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A scratch or template page in the temp files of the Spectral Theory Book folder, dated 3.26.05 and initialed PhL. It carries a placeholder title and a note that page numbering is on in the template. The body shows a dispersion relation for X(ω) as a principal-value integral and its real-part form with Im X, labeled equation 31.3a. Some symbols are garbled in the extraction.
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1 This is the Title PhL 3.26.05
Note that page numbering is tu rned on in this template.
0 ∞
Pm
l(ρr) ∫∞0(ρr) ∫--∞0 dx
X(ω) = X(∞) + (1/iπ) ∫-- ∞
-∞ dω' X(ω')
ω'-ω
Re[X( ω)] = Re[X( ∞)] + (1/π) ∫-∞ ∞ dω' Im[X(ω ')]
ω'-ω ( 3 1 . 3 a )