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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


             ole.gif  


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


ole1.gif


Let us now take the limit as s ole2.gif 0.


ole3.gif


Now


ole4.gif


                                                 ole5.gif


Substituting 3) into 2) we have


             ole6.gif


or


             ole7.gif  



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