Question

Suppose f (t) is of exponential order c where F (s) is its

Laplace transform. If a ∈ R+, show that the Laplace transform of f
(at) is 1/a F (s/a)

Answer 1

The Laplace transform of f(at) can be found by using a variable substitution in the Laplace integral definition. The substitution leads to a scaled Laplace transform of the original function, namely 1/a F(s/a).

Step-by-step explanation:

When f(t) is of exponential order c and F(s) is its Laplace transform, for a ∈ R+, the Laplace transform of f(at) is 1/a F(s/a). We start with the definition of the Laplace transform, which for f(at) is:

• ℒ{f(at)} = ∫₀¹ ℓ f(at)e-stdt

To find the Laplace transform of f(at), we make a substitution u = at, which implies du = a dt. We adjust the limits of integration accordingly and divide by a, yielding:

• ℒ{f(at)} = (1/a) ∫adjusted limits ℓ f(u)e-(s/a)udu

This integral is the definition of F(s/a), so we can conclude that ℒ{f(at)} = 1/a F(s/a), completing the proof.