13.1.5. Properties of Convolution

Try these first

1. 1.

   Define the term convolution in the context of Laplace Transforms.

   Hint

   Think about how functions interact over time.

2. 2.

   What property states that (𝑓 ∗ 𝑔)(𝑡) = (𝑔 ∗ 𝑓)(𝑡)?

   Hint

   Reflect on the order of operations.

3. 3.

   What is the definition of convolution?

   Hint

   Think about integrating products over an interval.

4. 4.

   True or False: The convolution of two functions is non-commutative.

   • True
   • False

   Hint

   Recall the commutative property you learned earlier.

5. 5.

   Find the convolution of two functions 𝑓(𝑡) = e^{-at} and 𝑔(𝑡) = e^{-bt}. Show all steps.

   Hint

   Set up your integral and consider how each function behaves over time.

6. 6.

   Explain and derive the inverse Laplace transform for 1/(s^2 + 1) using convolution.

   Hint

   Look for pairs of known Laplace transforms that will yield the function when combined.