Fourier integral representation

For f(x) defined on (−∞, ∞), the Fourier integral is f(x) = (1/π)∫₀^∞ ∫₋∞^∞ f(v) cos α(x − v) dv dα. Because the given f is even, this reduces to f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos αv dv] cos αx dα.

Inner integral. f(v) = 1 for |v| < 1 and 0 otherwise:

∫₀^∞ f(v) cos αv dv = ∫₀¹ cos αv dv = sin α / α

Representation

f(x) = (2/π) ∫₀^∞ (sin α cos αx / α) dα

This equals 1 for |x| < 1, 0 for |x| > 1, and 1/2 at x = ±1 (the average at the jump).

Putting x = 0 gives the useful result ∫₀^∞ (sin α / α) dα = π/2.