2.3 Laplace Transform Calculations

Problem 2.1

Calculate the inverse Laplace transform of the following function:

Solution 2.1

Let us consider the following Laplace transforms:

Their inverse Laplace transforms will be denoted by y ₁( t) and y ₂( t), respectively. It follows that:

Taking into account the fact that:

we derive:

Problem 2.2

Calculate the inverse Laplace transform of the following function:

Solution 2.2

The poles (natural modes) of this function are:

Let us now calculate the residues associated with these singularities. The residue related to the pole p ₁ = 0 is given by:

The residue associated with the pole p ₂ = j is:

Finally, the residue associated with the pole p ₃ = ? j is:

Therefore:

Problem 2.3

Calculate the inverse Laplace transform of the following function:

Solution 2.3

Let us calculate the roots of the denominator of this function. Observe that s = ?1 is a root. Therefore, we obtain:

Let us now determine the partial fraction expansion ^[5] of Y( s):

Multiplying both sides of this equation by ( s + 1) and setting s = ?1, we obtain:

Multiplying both sides of the equation by ( s + 2) and setting s = ?2, we obtain:

Now, proceeding as before for the pole s = ?5, we derive:

Finally, the partial fraction expansion of Y( s) is:

from which...