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Fourier cosine integral

Short answer

The Fourier cosine integral of e^(-ax) for x>0 is e^(-ax) = (2/π) ∫_0^∞ [a/(a²+ω²)] cos ωx dω. Equivalently, ∫_0^∞ cos ωx/(a²+ω²) dω = π e^(-ax)/(2a).

Find the Fourier cosine integral of f(x) = e^(−ax) for x > 0, a > 0.

For x > 0 the Fourier cosine integral is f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos ωv dv] cos ωx dω.

With f(v) = e^(−av): ∫₀^∞ e^(−av) cos ωv dv = a/(a² + ω²).

Answer:

e^(−ax) = (2/π) ∫₀^∞ [a / (a² + ω²)] cos ωx dω, x > 0, a > 0

(Equivalently ∫₀^∞ cos ωx/(a² + ω²) dω = π e^(−ax)/(2a).)

✓ Answer checked by Claude Opus 5.5 (high effort)

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