Math 2.0

Functions of a complex variable Proposition

Residue at a simple pole

Builds on ResidueOrder of a pole

Statement

Let c∈ℂc \in \mathbb{C}, let r>0r > 0, and let ff be holomorphic on 0<∣z−c∣<r0 < |z - c| < r. Assume that ff has a simple pole at cc, that is, a pole of order 11. Then Res⁡(f,c)=lim⁡z→c(z−c)f(z),\operatorname{Res}(f, c) = \lim_{z \to c} (z - c) f(z), where Res⁡(f,c)\operatorname{Res}(f, c) is the residue of ff at cc. In particular, let f=g/hf = g / h, where gg and hh are holomorphic near cc, g(c)≠0g(c) \neq 0, h(c)=0h(c) = 0 and h′(c)≠0h'(c) \neq 0. Then Res⁡(f,c)=g(c)h′(c)\operatorname{Res}(f, c) = \frac{g(c)}{h'(c)}.

Remarks

Proof: near cc, f(z)=a−1z−c+a0+a1(z−c)+⋯f(z) = \frac{a_{-1}}{z - c} + a_0 + a_1 (z - c) + \cdots, so (z−c)f(z)→a−1(z - c) f(z) \to a_{-1}. For example, Res⁡(1z2+1,i)=12i\operatorname{Res}\left(\frac{1}{z^2 + 1}, i\right) = \frac{1}{2i}, with g=1g = 1 and h=z2+1h = z^2 + 1. At a pole of order mm, the formula is Res⁡(f,c)=1(m−1)!lim⁡z→cdm−1dzm−1((z−c)mf(z))\operatorname{Res}(f, c) = \frac{1}{(m-1)!} \lim_{z \to c} \frac{d^{m-1}}{dz^{m-1}} \left( (z - c)^m f(z) \right).

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

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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 integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionLaurent 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 functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementIsolated singularity, definition, no evidence levelIsolated singularitydefinitionZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definitionExistence of Laurent expansions on an annulus, statement, no evidence levelExistence of Laurent …statementPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementPole, definition, no evidence levelPoledefinitionHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementResidue, definition, no evidence levelResiduedefinitionHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementOrder of a zero, definition, no evidence levelOrder of a zerodefinitionOrder of a pole, definition, no evidence levelOrder of a poledefinitionResidue at a simple pole, statement, no evidence levelResidue at a simple p…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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. Disc of convergence definition
  12. Laurent series definition
  13. Completeness axiom statement
  14. Existence of square roots of nonnegative reals statement
  15. Square root definition
  16. Modulus of a complex number definition
  17. Complex differentiability definition
  18. Annulus definition
  19. Holomorphic function definition
  20. Cauchy's integral formula statement
  21. Isolated singularity definition
  22. Zero of a holomorphic function definition
  23. Existence of Laurent expansions on an annulus statement
  24. Power series are holomorphic in the disc of convergence statement
  25. Pole definition
  26. Holomorphic functions are analytic statement
  27. Residue definition
  28. Holomorphic functions are infinitely differentiable statement
  29. Order of a zero definition
  30. Order of a pole definition
  31. Residue at a simple pole statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Residue_(complex_analysis), Simple poles
About this page
Tag01991NCP7H
URIm2://01991NCP7H
Aliasesresidue-formula-for-simple-poles
Kindstatement, proposition
WritingW0 stub
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 01991NCP7H.
UsesResidue, Order of a pole

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