Example 12.24.  Let  .  Find  .

Explore Solution 12.24.

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

Traditionally. The partial fraction expansion can be obtained with the residue calculus.
  
At  s = 1, we compute    as follows.

At  s = 0, we compute    as follows.

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 use the partial fraction expansion that was developed in Section 8.1.

            

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

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