Manuscript IX-E-2@1 · IX-E-2.tex · proposition · e70d9077

Proposition: Exact subtraction

Exact direct-plus-reflected subtraction · lines 617–666 · proof included or linked

IX-E-2@1#eq:exact-subtraction, IX-E-2@1#eq:imag-decomposition

With the convention (eq:variables),

K - K = \frac{i t}{2} D (τ, z) + \frac{i}{4} H (τ, z) - \frac{i}{4} H (u, z) .

Equivalently,

Im K = \frac{t}{4} D (τ, z) + \frac{1}{8} H (τ, z) - \frac{1}{8} H (u, z) .

Proof. Conjugate (eq:locked-kernel) at fixed observation arguments. The spatial phase becomes $e^{- i p \cdot z}$ . Replacing $p$ by $- p$ restores the original spatial phase without changing $∣ p ∣$ .

The direct difference is

- \frac{t}{4∣ p ∣} (e^{- i ∣ p ∣ τ} - e^{i ∣ p ∣ τ}) + \frac{i}{8∣ p ∣ ^{2}} (e^{- i ∣ p ∣ τ} + e^{i ∣ p ∣ τ}) = \frac{i t}{2∣ p ∣} sin (∣ p ∣ τ) + \frac{i}{4∣ p ∣ ^{2}} cos (∣ p ∣ τ) .

The reflected difference is

- \frac{i}{8∣ p ∣ ^{2}} (e^{i ∣ p ∣ u} + e^{- i ∣ p ∣ u}) = - \frac{i}{4∣ p ∣ ^{2}} cos (∣ p ∣ u) .

These identities hold with the common Abel regulator, and hence after its distributional limit. Division by $2 i$ gives (eq:imag-decomposition).

Displayed formulas in this passage

IX-E-2@1#eq:exact-subtraction · line 619
$K - K = \frac{i t}{2} D (τ, z) + \frac{i}{4} H (τ, z) - \frac{i}{4} H (u, z) .$
IX-E-2@1#eq:imag-decomposition · line 629
$Im K = \frac{t}{4} D (τ, z) + \frac{1}{8} H (τ, z) - \frac{1}{8} H (u, z) .$
unlabeled · line 647
$- \frac{t}{4∣ p ∣} (e^{- i ∣ p ∣ τ} - e^{i ∣ p ∣ τ}) + \frac{i}{8∣ p ∣ ^{2}} (e^{- i ∣ p ∣ τ} + e^{i ∣ p ∣ τ}) = \frac{i t}{2∣ p ∣} sin (∣ p ∣ τ) + \frac{i}{4∣ p ∣ ^{2}} cos (∣ p ∣ τ) .$
unlabeled · line 657
$- \frac{i}{8∣ p ∣ ^{2}} (e^{i ∣ p ∣ u} + e^{- i ∣ p ∣ u}) = - \frac{i}{4∣ p ∣ ^{2}} cos (∣ p ∣ u) .$

Proposition: Exact subtraction

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