Example 12.25.  Let  .  Find  .

Explore Solution 12.25.

Enter Y[s] and find its partial fraction expansion.

Traditionally. The partial fraction expansion can be obtained with the residue calculus.
The simple poles in the upper half plane occur at  .  First determine their residues.

Using the residue calculus, we construct the partial fraction expansion.

Use a table of Laplace transforms to solve for  .  

Aside. We can check this with Mathematica's  result using the LaplaceTransform package.

Alternately.  We could the methods of partial fraction expansion that were developed in Section 8.1.  
All we would need to do is consider complex terms in the denominator.

            

Where

              

It is useful to notice that Mathematica can easily find these residues using the built in command  .

Then it is easy to form with the calculation

It takes a bit of work to collect the two fractions involving the irreducible quadratics.

(c) 2006 John H. Mathews, Russell W. Howell