Periodic Delta Function and Poisson Integral for - Gauge-institute.org

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Gauge Institute JournalH. Vic Dannon= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) + ...= δ ( ξ − x).Periodic8.5.. + δξ ( − x + 4) + δξ ( − x + 2) + δξ ( − x) + δξ ( −x − 2) + δξ ( −x− 4) + .. ==dr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)30
Gauge Institute JournalH. Vic Dannon9.Poisson Integral and δ ( ξ − x)Periodic9.1 Poisson Integral of a Hyper-real Function f ( x )Let f ( x ) be a hyper-real function integrable on [ −1,1], so thatf (1) = f ( − 1) .Then, for eachn = ..., −3, −2, −1, 0,1,2, 3,... , the integrals12ξ=1∫ξ=−1−inπξf () ξ e dξ≡ c nexist, [Dan8], with finite, or infinite hyper-real values. Thec narethe Fourier Coefficients off ( x ).The Fourier Series associated with f ( x ) isi( −n) πx i( −1) πx i(1) πx i( n)πx−1 0 1n.. + c e + .. + c e + c + ce + .. + c e + .. =−nIt may equal The Poisson Integral associated with f ( x ) atr= 1 −dr,PoissonSξ=121 1 − r{()} ≡ ∫ () ξ2 21 1 + r − 2 r cos πξ ( −xξ=−)fx f dξr=−1 dr=ξ=1∫ξ=−1dr(2 − dr)f () ξ2(1 −dr )(1 − cos πξ [ − x ]) + ( dr )12 2dξFor each x , it may assume finite or infinite hyper-real values.31
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Periodic Delta Function and Poisson Integral for - Gauge-institute.org

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