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Using Fourier inversion with the stated convention,Hence applying the transform twice gives the duality property of the Fourier transform:The formula holds directly for rapidly decreasing smooth functions, and extends through the usual L2 space or tempered distribution formulations when absolute convergence is unavailable.

Set $\alpha =1$ in part (d), obtaining $\mathcal{F}[e^{-|t|}](\omega )=2/(1+{\omega }^2)$. Apply the duality property of the Fourier transform from part (a):Therefore