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

Gauge Institute JournalH. Vic Dannon9.2 P S{ x }Proof:δ ( ξ − ) = δ ( ξ −x)oisson Periodic Periodicξ = 11dr(2 − dr)P S{ δ ( ξ − x) } = ∫ δ ( ξ) dξ2 2(1 )(1 cos [ ]) ( )2−dr − πξ− x + droisson Periodic Periodicξ=−1ξ = 11dr(2 − dr)= ∫ {.. + δξ ( + 2) + δξ ( ) + δξ ( − 2) + ..} dξ2 2(1 )(1 cos [ ]) ( )2−dr − πξ− x + drξ=−11dr(2 − dr)= ... + +2 22(1 −dr)(1 − cos π[2 + x]) + ( dr)1dr(2 − dr)+ +2 22(1 −dr)(1 − cos πx) + ( dr)1dr(2 − dr)+ + ..2 22(1 −dr)(1 −cos π[2 − x]) + ( dr)= ... + .. + δ( x + 2) + δ( x) + δ( x − 2) + ..= δ Periodic( x ) . 32