Delta Function - Gauge-institute.org

More documents

Recommendations

Info

<strong>Gauge</strong> Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon4.δ ( x)and the Fourier Transform4.1 F { x }δ ( ) = 1Proof:For any infinitesimal dx , the Integration Sum for thefunction−iωx 1 −iωx dx dxδ( xe ) = e χ[ − , ]dx2 2has only the unique hyper-real term1 −iωx[dx,dx −iωxe χ− ] dx = e χ[ −dx,dx].dx2 2 2 2Therefore, the Fourier Transformexists.F{ δ }x =∞−( x) = ∫ δ( x)e ωx =−∞i xdxSince−e − ] is bounded by 1, it is a finite hyper-real.i ω x dx dxχ[ ,2 2Therefore, the Fourier Transform equals to the constant part ofthis hyper-real.Since the constant hyper-real in[ − ,dx dx2 2] is zero, the constanthyper-real part of−e − ] isi ω x dx dxχ[ ,2 216
<strong>Gauge</strong> Institute Journal, Volume 8, No.2, May 2012H. Vic Dannoni ω 01e− = .That is{ x }F δ ( ) = 1.Consequently,4.2 δ( x)= the inverse Fourier Transform of the unit function 1ω=∞1=2π∫ν=∞ω=−∞e2πixiωxdω= ∫ e dν, ω = 2πνν=−∞Thus,4.312πω=∞∫ω=−∞eiωxdωx = 0=1 = an infinite hyper-realdxω=∞∫ω=−∞eiωxdωx ≠0=0Proof:1δ (0) = .dxδ( x) 0x≠ = . 170
Page 4: Gauge Institute Journal, Volume 8,
Page 9 and 10: Gauge Institute Journal, Volume 8,

Delta Function - Gauge-institute.org

Create successful ePaper yourself

Delete template?

Save as template?