Manuscript IX-B-ST@1 · IX-B-ST.tex · section · f6af1523

Parent finite part

Chart formulae and the exact dependency ledger › Parent finite part · lines 595–621

IX-B-ST@1#eq:residue

Let

Z_{U} Ψ (z, n, t, s, t^{'}) d ν r = {(z, n, t, s, t^{'}) : Ψ (z, n, t, s, t^{'}) \in X}, = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)), = d^{3} z d Ω (n) d t d s d t^{'}, \int_{S^{2}} d Ω = 4 π, = c_{0} Ψ_{*} (ν ∣_{Z_{U}}), c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, R = Ψ (Z_{U}) .

For compact $A ⋐ X$ , write

D_{A} (X) = {Φ \in C_{c}^{\infty} (X) : supp Φ \subset A}, p_{A, N} (Φ) = ∣ β ∣ \leq N max X sup ∣ \partial^{β} Φ∣.

Distributional pairings are complex linear in the test.

Displayed formulas in this passage

IX-B-ST@1#eq:residue · line 598
$Z_{U} Ψ (z, n, t, s, t^{'}) d ν r = {(z, n, t, s, t^{'}) : Ψ (z, n, t, s, t^{'}) \in X}, = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)), = d^{3} z d Ω (n) d t d s d t^{'}, \int_{S^{2}} d Ω = 4 π, = c_{0} Ψ_{*} (ν ∣_{Z_{U}}), c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, R = Ψ (Z_{U}) .$
unlabeled · line 615
$D_{A} (X) = {Φ \in C_{c}^{\infty} (X) : supp Φ \subset A}, p_{A, N} (Φ) = ∣ β ∣ \leq N max X sup ∣ \partial^{β} Φ∣.$