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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


             ole.gif  


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


ole1.gif  

 

and 

 

2)        L[F '(t)] = sf(s) - F(0+) .


Also, by definition,


ole2.gif


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


ole3.gif


If we now take the limit in 2) as s ole4.gif , we get


ole5.gif

 

Then


             ole6.gif




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