Nonextensive Statistical Mechanics

Appendix A Useful Mathematical Formulae 331eq x = ex [1 − 1 2 (1 − q)x 2 + 1 3 (1 − q)2 x 3 (1 + 3 8 x) − 1 4 (1 − q)3 x 4 (1 + 2 3 x + 112 x 2 )+ 1 5 (1 − q)4 x 5 (1 + 6572 x + 524 x 2 + 5384 x 3 )− 1 6 (1 − q)5 x 6 (1 + 1110 x + 1748 x 2 + 1 24 x 3 + 1640 x 4 ) + ...] (q → 1; ∀x)(A.19)ln q x = ln x [1 + 1 2 (1 − q)lnx + 1 6 (1 − q)2 ln 2 x + 124 (1 − q)3 ln 3 x+ 1120 (1 − q)4 ln 4 x + 1720 (1 − q)5 ln 5 x + ...] (q → 1; x > 0)(A.20)[x ⊗ q y = xy 1 − (1 − q)(ln x)(ln y)+ 1 2 (1 − q)2 [(ln 2 x)(ln y) + (ln x)(ln 2 y) + (ln 2 x)(ln 2 y)]− 112 (1 − q)3 [2(ln 3 x)(ln y) + 9(ln 2 x)(ln 2 y) + 2(ln x)(ln 3 y)+6(ln 3 x)(ln 2 y) + 6(ln 2 x)(ln 3 y) + 2(ln 3 x)(ln 3 y)]+ 124 (1 − q)4 [(ln 4 x)(ln y) + 14(ln 3 x)(ln 2 y)+14(ln 2 x)(ln 3 y) + (ln x)(ln 4 y)+7(ln 4 x)(ln 2 y) + 24(ln 3 x)(ln 3 y) + 7(ln 2 x)(ln 4 y)]+6(ln 4 x)(ln 3 y) + 6(ln 3 x)(ln 4 y) + (ln 4 x)(ln 4 y)] + ... (A.21)e x q = 1 + x + 1 2 x 2 q + 1 6 x 3 q(2q − 1) + 124 x 4 q(2q − 1)(3q − 2)+ 1120 x 5 q(2q − 1)(3q − 2)(4q − 3) + ... (x → 0; ∀q) (A.22)ln q (1 + x) = x − 1 2 x 2 q + 1 6 x 3 q(1 + q) − 124 x 4 q(1 + q)(2 + q)+ 1120 x 5 q(1 + q)(2 + q)(3 + q) + ... (x → 0; ∀q) (A.23)[x a e − x bq =bq − 1] 1q−1x a− 1−q−1 e b/(q−1)2xq (q > 1; b > 0) (A.24)