6.3. Proof of the Theorem

Try these first

1. 1.

   What is the definition of the Laplace Transform?

   Hint

   What does it do to a time domain function?

2. 2.

   What theorem allows the interchange of integration order?

   Hint

   Recall its significance in multivariable calculus.

3. 3.

   What is the formula for the Laplace Transform of an integral?

   • L{∫f(τ)dτ} = F(s)
   • L{∫f(τ)dτ} = F(s)/s
   • L{∫f(τ)dτ} = s*F(s)

   Hint

   Recall how integration affects the transformation.

4. 4.

   True or False: The Laplace Transform can only be applied to continuous functions.

   • True
   • False

   Hint

   Think about the types of functions we defined earlier.

5. 5.

   Prove the theorem that L{f(t)} = F(s) implies L{∫f(τ)dτ} = F(s)/s by evaluating in detail each step including usage of Fubini's theorem.

   Hint

   Carefully track changes in limits during integration and remind yourself of the definition throughout.

6. 6.

   How would you apply the theorem to analyze a capacitor charging described by a differential equation?

   Hint

   Remember to define your variables clearly before applying any transforms.