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

Gauge Institute JournalH. Vic Dannonξ− x = 2m ⇒ cos π( ξ − x)= 1,21r12 221− 1+r⇒ =1+ r −2r cos πξ ( −x)1 − r,11+1⎯⎯⎯→ =r ↑1 20 +∞Thus, the Poisson Integral diverges in the Calculus of Limits.Avoiding the singularity atξ = x , by using the Cauchy PrincipalValue of the integral does not recover the Theorem, because at anyξ −x≠2m,2 211−r11−r=2 2 2 2πξ1221+ r −2r cos ( −x) (1 − r) + 4r sin [ πξ ( −x)]0⎯⎯⎯→ =2sin [ πξ ( − x)]r ↑1 2 120That is, the Poisson Kernel vanishes, and the integral isidentically zero, for any function f ( x ).Thus, the Poisson Integral Theorem does not hold in the Calculusof Limits.10.2 Calculus of Limits Conditions are irrelevant toPoisson Integral TheoremProof:The Poisson Conditions are34