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Prove: The Complex Inversion Formula. If L[F(t)] = f(s), then


ole.gif



Proof. By definition


ole1.gif


We now substitute 2) into 1). This should give F(t).


ole2.gif



                                                                         ole3.gif



We now make the substitution s = α + iy, ds = iy. This gives


ole4.gif


We now appeal to the Fourier Integral Theorem. It states:


ole5.gif


Comparing 4) to 5) we see that the variable y in 4) corresponds to ω in 5) and the quantity e-αuF(u) shown in brackets in 4) corresponds to f(u) in 5). Thus by the Fourier Integral Theorem the quantity


             ole6.gif


in 4) is equal to e-αuF(t) for an F(t) defined for t > 0 as in the case of a Laplace transform. Thus 4) becomes


ole7.gif  


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