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

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Gauge Institute JournalH. Vic DannonAs ξ −x→ 2m,k −1 k −1→ =2 kk ( − 1) + 1−2 kk ( −1)cosπx 2 kk ( − 1) + 1−2 kk ( −1)k− 1.7.3 Poisson Sequence Represents δ ( ξ − x)PeriodicδPeriodic( ξ − x)=k − 12( kk−1)[1−cos πξ ( − x)] + 126
Gauge Institute JournalH. Vic Dannon8.Poisson Kernel and δ ( ξ − x)Periodic8.1 Poisson Kernel in the Calculus of LimitsThe Sequence of Poisson Integrals at1r = 1 −kkξ=1 211 − rk∫ f ξdξk→∞ 2 2r1krkπ x ξξ =−+ − −r =− 11lim ( )1 2 cos ( )k k==ξ=1∫ξ=−1k − 1f () ξdξ.2( kk− 1) + 1 − 2( kk− 1)cos π ( x−ξ)Poisson Sequencegives rise to a Poisson Sequence.Alternatively, we could use r = 1 − 1, r = 1 − 1,…23kkkkNote that the hyper-realr =1 −drincludes all such sequences.The limit of the Poisson Sequence is the Poisson Kernelk − 1klim→∞ 2( kk− 1)[1 − cos π ( x− ξ )] + . 18.2 In the Calculus of Limits, the Poisson Kernel does not haveProof:the sifting property27
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