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 Incoming links: Fourier transform of a two-sided exponential

Past exam of the short options course of the University of Oxford / 2017 / 5 / a / ii / Solution 2026-10-31

Split the integral at the origin and reflect the negative half:

e^{- a ∣ x ∣} (s) = \int_{0}^{\infty} e^{- (a + i s) x} d x + \int_{0}^{\infty} e^{- (a - i s) x} d x = \frac{1}{a + i s} + \frac{1}{a - i s} = \frac{2 a}{a ^{2} + s ^{2}} .

Thus the Fourier transform of a two-sided exponential is

e^{- a ∣ x ∣} (s) = \frac{2 a}{a ^{2} + s ^{2}} (a > 0) .

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