Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

The reflected radial family: normalization and Lemma A

The reflected radial family: normalization and Lemma A · lines 824–864

VII-D@1#eq:Aintrinsic, VII-D@1#eq:Aepsintrinsic, VII-D@1#eq:rawsplit

For $q = r ν$ , $r \geq 1$ , set

P = ∣ α - r ν ∣, P^{'} = ∣ β - r ν ∣, ω (r) = r^{2} + m^{2},

in accordance with (eq:substitution).

The radial families are defined intrinsically from the exact branch integrals:

⟨ A (r), Φ ⟩ ⟨ A_{ε}^{ρ} (r), Φ ⟩ = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{0}^{R R} (r ν; Φ) d Ω (ν), = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{ε}^{R R} (r ν; Φ) d Ω (ν) .

No residue constant has been inserted in these definitions. They give the exact decomposition

T_{ε}^{ρ} = S_{ε}^{ρ} + \int_{1}^{\infty} M_{ε} (r)^{2} A_{ε}^{ρ} (r) \frac{d r}{r} .

The use of $r$ as the radial integration variable is distinct from its use for the fixed residue distribution. In expressions such as $A (r) - r$ , the second term is the distribution (eq:rdef).

Displayed formulas in this passage

VII-D@1#eq:Aintrinsic · line 836
$⟨ A (r), Φ ⟩ ⟨ A_{ε}^{ρ} (r), Φ ⟩ = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{0}^{R R} (r ν; Φ) d Ω (ν), = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{ε}^{R R} (r ν; Φ) d Ω (ν) .$
VII-D@1#eq:Aepsintrinsic · line 836
$⟨ A (r), Φ ⟩ ⟨ A_{ε}^{ρ} (r), Φ ⟩ = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{0}^{R R} (r ν; Φ) d Ω (ν), = \frac{r ^{3} ω ( r )}{( 2 π ) ^{3}} \int_{S^{2}} Q_{ε}^{R R} (r ν; Φ) d Ω (ν) .$
VII-D@1#eq:rawsplit · line 852
$T_{ε}^{ρ} = S_{ε}^{ρ} + \int_{1}^{\infty} M_{ε} (r)^{2} A_{ε}^{ρ} (r) \frac{d r}{r} .$
unlabeled · line 827
$P = ∣ α - r ν ∣, P^{'} = ∣ β - r ν ∣, ω (r) = r^{2} + m^{2},$