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

Definition: Prescribed VII-C residue

The locked residue · lines 546–566

VII-D@1#def:residue, VII-D@1#eq:rdef, VII-D@1#eq:rjoint

The subtraction distribution is

r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}} .

Equivalently, for an arbitrary joint test,

⟨ r, Φ ⟩ = c_{0} \int d s d^{3} y d t d t^{'} \int_{S^{2}} Φ (t, y + (t + s) n; s, y; t^{'}, y + (t^{'} + s) n) d Ω (n) .

Displayed formulas in this passage

VII-D@1#eq:rdef · line 549
$r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}} .$
VII-D@1#eq:rjoint · line 554
$⟨ r, Φ ⟩ = c_{0} \int d s d^{3} y d t d t^{'} \int_{S^{2}} Φ (t, y + (t + s) n; s, y; t^{'}, y + (t^{'} + s) n) d Ω (n) .$