Math 2.0

Integral transforms, operational calculus Definition

Residue calculus for Laplace inversion

Builds on Meromorphic functionResidueResidue theoremBromwich integralLaplace transformInverse Laplace transform

Definition

Let FF be a meromorphic function on ℂ\C with finitely many poles s1,…,sns_1, \ldots, s_n. Let γ\gamma be a real number with γ>Re⁡sk\gamma > \operatorname{Re} s_k for every kk. Assume that there are constants C>0C > 0, p>0p > 0 and R0>0R_0 > 0 with ∣F(s)∣≤C∣s∣−p\abs{F(s)} \le C \abs{s}^{-p} for all ss with ∣s∣≥R0\abs{s} \ge R_0 and Re⁡s≤γ\operatorname{Re} s \le \gamma. Residue calculus for Laplace inversion is the evaluation, for t>0t > 0, of the Bromwich integral of FF along Re⁡s=γ\operatorname{Re} s = \gamma as 12πi∫γ−i∞γ+i∞estF(s) ds=∑k=1nRes⁡s=sk(estF(s)),\frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{s t} F(s) \, ds = \sum_{k=1}^{n} \operatorname{Res}_{s = s_k} \left( e^{s t} F(s) \right), where Res⁡\operatorname{Res} is the residue. Under these assumptions the identity holds for every t>0t > 0. Let FF also be the Laplace transform of a function ff as in Mellin's inverse formula. Then the right side is the inverse Laplace transform f(t)f(t) at each t>0t > 0 where ff is continuous.

Remarks

The identity follows from the residue theorem on the region bounded by the segment from γ−iR\gamma - i R to γ+iR\gamma + i R and a left arc of radius about RR. A Jordan-type estimate shows that the arc contributes 00 in the limit when t>0t > 0. The hypotheses hold for every rational F=P/QF = P/Q with deg⁡P<deg⁡Q\deg P < \deg Q. Example: F(s)=1/(s+1)2F(s) = 1/(s + 1)^2 has one double pole, and Res⁡s=−1est/(s+1)2=ddsest∣s=−1=te−t\operatorname{Res}_{s=-1} e^{s t}/(s+1)^2 = \frac{d}{ds} e^{s t} \big|_{s = -1} = t e^{-t}. Non-example: F(s)=e−s/sF(s) = e^{-s}/s is not small on left arcs, because ∣e−s∣=e−Re⁡s\abs{e^{-s}} = e^{-\operatorname{Re} s} grows there. The residue sum would give 11 for all t>0t > 0, while the true inverse is the delayed step u(t−1)u(t - 1), which is 00 for 0<t<10 < t < 1. Transforms that come from partial differential equations often have infinitely many poles. For them one uses a sequence of contours that avoid the poles, and one gets a series over all poles.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptFunction, concept, no evidence levelFunctionconceptReal 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…definitionLaplace transform, definition, no evidence levelLaplace transformdefinitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionWinding number, definition, no evidence levelWinding numberdefinitionLaurent series, definition, no evidence levelLaurent seriesdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementInverse Laplace transform, definition, no evidence levelInverse Laplace trans…definitionExistence 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 …statementBromwich integral, definition, no evidence levelBromwich integraldefinitionPole, definition, no evidence levelPoledefinitionResidue, definition, no evidence levelResiduedefinitionMeromorphic function, definition, no evidence levelMeromorphic functiondefinitionResidue theorem, statement, no evidence levelResidue theoremstatementResidue calculus for Laplace inversion, definition, no evidence levelResidue calculus for …definition
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.
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  1. Complex numbers object
  2. Complex plane concept
  3. Function concept
  4. Real numbers object
  5. Set bounded above definition
  6. Complex exponential definition
  7. Contour integral definition
  8. Power series in the complex plane definition
  9. Real and imaginary parts definition
  10. Laplace transform definition
  11. Supremum definition
  12. Circle integral definition
  13. Winding number definition
  14. Laurent series definition
  15. Completeness axiom statement
  16. Inverse Laplace transform definition
  17. Existence of square roots of nonnegative reals statement
  18. Square root definition
  19. Modulus of a complex number definition
  20. Complex differentiability definition
  21. Annulus definition
  22. Holomorphic function definition
  23. Isolated singularity definition
  24. Existence of Laurent expansions on an annulus statement
  25. Bromwich integral definition
  26. Pole definition
  27. Residue definition
  28. Meromorphic function definition
  29. Residue theorem statement
  30. Residue calculus for Laplace inversion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Residue_theorem
About this page
TagZZDMHH8V1K
URIm2://ZZDMHH8V1K
Kinddefinition
WritingW0 stub
MSC44A10, 30E20
DefinesResidue calculus for Laplace inversion
UsesMeromorphic function, Residue, Residue theorem, Bromwich integral, Laplace transform, Inverse Laplace transform

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