- #1

roam

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## Homework Statement

Given the Laplace transform

$$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$

by using the

*complex inversion formula*compute the inverse Laplace transform, ##f(t),## for the following regions of convergence:

**(i)**##Re(s)<-2;##

**(ii)**##-2<Re(s)<0;##

**(iii)**##Re(s)>0.##

## Homework Equations

Inverse Laplace transform relationship:

$$f(t) = \frac{1}{j2\pi} \int^{\sigma + j \infty}_{\sigma-j\infty} F_L (s) \exp(st) \ ds \tag{1}$$

Where ##s=\sigma + j \omega,## and ##\sigma## must be chosen to lie within the region of absolute convergence of ##F_L.##

## The Attempt at a Solution

So, using equation

**(1)**, how do I exactly choose the values of ##\sigma## for each case? I am very confused about this part.

I tried to solve this without the complex inversion formula (just to see what the solution has to look like). I started out by expanding using partial fractions as:

$$F_L(s) = \frac{1}{(s+2)(s^2+4)} = \frac{1}{8(s+2)} + \frac{1}{8(s^2 +4)}$$

There is a pole at ##s=-2## due to the first term. The first term has the form ##1/(s-a),## so its transform can be written as ##\frac{1}{8} e^{-2t}.## However I am unable to proceed further because I don't see in Laplace transform tables what the transform of the form ##1/(s^2 +a)## looks like.

Any help would be appreciated.