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

Proof. By definition

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

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

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

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

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

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