Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · section · 6d700be9

Boundary calculations for the finite conversions

Boundary calculations for the finite conversions · lines 1497–1524

We give the local transverse calculation; longitudinal integration is passive. Write $⟨ \cdot ⟩_{φ}$ for an unnormalized angular integral, and use the constant density factor $α$ . All remainder estimates are uniform in the required compact longitudinal set.

For a smooth test,

\frac{1}{2 π} ⟨ f (ε ω) ⟩_{φ} = f (0) + \frac{ε ^{2}}{4} Δ f (0) + O (ε^{4} p_{4} (f)) .

The angular identities used below are

\int_{0}^{2 π} ω_{i} ω_{j} d φ = π δ_{ij},

\int_{0}^{2 π} ω_{i} ω_{j} ω_{k} ω_{l} d φ = \frac{π}{4} (δ_{ij} δ_{k l} + δ_{ik} δ_{j l} + δ_{i l} δ_{j k}) .

Displayed formulas in this passage

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$\frac{1}{2 π} ⟨ f (ε ω) ⟩_{φ} = f (0) + \frac{ε ^{2}}{4} Δ f (0) + O (ε^{4} p_{4} (f)) .$
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$\int_{0}^{2 π} ω_{i} ω_{j} d φ = π δ_{ij},$
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$\int_{0}^{2 π} ω_{i} ω_{j} ω_{k} ω_{l} d φ = \frac{π}{4} (δ_{ij} δ_{k l} + δ_{ik} δ_{j l} + δ_{i l} δ_{j k}) .$