Periodic Delta Function and Poisson Integral for - Gauge-institute.org
Periodic Delta Function and Poisson Integral for - Gauge-institute.org
Periodic Delta Function and Poisson Integral for - Gauge-institute.org
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<strong>Gauge</strong> Institute JournalH. Vic DannonThe <strong>Delta</strong> <strong>Function</strong>, the idealization of an impulse in Radarcircuits, is a Discontinuous Hyper-Real function which definitionrequires Infinite Hyper-reals, <strong>and</strong> which analysis requiresInfinitesimal Calculus.In [Dan5], we show that in infinitesimal Calculus, the hyper-realω=∞1( x)e i ωδ = x ω2π∫ dω=−∞is zero <strong>for</strong> any x ≠ 0 ,it spikes atx = 0 , so that its Infinitesimal Calculusx =∞∫integral is δ( xdx ) = 1,<strong>and</strong>x =−∞1δ (0) = < ∞.dxHere, we show that in Infinitesimal calculus, the <strong>Poisson</strong> Kernel isa periodic hyper-real <strong>Delta</strong> <strong>Function</strong>: A periodic train of <strong>Delta</strong><strong>Function</strong>s. And the <strong>Poisson</strong> <strong>Integral</strong> associated with a Hyper-realperiodic function f ( x ), atr = 1 −dr , equals f ( x ).12
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