Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Image–diffraction representation

Euclidean kernels, diffraction, and local Hadamard form › Angular summation · lines 834–860 · proof included or linked

I-B-VP@3#prop:contour, I-B-VP@3#eq:contour

At generic pairs,

G_{F, E} = k \in K (ϑ) \sum \frac{e ^{2 π i β k}}{4 π ^{2} ( R _{k} ^{2} + Δ τ ^{2} )} - \frac{1}{π α} \int_{0}^{\infty} \frac{Φ _{β} ( u , ϑ )}{4 π ^{2} ( ρ ( u ) ^{2} + Δ τ ^{2} )} d u .

Across image boundaries the right-hand side denotes the total joint distribution fixed by the heat kernel.

Proof. Apply Lemma lem:angular inside the four-dimensional heat integral and use

\int_{0}^{\infty} (4 π s)^{- 2} e^{- D^{2} / (4 s)} d s = \frac{1}{4 π ^{2} D ^{2}} .

At generic pairs the integrals converge absolutely. The joint local bound gives the tested boundary prescription.

Displayed formulas in this passage

I-B-VP@3#eq:contour · line 837
$G_{F, E} = k \in K (ϑ) \sum \frac{e ^{2 π i β k}}{4 π ^{2} ( R _{k} ^{2} + Δ τ ^{2} )} - \frac{1}{π α} \int_{0}^{\infty} \frac{Φ _{β} ( u , ϑ )}{4 π ^{2} ( ρ ( u ) ^{2} + Δ τ ^{2} )} d u .$
unlabeled · line 854
$\int_{0}^{\infty} (4 π s)^{- 2} e^{- D^{2} / (4 s)} d s = \frac{1}{4 π ^{2} D ^{2}} .$