Nonextensive Statistical Mechanics

332 Appendix A Useful Mathematical Formulaeln q,q ′ x ≡ ln q ′ e ln q x(x > 0, (q, q ′ ) ∈ R 2 ) (A.25)ln q,q ′(x ⊗ q y) = ln q,q ′ x ⊕ q ′ ln q,q ′ y (x > 0, (q, q ′ ) ∈ R 2 ) (A.26)e −β zq =1 ( 1q−1)1∫ ∞dαα 2−qq−1 e−αβ(q−1)e −α z[β(q − 1)] 1q−1 0(α >0; β>0; 1 < q < 2)(A.27)The following relations are useful for the Fourier transform of q-Gaussians (withβ>0):F q (p) ≡= 2∫ ∞⎪⎨=dx−∞∫ ∞0⎧ √√πβe ixp[1 + (q − 1) β x 2 ] 1/(q−1)cos (xp)[1 + (q − 1) β x 2 ] 1/(q−1)dxπ(1−q)β(2−q(1−q ) √ 2 β(1−q)p) 3−q2(1−q)J 3−q1−q()√pβ(1−q)if q < 1 ,e−p24βif q = 1 ,(A.28)⎪⎩(21q−1) √ (πβ(q−1)|p|2 √ β(q−1)) 3−q2(q−1)K 3−q2(q−1)()√|p|β(q−1)if 1 < q < 3 ,where J ν (z) and K ν (z) are, respectively, the Bessel and the modified Bessel functions.For the three successive regions of q we have respectively used formulae3.387-2 (page 346), 3.323-2 (page 333) and 8.432-5 (page 905) of [228] (see also[868]). For the q < 1 result we have taken into account the fact that the q-Gaussian1identically vanishes for |x| > √ β(1−q).F q [ f ](ξ) ≡∫ ∞−∞∫ ∞dx eq iξ x ⊗ q f (x) =−∞iξ x[ f (x)]q−1dx eq f (x) (q ≥ 1) (A.29)∫ ∞F q [ f ](0) =dF q [ f ](ξ)dξ ∣ = iξ=0d 2 ∣F q [ f ](ξ) ∣∣∣ξ=0=−qdξ 2−∞∫ ∞dx f(x) (q ≥ 1) (A.30)−∞∫ ∞dx x [ f (x)] q (q ≥ 1) (A.31)−∞dx x 2 [ f (x)] 2q−1 (q ≥ 1) (A.32)