Delta Function - Gauge-institute.org

Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon6.2 Cosine representation of the Fourier IntegralTheoremω=∞ ξ=∞1f ( x) = f( ξ)cos ω( x −ξ)ξπ∫ ∫ d dωω= 0 ξ=−∞ν=∞ ξ=∞∫ ∫ ν, ω = 2πν= 2 f ( ξ)cos2 πν( x −ξ)dξdξ=∞ν= 0 ξ=−∞Proof: f ( x) = f( ξδ ) ( x −ξ)d ξ∫ξ=−∞ξ=∞ ⎧ω=∞⎫1= f () ξ ⎪⎨ cos ω( x −ξ)d ⎪⎬dπξ=−∞ ⎪⎩ω=0⎪⎭∫ ∫ ω ξω=∞ ξ=∞= 1f ()cos ξ ω( x ξ)dξdωπ∫ ∫− .ω= 0 ξ=−∞25