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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)] = s*f*(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

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