Proving the Vanishing Integral in Inverse Laplace Transform by Residue Method

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SUMMARY

The discussion focuses on finding the inverse Laplace transform of the function s/(s^2+a^2) using the residue method. The key challenge is to demonstrate that the integral over the contour, excluding the straight line from -R to R, approaches zero as R approaches infinity. Participants emphasize the necessity of applying techniques similar to those in the proof of Jordan's theorem, particularly considering the behavior of the integrand, which resembles 1/s, and the proportionality of the circle's radius to s.

PREREQUISITES
  • Understanding of inverse Laplace transforms
  • Familiarity with the residue theorem in complex analysis
  • Knowledge of Jordan's theorem and its applications
  • Basic concepts of contour integration
NEXT STEPS
  • Study the residue theorem in detail
  • Explore Jordan's theorem and its implications for contour integrals
  • Practice calculating inverse Laplace transforms using various functions
  • Investigate the behavior of integrals at infinity in complex analysis
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Students and professionals in mathematics, particularly those studying complex analysis, as well as anyone involved in solving problems related to Laplace transforms and contour integration.

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Homework Statement

I need to find the inverse Laplace transform of s/(s^2+a^2), where a is a constant, by the method of residues. I need to prove the part of the contour not actually relating to the desired integral tends to zero as R---> infinity.



Homework Equations





The Attempt at a Solution

I have calculated the residues, however I am having trouble proving the integral of the contour which isn't the straight line from -R to R vanishes. Any help would be appreciated.
 
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You need to use the same methods here that are used in the proof of Jordan's theorem. The large s behavior is like 1/s, while the radius of the circle will be proportional to s, so a more straightforward way of estimating the integrand won't do.
 

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