Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · section · f4bd2b11

Comparison with the image–diffraction representation

The diffracted coefficient and its nonvanishing criterion › Comparison with the image–diffraction representation · lines 1165–1231

I-B-VP-2@1#eq:contourW

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For orientation and normalization, define

ρ (u)^{2} = r^{2} + r^{'2} + 2 r r^{'} cosh u + Z^{2},

Φ_{β} (u, δ) = n \in Z \sum sin (π ν_{n}) e^{- ν_{n} u} e^{i (n + β) δ}, u > 0.

The notation here uses $δ$ , rather than the developing angle $α δ$ , as its second argument.

The Bessel angular identity gives, off the transition angles,

W_{β} = k \in Z ∣ θ_{k} ∣ < π \sum \frac{e ^{2 π i β k}}{4 π ^{2} ( R _{k} ^{2} - ( T - i 0 ) ^{2} )} - \frac{1}{4 π ^{3} α} \int_{0}^{\infty} \frac{Φ _{β} ( u , δ )}{ρ ( u ) ^{2} - ( T - i 0 ) ^{2}} d u .

One obtains this formula by applying the heat-kernel angular summation before integrating the Euclidean time heat factor, and then continuing its total kernel. It is the covariance counterpart of the Sommerfeld–Carslaw–Dowker formulas in I-B-G and I-B-VP CheungIBG,CheungIBVP. At transition angles the joint distribution, not separate pointwise sector values, is the object.

The coefficient in (eq:A) can be checked without assigning a value to a divergent angular series:

Φ_{β} (u, δ) = \frac{S _{β} ( u - i π , δ ) - S _{β} ( u + i π , δ )}{2 i}, u > 0.

Its transition-free limit is therefore

Φ_{β} (0, δ) = - A_{α, β} (δ) .

Since

ρ (u)^{2} - (T - i 0)^{2} = F_{ap} + r r^{'} u^{2} + O (u^{4}) + i 0 sgn T,

the endpoint model

\int_{0}^{\infty} \frac{d u}{s + r r ^{'} u ^{2}} = \frac{π}{2 r r ^{'}} s^{- 1/2}

recovers (eq:diffcoefficient). The full wavefront proof at the angular transition was instead given by the reduced-variable argument of Proposition prop:upper; this endpoint check is not used to differentiate a singular moving image sector.

Displayed formulas in this passage

I-B-VP-2@1#eq:contourW · line 1181
$W_{β} = k \in Z ∣ θ_{k} ∣ < π \sum \frac{e ^{2 π i β k}}{4 π ^{2} ( R _{k} ^{2} - ( T - i 0 ) ^{2} )} - \frac{1}{4 π ^{3} α} \int_{0}^{\infty} \frac{Φ _{β} ( u , δ )}{ρ ( u ) ^{2} - ( T - i 0 ) ^{2}} d u .$
unlabeled · line 1168
$ρ (u)^{2} = r^{2} + r^{'2} + 2 r r^{'} cosh u + Z^{2},$
unlabeled · line 1171
$Φ_{β} (u, δ) = n \in Z \sum sin (π ν_{n}) e^{- ν_{n} u} e^{i (n + β) δ}, u > 0.$
unlabeled · line 1205
$Φ_{β} (u, δ) = \frac{S _{β} ( u - i π , δ ) - S _{β} ( u + i π , δ )}{2 i}, u > 0.$
unlabeled · line 1212
$Φ_{β} (0, δ) = - A_{α, β} (δ) .$
unlabeled · line 1216
$ρ (u)^{2} - (T - i 0)^{2} = F_{ap} + r r^{'} u^{2} + O (u^{4}) + i 0 sgn T,$
unlabeled · line 1221
$\int_{0}^{\infty} \frac{d u}{s + r r ^{'} u ^{2}} = \frac{π}{2 r r ^{'}} s^{- 1/2}$

Nearby in this section

corollary: Corollary: Relative strength away from transitions