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Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon9.Delta Sequenceδn ( x )n⎛sin( nx)⎞= ⎜π ⎜⎝ nx ⎠⎟2We show that the Hyper-real Delta Function is represented by theDelta Sequencesin( )n ( ) n⎛nx ⎞δ x = ⎜π ⎜⎝ nx ⎟⎠ .The n th component of the Hyper-real Delta is2n⎛sin( nx)⎞ ⎜. That is,π ⎜⎝ nx ⎠ ⎟2δ( x)n⎛sin( nx)⎞= ⎜π ⎜⎝ nx ⎠⎟2.9.1 Eachsin( )n ( ) n⎛nx ⎞δ x = ⎜π ⎜⎝ nx ⎟⎠• has the sifting property δ ( xdx ) = 1,2x =∞∫x =−∞• is a continuous Hyper-real function,n• peaks atx = 0 to1δ (0) = n .nπProof:30
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonx =∞ x =∞2 x =∞2n sin ( nx) 1 sin ( nx)δ ( xdx ) = dx=2 dx= 1.∫ ∫ ∫nπ2 2 2nx πnxx=−∞ x=−∞ x=012πn,by ⎡Spiegel2⎤⎣ ⎦To see that1δ (0) = n , we use the infinitesimal2nπ1n. Then,δ211 n⎛sin( ) ⎞ nn( ) = n2π⎜ 1⎜⎝ ⎟n ⎠.sin( ο)Since for any infinitesimal ο , = 1, we haveο1 nδ n( ) = ,n2πand1δ (0) = n .nπTherefore,9.2 The sequence represents the Hyper-real Delta2 2 2sin ( x) sin (2 x) sin (3 x)δ( x) = , , ,... .2 2 2πx 2πx 3πx9.3plots in Maple the 1000 th component, that peaks at 1000 318π ≈ .31
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