Website owner:  James Miller

Prove: If L[F(t)] = f(s), then

Proof. Let

Then

1)        G'(t) = F(t)

and G(0) = 0. Taking the Laplace transform of both sides gives

2)        L[G'(t)] = L[F(t)] = f(s)

Also, from the formula for the transform of a derivative we have

3)        L[G'(t)] = sL[G(t)] - G(0) = sL[G(t)]

From 2) and 3) we get

4)        sL[G(t)] = f(s)

Thus

Website owner:  James Miller