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Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonsequence 2 n. Once we determined the basic infinitesimaldx , we will use it in the Infinite Riemann Sum that definesan Integral in Infinitesimal Calculus.3. The Delta Function is strictly smaller than ∞4. We define,1χ δ ( x) ≡ dx ( ),dx xdx⎡ ⎤ ,⎢−⎣ 2 2 ⎥⎦whereχ ⎡⎢−⎣dx,dx2 2⎧ dx dx1, x ∈ ⎡−, ⎤( x)= ⎪ ⎢ 2 2 ⎥⎨ ⎣ ⎦ .⎪⎪ 0, otherwise⎩⎤⎥⎦5. Hence, for x < 0 , δ ( x) = 0 at fordxx =− , δ( x)jumps from 0 to2dx dx⎢ ⎣,2 2 ⎥ ⎦ , 1( x)x ∈ ⎡−⎤δ = .dx1dx , at x = 0 ,δ (0) =1dx atdxx = , δ( x)drops from2 for x > 0 , δ ( x) = 0. xδ ( x) = 01dx to 0.6. If dx =1, ( x) = 1 1( x),2 1 1( x),3 1 1( x )...n[ − , ] [ − , ] [ − , ]δ χ χ χ2 2 4 4 6 67. If dx =2,n1 2 3δ ( x) = , , ,...2 2 22 cosh x 2 cosh 2x 2 cosh 3x14
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon8. If dx =1,n− x − 2x − 3x[0, ∞) [0, ∞) [0, ∞)δ( x) = e χ ,2 e χ , 3 e χ ,...x =∞∫9. δ( xdx ) = 1.x =−∞k =∞1 ik( x−ξ)10. δ( x − ξ)= e2π∫ dkk =−∞15
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