DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The residue of an isolated singularity

Definition

Let $f$ have an isolated singularity at $a$ . By Laurent expansion on an annulus, on every sufficiently small punctured disc about $a$ it has a Laurent expansion

$f (z) = \sum_{n \in Z} c_{n} (z - a)^{n} .$

By Laurent coefficients are given by contour integrals and are unique, the coefficient $c_{- 1}$ is uniquely determined on that disc, and Laurent coefficients are independent of the intermediate radius shows that shrinking the disc does not change it. It is the residue of $f$ at $a$ , written

$Res (f, a) : = c_{- 1} .$

Remarks

The residue depends only on the singularity of $f$ at $a$ , not on which small punctured disc is used to write the Laurent series.

Depends on

Used by

The residue is the normalized small-circle integral Corollary
Zero residue does not imply a removable singularity Counterexample
At a simple pole the residue is the limit of (z-a)f(z) Lemma
Residue formula for a pole of order m Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Baptiste Campesato, MAT334 course page and notes index (standard reference, not scraped)
Jeremy Orloff, MIT 18.04 Topic 7: Taylor and Laurent Series (standard reference, not scraped)