Delta Function - Gauge-institute.org

Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic DannonSince Delta is the Inverse Fourier Transform of the function 1,δ(x− ξ ) equals the Integration Sumwhich vanishes at any x12πk =∞ik( x−ξ)∫ e dk,k =−∞≠ ξ , and equalsSubstituting in the Integration Sum for f ( x ),1dx at x = ξ .⎛ k⎞ ⎟f ( x) = f( ) ⎜ e d2⎟⎝⎠By changing the Summation order,ξ=∞ =∞1 ik( x ξ)ξ ⎜−∫ ξπ∫ k d .ξ=−∞⎜ k =−∞ ⎟k =∞ ⎛ ξ=∞⎞1−ikξikxf ( x) = f( ξ)e dξe dk2π.k =−∞ ⎜⎝ξ=−∞⎠⎟∫ ∫ Then, the Fourier transform of f ( x )x =∞−iαx∫ f ( xe ) dx,x =−∞converges to a hyper-real functionF( α)infinite hyper-reals, like the Delta Function.And the Inverse Fourier Transform of F( α)12πα=∞∫α=−∞−iαxF( α)e dα, some of its values may be21