Math 2.0

Functions of a complex variable Definition

Residue

Builds on Isolated singularityExistence of Laurent expansions on an annulusCircle integral

Definition

Let c∈ℂc \in \mathbb{C}, let r>0r > 0, and let ff be holomorphic on the punctured disc 0<∣z−c∣<r0 < |z - c| < r, so that cc is an isolated singularity of ff. Let f(z)=∑n∈ℤan(z−c)nf(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n be the Laurent expansion of ff on this punctured disc. The residue of ff at cc is the coefficient Res⁡(f,c)=a−1.\operatorname{Res}(f, c) = a_{-1}.

Remarks

For each ρ\rho with 0<ρ<r0 < \rho < r, Res⁡(f,c)=12πi∮∣z−c∣=ρf(z) dz\operatorname{Res}(f, c) = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} f(z) \, dz. This follows from the coefficient formula for n=−1n = -1. The residue is 00 at a removable singularity. It can be 00 at a pole too: Res⁡(1/z2,0)=0\operatorname{Res}(1/z^2, 0) = 0. For example, Res⁡(1/z,0)=1\operatorname{Res}(1/z, 0) = 1 and Res⁡(e1/z,0)=1\operatorname{Res}(e^{1/z}, 0) = 1. Residues at simple poles are easy to compute, see residue at a simple pole.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionLaurent series, definition, no evidence levelLaurent seriesdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionAnnulus, definition, no evidence levelAnnulusdefinitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionIsolated singularity, definition, no evidence levelIsolated singularitydefinitionExistence of Laurent expansions on an annulus, statement, no evidence levelExistence of Laurent …statementResidue, definition, no evidence levelResiduedefinition
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  1. Complex numbers object
  2. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Contour integral definition
  7. Power series in the complex plane definition
  8. Real and imaginary parts definition
  9. Supremum definition
  10. Circle integral definition
  11. Laurent series definition
  12. Completeness axiom statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Complex differentiability definition
  17. Annulus definition
  18. Holomorphic function definition
  19. Isolated singularity definition
  20. Existence of Laurent expansions on an annulus statement
  21. Residue definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Residue_(complex_analysis), Definition
  2. Web https://en.wikipedia.org/wiki/Residue_(complex_analysis), Generalization to Laurent series
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Kinddefinition
WritingW1 stated
MSC30E20
DefinesResidue
UsesIsolated singularity, Existence of Laurent expansions on an annulus, Circle integral

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