Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Reported Laplace-domain wave check

Retained numerical data from the supplied draft › Reported Laplace-domain wave check · lines 2690–2731

At $r = 1, r^{'} = 1.5, Δ φ = 2.9, Δ z = 0$ , the supplied draft compares the Laplace transform of the massless retarded formula with a mode-built longitudinal Fourier integral. It reports:

Case	$s$	Reported Laplace value	Reported relative discrepancy
$μ = π /2$	1	$5.54410642265313 \times 1 0^{- 3}$	$1.5 \times 1 0^{- 19}$
$μ = π /2$	3	$6.28663902035412 \times 1 0^{- 5}$	$4.2 \times 1 0^{- 19}$
$μ = π /2, β = 0.3$	1	$(2.16479458346901 + 1.65283850075681 i) \times 1 0^{- 3}$	$1.6 \times 1 0^{- 19}$
$μ = π /2, β = 0.3$	3	$(3.34886969040763 + 1.77134457766163 i) \times 1 0^{- 5}$	$1.2 \times 1 0^{- 18}$

The stated comparison integral was

\frac{1}{π} \int_{0}^{40} R_{2 D, β} (s^{2} + k^{2}; x, x^{'}) d k .

Its finite upper limit introduces a tail error. Without a supplied tail certificate and rounding analysis, the table is a discrepancy comparison, not a rigorous error enclosure. Agreement at these finitely many Laplace parameters is not an application of Laplace-transform uniqueness to all parameters.

Displayed formulas in this passage

unlabeled · line 2721
$\frac{1}{π} \int_{0}^{40} R_{2 D, β} (s^{2} + k^{2}; x, x^{'}) d k .$