Manuscript Em02-D@4 · Em02-D-v4.tex · lemma · fcf2fac7

Lemma: Signed internal integration

Exact signed internal convolution for 5z › Frozen-weight identity · lines 1533–1560 · proof included or linked

Em02-D@4#lem:internal-identity, Em02-D@4#eq:internal-identity

With propagation derivatives freezing the weight,

\int R^{'} (S - r) R^{''} (r) v (r) d^{4} r = D_{p}^{3} J_{int} (S; v) - 2 D_{p}^{2} J_{int} (S; \partial_{r^{0}} v) + D_{p} J_{int} (S; \partial_{r^{0}}^{2} v) .

The identity includes the internal cone vertex.

Proof. Transfer the two derivatives on $R^{''} (r)$ :

\partial_{r^{0}}^{2} [R^{'} (S - r) v (r)] = R^{'''} (S - r) v - 2 R^{''} (S - r) \partial_{r^{0}} v + R^{'} (S - r) \partial_{r^{0}}^{2} v .

Pair with $R (r)$ . Each resulting term is a frozen propagation derivative of $J_{int}$ . The computation is an identity for the proper pushforward of an independent two-cone distribution and remains valid at the vertex.

Displayed formulas in this passage

Em02-D@4#eq:internal-identity · line 1536
$\int R^{'} (S - r) R^{''} (r) v (r) d^{4} r = D_{p}^{3} J_{int} (S; v) - 2 D_{p}^{2} J_{int} (S; \partial_{r^{0}} v) + D_{p} J_{int} (S; \partial_{r^{0}}^{2} v) .$
unlabeled · line 1549
$\partial_{r^{0}}^{2} [R^{'} (S - r) v (r)] = R^{'''} (S - r) v - 2 R^{''} (S - r) \partial_{r^{0}} v + R^{'} (S - r) \partial_{r^{0}}^{2} v .$

Nearby in this section

section: Frozen-weight identity