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Fourier cosine transform

Short answer

F_c(ω) = ∫_0^a cos ωx dx = sin(aω)/ω.

Determine the Fourier cosine transform of f(x) = 1 for 0 < x < a; 0 otherwise.

The Fourier cosine transform of f(x) on x > 0 is F_c(ω) = ∫₀^∞ f(x) cos ωx dx.

For f(x) = 1 on 0 < x < a and 0 for x > a:

F_c(ω) = ∫₀^a cos ωx dx = [ sin ωx / ω ]₀^a

Answer: F_c(ω) = sin(aω) / ω

In the symmetric (normalised) convention, F_c(ω) = √(2/π) · sin(aω)/ω. The inverse is f(x) = (2/π)∫₀^∞ [sin(aω)/ω] cos ωx dω.

✓ Answer checked by Claude Opus 5.5 (high effort)

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