Appendix a Properties of the Gamma Functions
Appendix A Properties of the Gamma Functions We list here some basic properties of the Gamma function (see, e.g., Abramowitz and Stegun (1964)), defined by ∞ Γ (z)= tz−1e−tdt, ∀z ∈ C with Re(z) > 0. (A.1) 0 √ In particular, we have Γ (1)=1andΓ (1/2)= π. • Recursion formula: Γ (z + 1)=zΓ (z), Γ (n + 1)=n!, (A.2) and − / − / Γ (2z)=(2π) 1 222z 1 2Γ (z)Γ (z + 1/2). (A.3) • Connection with binomial coefficient: , - z Γ (z + 1) = . (A.4) w Γ (w + 1)Γ (z − w + 1) • Relation with Beta function: 1 Γ (x)Γ (y) B(x,y)= tx−1(1 −t)y−1dt = , x,y > 0. (A.5) 0 Γ (x + y) In particular, for α,β > −1, 1 1 (1 − x)α(1 + x)β dx = 2α+β +1 tβ (1 −t)αdt −1 0 (A.6) Γ (α + 1)Γ (β + 1) = 2α+β +1 . Γ (α + β + 2) J. Shen et al., Spectral Methods: Algorithms, Analysis and Applications, 415 Springer Series in Computational Mathematics 41, DOI 10.1007/978-3-540-71041-7, c Springer-Verlag Berlin Heidelberg 2011 416 A Properties of the Gamma Functions • Stirling’s formula: √ ! − / − 1 1 − Γ (x)= 2πxx 1 2e x 1 + + + O(x 3) , x 1. (A.7) 12x 288x2 Moreover, we have √ √ + / + / 1 2πnn 1 2 < n!en < 2πnn 1 2 1 + , n ≥ 1. (A.8) 4n Appendix B Essential Mathematical Concepts We provide here some essential mathematical concepts which have been used in the mathematical analysis throughout the book.
[Show full text]