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Fourier integral representation

Short answer

Since e^{-|x|} is even, its Fourier integral is (2/π)∫_0^∞ cos(αx)/(1+α²) dα, giving the representation e^{-|x|} = (2/π)∫_0^∞ cos(αx)/(1+α²) dα.

Find the Fourier integral representation of f(x) = e^(−|x|).

The function e^(−|x|) is even, so its Fourier integral is f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos αv dv] cos αx dα.

∫₀^∞ e^(−v) cos αv dv = 1/(1 + α²)

Fourier integral representation

e^(−|x|) = (2/π) ∫₀^∞ cos αx / (1 + α²) dα, −∞ < x < ∞

The function is continuous everywhere, so the integral equals e^(−|x|) at every x. At x = 0 it gives ∫₀^∞ dα/(1 + α²) = π/2.

✓ Answer checked by Claude Opus 5.5 (high effort)

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