Manuscript XV-B@2 · XV-B.tex · section · 4d3f12e8

The fixed internal pole

The retained target and the available inherited formulas › The fixed internal pole · lines 1906–1934

XV-B@2#eq:pole

For $s \in {+ 1, - 1}$ , let

ρ > 0, λ_{s} = ω - s ρ, ∣ λ_{s} ∣ < c ρ, 0 < c < 1.

The supplied retarded multiplier has the exact decomposition

M_{R} = - \frac{s}{2 ρ} (λ_{s} - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, (λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ) .

Both the singular and regular companion terms are retained. The boundary value is fixed by the already chosen internal retarded line CheungXIVI.

The delta part in (eq:pole) is not an adjustable graph counterterm. This Fourier representation of one existing line does not choose a characteristic value for the complete graph. Likewise, its internal frequency estimates cannot be relabeled as target transverse estimates without the missing operation and a domain-sensitive transfer theorem.

Displayed formulas in this passage

XV-B@2#eq:pole · line 1915
$M_{R} = - \frac{s}{2 ρ} (λ_{s} - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, (λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ) .$
unlabeled · line 1909
$ρ > 0, λ_{s} = ω - s ρ, ∣ λ_{s} ∣ < c ρ, 0 < c < 1.$