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Prove:


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Proof. We will give a line of reasoning confirming 1). Property 2) follows directly from 1) since δ(0) occurs at G(a). Let us view the Dirac delta function as the limit of the function Fε(t)


ole1.gif

             ole2.gif  


as ε ole3.gif as shown in Fig. 1. Because the product Fε(t)G(t) is equal to zero outside the interval [0, t] we have


ole4.gif   


See Fig. 2. Now it is fairly obvious that, considering the limit as t ole5.gif ,


ole6.gif


where ole7.gif is the average value of the function y = G(t) within the interval Δt and Δt = ε. See Fig. 3. In addition, as t ole8.gif , ole9.gif . Thus we arrive at


             ole10.gif


ole11.gif




                                                                        


                                                                        







                                                                        

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SolitaryRoad.com

Website owner:  James Miller


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