The bookUnit 2 · Singularities and Residues34 / 83

Laurent series, classification of singularities, Casorati–Weierstrass

Why this is asked: Classify a singularity from the Laurent tail or from the behaviour of |f| nearby: bounded ⇒ removable, → ∞ ⇒ pole, neither ⇒ essential (and then the image of every punctured neighbourhood is dense).

Singularities — three types, three tests

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Classifying an isolated singularity in one lookinteractive

Removable, pole or essential — decided by the Laurent tail, with the test for each.

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Let $f(z) = \exp(z + 1/z), z \in \mathbb{C}$ ∖{0}. The residue of f at z = 0 is

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Open this in the full syllabus view · Unit 2