Laplace transform of derivative/shift property

First shifting theorem. If L{f(t)} = F(s), then L{e^(at) f(t)} = F(s − a).

Proof. By definition of the Laplace transform,

L{e^(at) f(t)} = ∫₀^∞ e^(−st) e^(at) f(t) dt
               = ∫₀^∞ e^(−(s − a)t) f(t) dt

The last integral is the same integral that defines F(s), with s replaced by (s − a). Hence

L{e^(at) f(t)} = F(s − a)

This is valid for s − a > k, where f is of exponential order k (|f(t)| ≤ M e^(kt)), so that F(s − a) exists. ∎

Examples.

• L{e^(at)} = 1/(s − a), since L{1} = 1/s.
• L{e^(−2t) cos 3t} = (s + 2)/((s + 2)² + 9), since L{cos 3t} = s/(s² + 9).
• L{e^(3t) t²} = 2/(s − 3)³.