Nonextensive Statistical Mechanics

330 Appendix A Useful Mathematical Formulaex ⊕ q y ≡ x + y + (1 − q) xy(A.9)For x ≥ 0 and y ≥ 0:x ⊗ q y ≡ [x 1−q + y 1−q − 1]11−q+ ≡⎧0 if q < 1andx 1−q + y 1−q < 1 ,⎪⎨⎪⎩[x 1−q + y 1−q − 1] 11−qif q < 1andx 1−q + y 1−q ≥ 1 ,xy if q = 1 ∀(x, y) ,[x 1−q + y 1−q − 1] 11−qif q > 1andx 1−q + y 1−q > 1 .(A.10)x ⊗ q y = [1 + (1 − q)(ln q x + ln q y)] 11−q(A.11)e x⊕ q yq = e x q ey q (∀q) (A.12)e x+yq = e x q ⊗ q e y q (∀q) (A.13)d ln q xdx= 1x q (x > 0; ∀q) (A.14)de x qdx = (ex q )q (∀q) (A.15)(e x q )q = e qx2−(1/q)(∀q) (A.16)(e x q )a = e ax1−(1−q)/a (∀q) (A.17)[x a e − x bbq =q − 1] 1/(q−1)xa− 1 −q−1 e b/(q−1)2xq (b > 0; q > 1) (A.18)