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

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Gauge Institute JournalH. Vic Dannon7.Poisson Sequence and δ ( ξ − x)Periodic7.1 Poisson Sequence DefinitionLetr = 1 − , k = 1, 2, 3,...k1kThe Sequence of Poisson Integrals at r = 11 − ,kkξ=1∫ξ=−1f211 − r() ξdξ2 12+ r − 2 r cos π ( x −ξ)r =− 11k==ξ=1∫ξ=−1k − 1f () ξdξ.2( kk− 1) + 1 − 2( kk− 1)cos πξ ( −x)Poisson Sequencegives rise to the Poisson SequenceP ( ξ − x)=kk − 12( kk− 1) + 1−2( kk−1)cos πξ ( − x).7.2 Poisson Sequence is a Periodic Delta Sequence andrepresents the Periodic Delta Function,δPeriodic( ξ− x) = ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) +...24
Gauge Institute JournalH. Vic DannonEachP ( x)k=k − 12( kk− 1) + 1−2( kk−1)cosπx, k = 1, 2, 3,...1. has the sifting property on each interval,x =−3∫.. P ( x) dx = 1; P ( x) dx = 1; P ( x) dx = 1..x =−52. is a continuous functionkx =−1∫x =−3kx = 1∫x =−13. peaks on each of these interval to lim P ( ξ − x ) = k −1.Proof of (1)x= 1 x=1ξ−x→2m1Pk( x) dx = ( k −1)dx2( kk− 1) + 1−2( kk−1)cosπx∫ ∫ x=− 1 x=−1By [Spiegel, p.78],p−q2⎡ p − q= ( k −1) arctan tana ( p − q)( p + q)⎢⎣p + qk12a⎤ax⎥⎦kx = 1x =−121= ( k −1) arctan[ 4 k( k − 1) + 1tan πx]2π 4( kk− 1) + 12k−11/ πx = 1x =−11 1= 1 {arctan[(2 k − 1)tan π ] − arctan[(2 1)tan( )]}2 k − ππ − 2= 1.Proof of (3)1π−12 2π25
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