Proving the Vanishing Integral in Inverse Laplace Transform by Residue Method

[NT123]

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.

 

[Count Iblis]

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.