Delta Function - Gauge-institute.org

More documents

Recommendations

Info

Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonconverges to the hyper-real function f ( x ).5.4 If f ( x ) is hyper-real function,Then,x =∞−iαx the hyper-real integral ∫ f ( xe ) dx converges to F( α)x =−∞α=∞ the hyper-real integral 1 iαxF( α)e d2π∫ α converges to f ( x )α=−∞The value of the Hyper-real Fourier Integral Theorem isdemonstrated by the following examples:5.5 The Hyper-real Delta δ( x)is not defined in the Calculus ofLimits, and is not Piecewise Continuous.δ'( x)is not defined, andis not Piecewise Continuous in any bounded interval.Butδ( x)satisfies the Hyper-real Fourier integral Theoremk =∞ ⎛ ξ=∞⎞1−ikξδ( x) = δ( ξ)eξ2πk =−∞ ⎜⎝ξ=−∞⎠⎟ikx∫ ∫ d e dkProof:By the sifting property of the Hyper-real Delta22
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonξ=∞−ikξ∫ δξ () e dξ=1.ξ=−∞Substituting this representation of 1 intoδ( x)k =∞1 ikx= e2π∫ dk ,k =−∞k =∞ ⎛ ξ=∞⎞1−ikξikxδ( x) = δ( ξ)e dξ e dk2π.k =−∞ ⎜⎝ξ=−∞⎠⎟∫ ∫ x =∞5.6 The Hyper-real f ( x) ≡ 1, does not satisfy ∫ fx ( ) dx
Page 4: Gauge Institute Journal, Volume 8,
Page 9 and 10: Gauge Institute Journal, Volume 8,

Delta Function - Gauge-institute.org

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?