Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Exact integral and density

Evaluated complete degenerate specialization › Exact integral and density · lines 1738–1777

Em01-E@2#eq:appF

Use the fixed-frame coordinates

b = a + B ℓ + u k + X, y = a + A ℓ + v k + Y, z = a + r .

Write

τ = (a, B, A, Y), ν = (r, u, v, X), d τ = d^{4} a d B d A d^{2} Y .

This linear coordinate system has constant absolute Jacobian four. The attachment is $ν = 0$ , of codimension eight. Moreover,

(b - a)^{2} = 4 B u - ∣ X ∣^{2}, s = 2 v, t = 2 u .

Define ambient test expressions

F_{0} [Φ] F_{1} [Φ] F_{2} [Φ] = Φ_{0}, = - \partial_{b^{0}} Φ_{0} + 2 (b^{0} + y^{0}) Φ_{0}, = - 2 y^{0} (\partial_{b^{0}} Φ_{0} - 2 b^{0} Φ_{0}),

and then evaluate them:

F_{j} (τ; u, v, X) = F_{j} [Φ] (a, a + B ℓ + u k + X, a, a + A ℓ + v k + Y) .

Ambient derivatives are taken before evaluation. Tests are extended by zero outside their compact support in $U$ .

Displayed formulas in this passage

unlabeled · line 1760
$F_{0} [Φ] F_{1} [Φ] F_{2} [Φ] = Φ_{0}, = - \partial_{b^{0}} Φ_{0} + 2 (b^{0} + y^{0}) Φ_{0}, = - 2 y^{0} (\partial_{b^{0}} Φ_{0} - 2 b^{0} Φ_{0}),$
Em01-E@2#eq:appF · line 1768
$F_{j} (τ; u, v, X) = F_{j} [Φ] (a, a + B ℓ + u k + X, a, a + A ℓ + v k + Y) .$
unlabeled · line 1741
$b = a + B ℓ + u k + X, y = a + A ℓ + v k + Y, z = a + r .$
unlabeled · line 1747
$τ = (a, B, A, Y), ν = (r, u, v, X), d τ = d^{4} a d B d A d^{2} Y .$
unlabeled · line 1755
$(b - a)^{2} = 4 B u - ∣ X ∣^{2}, s = 2 v, t = 2 u .$