12.3.4. Heaviside’s Expansion Formula (for distinct poles)

Try these first

1. 1.

   Identify the poles of F(s) = (5s + 10) / (s^2 - 5s + 6).

   Hint

   Set the denominator equal to zero and solve for 's'.

2. 2.

   What does the term P(a_i) represent in Heaviside's formula?

   Hint

   Think of it as the output of the polynomial when you substitute the pole.

3. 3.

   What does Heaviside's Expansion Formula help calculate?

   • Direct Laplace Transform
   • Inverse Laplace Transform
   • Fourier Transform

   Hint

   Think about the function we retrieve from the frequency domain.

4. 4.

   If F(s) = (3s + 4) / ((s - 1)(s - 2)), what are the poles?

   Hint

   Factor the denominator to find the roots.

5. 5.

   Using Heaviside's expansion, derive the inverse Laplace transform of F(s) = (2s + 3) / ((s - 1)(s - 2)^2). Consider both poles in your solution.

   Hint

   Remember to account for the double pole by differentiating.

6. 6.

   Explain why Heaviside's formula is not applicable to rational functions with non-distinct poles. Create an example to illustrate your point.

   Hint

   Reflect on how distinct effects help simplify.