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

Lemma: Sommerfeld summation and boundary control

Euclidean kernels, diffraction, and local Hadamard form › Angular summation · lines 770–832 · proof included or linked

I-B-VP@3#lem:angular, I-B-VP@3#eq:sommerfeld

For $ζ > 0$ , at generic angles,

n \sum I_{ν_{n}} (ζ) e^{i q_{n} ϑ / α} = α k \in K (ϑ) \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} - \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u, ϑ) d u .

The identity extends after joint distributional testing through image boundaries. Near such a boundary $ϑ_{j}$ ,

∣ Φ_{β} (u, ϑ) ∣ \leq C (1 + [u^{2} + (ϑ - ϑ_{j})^{2}]^{- 1/2}) .

The weight decays exponentially at large $u$ . For $β = 0$ ,

Φ_{0} (u, ϑ) = \frac{1}{2} \pm \sum \frac{sin (( π \pm ϑ ) / α )}{cosh ( u / α ) - cos (( π \pm ϑ ) / α )} .

Proof. Set $A = π / α$ , $b = u / α$ , $c = ϑ / α$ , and

F (γ, w) = \frac{1}{2 i} [\frac{e ^{γ (w + i A)}}{1 - e ^{w + i A}} - \frac{e ^{γ (w - i A)}}{1 - e ^{w - i A}}] .

Splitting positive and negative frequencies gives

Φ_{β} = F (β, - b + i c) + F (1 - β, - b - i c) .

The denominators give the asserted joint local bound. For $0 < β < 1$ , a decay rate is $min (β, 1 - β) / α$ . At $β = 0$ , the zero-order sine vanishes and rate $1/ α$ suffices.

Insert the integral representation

I_{ν} (ζ) = \frac{1}{π} \int_{0}^{π} e^{ζ c o s v} cos (ν v) d v - \frac{sin ( π ν )}{π} \int_{0}^{\infty} e^{- ζ c o s h u - ν u} d u

DLMF (Eq. 10.32.4). Twisted Fourier summation of the first term gives the image indicator and its phases. An Abel factor $e^{- ϵ ν_{n}}$ justifies summation first. The joint bound is locally integrable in $(u, ϑ)$ ; it and the large- $u$ decay justify the distributional limit. Combining fractions at $β = 0$ gives the last formula.

Displayed formulas in this passage

I-B-VP@3#eq:sommerfeld · line 773
$n \sum I_{ν_{n}} (ζ) e^{i q_{n} ϑ / α} = α k \in K (ϑ) \sum e^{2 π i β k} e^{ζ c o s (ϑ - 2 π α k)} - \frac{1}{π} \int_{0}^{\infty} e^{- ζ c o s h u} Φ_{β} (u, ϑ) d u .$
unlabeled · line 785
$∣ Φ_{β} (u, ϑ) ∣ \leq C (1 + [u^{2} + (ϑ - ϑ_{j})^{2}]^{- 1/2}) .$
unlabeled · line 792
$Φ_{0} (u, ϑ) = \frac{1}{2} \pm \sum \frac{sin (( π \pm ϑ ) / α )}{cosh ( u / α ) - cos (( π \pm ϑ ) / α )} .$
unlabeled · line 801
$F (γ, w) = \frac{1}{2 i} [\frac{e ^{γ (w + i A)}}{1 - e ^{w + i A}} - \frac{e ^{γ (w - i A)}}{1 - e ^{w - i A}}] .$
unlabeled · line 809
$Φ_{β} = F (β, - b + i c) + F (1 - β, - b - i c) .$
unlabeled · line 819
$I_{ν} (ζ) = \frac{1}{π} \int_{0}^{π} e^{ζ c o s v} cos (ν v) d v - \frac{sin ( π ν )}{π} \int_{0}^{\infty} e^{- ζ c o s h u - ν u} d u$

Lemma: Sommerfeld summation and boundary control

Displayed formulas in this passage

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