Website owner:  James Miller

Prove: Initial-value theorem. Let L[F(t)] = f(s) and let F(t) and F '(t) both be piecewise regular and of exponential order. Then

Proof. This theorem is valid whether F(t) is continuous at t = 0 or not. In either case

and 

2)        L[F '(t)] = sf(s) - F(0⁺) .

Also, by definition,

Now because F '(t) is piecewise regular and of exponential order (by definition of exponential order )

If we now take the limit in 2) as s , we get

Then

Website owner:  James Miller