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
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Gauge</strong> Institute JournalH. Vic DannonFor ξ − x = 2m,k −1 k −1=2 kk ( −1)[1 −cos π( x− ξ)] + 1 2 kk ( −1)[1 − cos( π2 m)] + 1= k −1→ ∞.k→∞8.3 Hyper-real <strong>Poisson</strong> Kernel in Infinitesimal Calculus<strong>Poisson</strong>⎧ k 1 , ξ x 2( ξ x)⎪− − = m− = ⎨ .⎪ 0, ξ − x ≠ 2m⎪⎩Proof: At any ξ − x = 2m, the Kernel is an infinite hyper-real.At any ξ −x≠ 2m,k − 1 1= =2( kk−1)[1−cos πξ ( − x)] + 1 2[1 k −cos πξ ( − x)]+1k−1infinitesimal .8.4 Let1N = be an infinite Hyper-real, Thendr<strong>Poisson</strong>( ξ − x)==211 − r2 21−2r cos πξ ( − x)+ rr=−1 drdr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) + ...= δ ( ξ −x).periodicProof:28
Change language