Density fluctuations in the 1D Bose gas

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Panfil, M.K.

Publication date

2013

Link to publication

Citation for published version (APA):

Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.

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Appendix B

Special Functions and their

Properties

Gamma Function

We follow the usual definition of Gamma function Γ(x)

Γ(z) = Z ∞

0

dt tz−1e−t, z∈ C/{. . . , −2, −1, 0}, (B.1)

with its almost defining properties

Γ(1) = 1, (B.2) Γ(z + 1) = zΓ(z). (B.3)

Values ofΓ(x) at large integer arguments follows the Stirling approximation

log (Γ(n)) = n− 1 2 log(n)− n +1 2log(2π) +O(1/n). (B.4)

Γ-function can also be used to express the following integrals Z ∞ 0 dz z x−1 ez_{− 1} = Z ∞ 0 dz zx−1e−z 1− e−z−1= Z ∞ 0 dz zx−1e−z ∞ X k=0 e−kz = Z ∞ 0 dz zx−1e−z× ∞ X k=1 1 kx = Γ(x)× ∞ X k=1 1 kx. (B.5) 139

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Appendix B. Special Functions 140

The infinite summation is equal to the Riemann ζ function

∞ X k=1 1 kx = ζ(x). (B.6) Therefore Z ∞ 0 dz z x−1 ez_{+ 1} = Γ(x)ζ(x). (B.7)

For the case considered in Chapter 1we need the following values of Γ and ζ functions

Γ(3/2) =√π/2, ζ(3/2)_{≈ 2.61.} (B.8)

Barnes Function

We define the BarnesG(x) function through the following relations

G(1 + m + a) G(1 + a) = m Y i=1 Γ(i + a), (B.9) G(1) = 1. (B.10)

BarnesG function also admits Stirling like approximation for large integer arguments

log (G(n + 1)) = n 2 2 log(n)− 3 4n 2₊n 2 log(2π)− 1 12log(n) + ζ 0₍ −1) + O(1/n), (B.11)

whereζ0(x) is the derivative of Riemann ζ(x) function. The Barnes G function fulfills a special property

G(1_{− z)} G(1 + z) = (2π) −z_expZ z 0 dx πx cot(πx) . (B.12)

This property allows for a simplification in computation of the themodynamic limit of the form factor in Chapter4. Consider the following expression and its transformation

exp Z q −q dλ πF (λ)F0(λ) cot(πF (λ) = exp Z F (q) F (−q) dx πx cot(πx) ! = G(1− F (q))G(1 + F (−q)) G(1 + F (q))G(1_{− F (−q))}(2π) F (q)−F (−q) . (B.13)

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Appendix B. Special Functions 141

Thus the following idenity follows

G(1 + F (q))G(1− F (−q)) exp Z q −q dλ πF (λ)F0(λ) cot(πF (λ) = = G(1 + F (_{−q))G(1 − F (q)) (2π)}F (q)−F (−q). (B.14)