Manuscript IX-D-ST@1 · IX-D-ST.tex · section · b1dc631a

Common-regulator evaluation

Common-regulator evaluation · lines 1421–1437

Define

D (v, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z} \frac{sin ( ∣ p ∣ v )}{∣ p ∣}, H (v, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z} \frac{cos ( ∣ p ∣ v )}{∣ p ∣ ^{2}} .

Both have the same common-Abel interpretation. The changes $v = t - s$ and $v = t + s$ are submersion pullbacks, equivalently tensor products followed by invertible linear coordinate changes.

Displayed formulas in this passage

unlabeled · line 1424
$D (v, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z} \frac{sin ( ∣ p ∣ v )}{∣ p ∣}, H (v, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z} \frac{cos ( ∣ p ∣ v )}{∣ p ∣ ^{2}} .$

Nearby in this section

section: Common-regulator evaluation
lemma: Lemma: Fourier existence
proposition: Proposition: Exact direct-plus-reflected subtraction
lemma: Lemma: Global radial representatives
theorem: Theorem: Exact imaginary part
section: Common-regulator evaluation
corollary: Corollary: Support, including the spatial apex
section: Common-regulator evaluation