Cosine transform of e^{-mx}; evaluate integral

Definition. F_c(s) = √(2/π) ∫₀^∞ f(x) cos sx dx, and the inversion formula is f(x) = √(2/π) ∫₀^∞ F_c(s) cos sx ds.

Cosine transform of e^{−mx}

Using ∫₀^∞ e^{−mx} cos sx dx = m/(m² + s²):

F_c(s) = √(2/π) · m/(m² + s²)

(Derivation: integrating I = ∫₀^∞ e^{−mx} cos sx dx by parts twice gives I = m/(m² + s²).)

Deducing the integral

By the inversion formula, for x > 0:

e^{−mx} = √(2/π) ∫₀^∞ [ √(2/π) m/(m² + s²) ] cos sx ds
        = (2m/π) ∫₀^∞ cos sx /(s² + m²) ds

So ∫₀^∞ cos sx/(s² + m²) ds = (π/2m) e^{−mx}.

Now rename: x → p, s → y and m → β (β > 0, p > 0):

∫₀^∞ cos py /(y² + β²) dy = (π/(2β)) e^{−pβ} ∎