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

Proposition: Logarithmic principal-value Wick square

Unstable extensions and the two projected alternatives › Principal-value reference · lines 1914–1956 · proof included or linked

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

Back to manuscript

For $L_{L}$ ,

δ F_{PV} = \frac{1}{4 π ^{2} α r ^{2}} PV \int_{0}^{\infty} \frac{x K _{0} ( x ) ^{2}}{lo g ( x / κ _{L} r )} d x = - \frac{1}{8 π ^{2} α r ^{2} ℓ} [1 + \frac{lo g 2 - 1 - γ _{E}}{ℓ} + O (ℓ^{- 2})], ℓ = lo g (κ_{L} r),

as $ℓ \to + \infty$ or $ℓ \to - \infty$ .

Proof. Set $y = lo g x$ and $w (y) = e^{2 y} K_{0} (e^{y})^{2}$ . This is a Schwartz function on $R$ . The principal-value integral is $PV \int w (y) / (y - ℓ) d y$ . Split into $∣ y ∣ < ∣ ℓ ∣/2$ , a neighbourhood of the moving pole, and the remaining tails. On the first region expand the reciprocal. Near the pole subtract $w (ℓ)$ symmetrically and use Schwartz bounds on $w^{'}$ ; all tail errors are smaller than any prescribed inverse power after enough moments are kept. Hence

PV \int \frac{w ( y )}{y - ℓ} d y = - \frac{1}{ℓ} \int w - \frac{1}{ℓ ^{2}} \int y w + O (∣ ℓ ∣^{- 3}) .

The Mellin formula in Appendix app:mellin gives

\int_{0}^{\infty} x K_{0}^{2} d x = \frac{1}{2}, \int_{0}^{\infty} x K_{0}^{2} lo g x d x = \frac{1}{2} (lo g 2 - 1 - γ_{E}) .

Displayed formulas in this passage

I-B-VP@3#eq:logPV · line 1917
$δ F_{PV} = \frac{1}{4 π ^{2} α r ^{2}} PV \int_{0}^{\infty} \frac{x K _{0} ( x ) ^{2}}{lo g ( x / κ _{L} r )} d x = - \frac{1}{8 π ^{2} α r ^{2} ℓ} [1 + \frac{lo g 2 - 1 - γ _{E}}{ℓ} + O (ℓ^{- 2})], ℓ = lo g (κ_{L} r),$
unlabeled · line 1945
$PV \int \frac{w ( y )}{y - ℓ} d y = - \frac{1}{ℓ} \int w - \frac{1}{ℓ ^{2}} \int y w + O (∣ ℓ ∣^{- 3}) .$
unlabeled · line 1951
$\int_{0}^{\infty} x K_{0}^{2} d x = \frac{1}{2}, \int_{0}^{\infty} x K_{0}^{2} lo g x d x = \frac{1}{2} (lo g 2 - 1 - γ_{E}) .$

Proposition: Logarithmic principal-value Wick square

Displayed formulas in this passage

Nearby in this section