Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Scalar coefficients, profile moments, and locality

Scalar coefficients, profile moments, and locality · lines 1223–1254

Em01-G@5#eq:Mj, Em01-G@5#eq:second-moment, Em01-G@5#eq:third-moment

Write $S = B_{2} - B_{1}$ , and define

M_{j} (n) = \int ρ_{n} (H) (S - i 0)^{- j} d H, j = 2, 3.

These are pullbacks by the submersion $H \mapsto S$ , paired with the compact joint profile.

Since $\partial_{H_{i}^{μ}} B_{i} = ℓ_{μ} /2$ , one obtains

v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} ρ} [A] v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} \partial_{H_{2}^{σ}} ρ} [A] = - \frac{M _{2}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} A (0, 0), = - \frac{M _{3}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} ℓ_{σ} A (0, 0) .

These follow by integration by parts from

\partial_{S}^{2} lo g (- S + i 0) = - (S - i 0)^{- 2}, \partial_{S}^{3} lo g (- S + i 0) = 2 (S - i 0)^{- 3} .

Displayed formulas in this passage

Em01-G@5#eq:Mj · line 1226
$M_{j} (n) = \int ρ_{n} (H) (S - i 0)^{- j} d H, j = 2, 3.$
Em01-G@5#eq:second-moment · line 1238
$v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} ρ} [A] v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} \partial_{H_{2}^{σ}} ρ} [A] = - \frac{M _{2}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} A (0, 0), = - \frac{M _{3}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} ℓ_{σ} A (0, 0) .$
Em01-G@5#eq:third-moment · line 1238
$v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} ρ} [A] v_{0}^{\partial_{H_{2}^{μ}} \partial_{H_{2}^{κ}} \partial_{H_{2}^{σ}} ρ} [A] = - \frac{M _{2}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} A (0, 0), = - \frac{M _{3}}{128 π ^{2} β} ℓ_{μ} ℓ_{κ} ℓ_{σ} A (0, 0) .$
unlabeled · line 1249
$\partial_{S}^{2} lo g (- S + i 0) = - (S - i 0)^{- 2}, \partial_{S}^{3} lo g (- S + i 0) = 2 (S - i 0)^{- 3} .$