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

Proposition: Euclidean mode sum

Euclidean kernels, diffraction, and local Hadamard form · lines 707–758 · proof included or linked

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

The heat-integrated kernel

G_{F, E} (X, X^{'}) = \int_{0}^{\infty} \frac{e ^{- Δ τ^{2} / (4 s)}}{4 π s} K_{β} (s; x, x^{'}) d s

is a fundamental distribution for $L_{E}$ in each variable. For $η > 0$ ,

G_{F, E} = \frac{1}{8 π ^{2} α r r ^{'} sinh η} n \in Z \sum e^{- ∣ q_{n} ∣ η / α} e^{i q_{n} (φ - φ^{'})} = \frac{1}{8 π ^{2} α r r ^{'} sinh η} [\frac{e ^{- β a + i β b}}{1 - e ^{- a + i b}} + \frac{e ^{- (1 - β) a - i (1 - β) b}}{1 - e ^{- a - i b}}],

where $a = η / α$ , $b = φ - φ^{'}$ . For $β = 0$ ,

G_{F, E} = \frac{sinh ( η / α )}{8 π ^{2} α r r ^{'} sinh η [ cosh ( η / α ) - cos ( φ - φ ^{'} )]} .

At $α = 1, β = 0$ , this is $(4 π^{2} σ_{E}^{2})^{- 1}$ .

Proof. Large-time convergence follows from Proposition prop:heat; the additional Euclidean factor gives $s^{- 2 - ν_{*}}$ . Integrating the heat equation proves the fundamental-distribution identities. Substitute $ζ = r r^{'} / (2 s)$ and use

\int_{0}^{\infty} e^{- ζ c o s h η} I_{ν} (ζ) d ζ = \frac{e ^{- ν η}}{sinh η}, η > 0, ν > - 1.

The constants become $(8 π^{2} α r r^{'})^{- 1}$ . Absolute convergence justifies the Friedrichs sum for $η > 0$ . Splitting $n \geq 0$ and $n \leq - 1$ gives the two geometric series Watson,DLMF.

Displayed formulas in this passage

I-B-VP@3#eq:modes · line 718
$G_{F, E} = \frac{1}{8 π ^{2} α r r ^{'} sinh η} n \in Z \sum e^{- ∣ q_{n} ∣ η / α} e^{i q_{n} (φ - φ^{'})} = \frac{1}{8 π ^{2} α r r ^{'} sinh η} [\frac{e ^{- β a + i β b}}{1 - e ^{- a + i b}} + \frac{e ^{- (1 - β) a - i (1 - β) b}}{1 - e ^{- a - i b}}],$
unlabeled · line 710
$G_{F, E} (X, X^{'}) = \int_{0}^{\infty} \frac{e ^{- Δ τ^{2} / (4 s)}}{4 π s} K_{β} (s; x, x^{'}) d s$
unlabeled · line 733
$G_{F, E} = \frac{sinh ( η / α )}{8 π ^{2} α r r ^{'} sinh η [ cosh ( η / α ) - cos ( φ - φ ^{'} )]} .$
unlabeled · line 749
$\int_{0}^{\infty} e^{- ζ c o s h η} I_{ν} (ζ) d ζ = \frac{e ^{- ν η}}{sinh η}, η > 0, ν > - 1.$

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section: Euclidean kernels, diffraction, and local Hadamard form