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Prove. Final value theorem. Let L[F(t)] = f(s). If F(t) and F '(t) are both piecewise regular and of exponential order, and if the abscissa of convergence of F '(t) is negative, then

provided these limits exist.

Proof. This theorem is valid whether F(t) is continuous at t = 0 or not. For generality we will assume that it is discontinuous at t = 0.

First note that

Let us now take the limit as s 0.

Now

Substituting 3) into 2) we have

or

Website owner:  James Miller