Section 34 special case 2
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Word-document draft dated 3.26.05 (initialed PhL), apparently material incorporated into Phil's Spectral Theory book. It starts from general results (34.18) and (34.20) for X(ω) and P(ω) as sums of delta functions at ω = mω1/2 with coefficients A and B. It then sets A=1, B=0 and simplifies using sinc functions of a pulse of width T1. Some summation symbols were lost in extraction.
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This is the Title PhL 3.26.05
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The general results are
X(ω) = Xpulse(ω) (1/2) ω1!Syntax Error, I [ A + B(-1)m ] δ(ω - mω1/2) . (34.18)
P(ω) = Ppulse(ω) (1/4) ω1!Syntax Error, I { |A|2 + |B|2 + 2Re(AB)(-1)m } δ(ω - mω1/2) (34.20)
For A = 1 and B = 0 these become
X(ω) = Xpulse(ω) (1/2) ω1!Syntax Error, I δ(ω - mω1/2)
P(ω) = Ppulse(ω) (1/4) ω1!Syntax Error, I δ(ω - mω1/2)
Xpulse(ω) = T1 sinc(ωT1/2)
Ppulse(ω) = |Xpulse(ω)|2/ (2πT1) = (1/ω1) sinc2(ωT1/2)
X(ω) = T1 (1/2) ω1!Syntax Error, I sinc(mπ/2)δ(ω - mω1/2)
P(ω) = (1/ω1) (1/4) ω1!Syntax Error, I sinc2(mπ/2) δ(ω - mω1/2)
X(ω) = T1 (1/2) ω1!Syntax Error, I (mπ/2)-1δ(ω - mω1/2) + T1 (1/2) ω1 δ(ω)
P(ω) = (1/ω1) (1/4) ω1!Syntax Error, I (mπ/2)-2 δ(ω - mω1/2) + (1/ω1) (1/4) ω1 δ(ω)
X(ω) = π!Syntax Error, I (mπ/2)-1δ(ω - mω1/2) + π δ(ω)
P(ω) = (1/4)!Syntax Error, I (mπ/2)-2 δ(ω - mω1/2) + (1/4) δ(ω)
X(ω) = 2!Syntax Error, I (m)-1δ(ω - mω1/2) + π δ(ω)
P(ω) = (1/π2)!Syntax Error, I (m)-2 δ(ω - mω1/2) + (1/4) δ(ω)