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The residue of an isolated singularity
Definition
Let have an isolated singularity at . By Laurent expansion on an annulus, on every sufficiently small punctured disc about it has a Laurent expansion
By Laurent coefficients are given by contour integrals and are unique, the coefficient 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 at , written
Remarks
The residue depends only on the singularity of at , 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
- Meromorphic differentials, orders and residues Definition
- Orders and residues under inversion on the sphere Example
- An indented arc around a simple singularity contributes the expected residue fraction Lemma
- At a simple pole the residue is the limit of (z-a)f(z) Lemma
- Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions Theorem
- Divisor and residue laws for elliptic functions Theorem
- 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)