Preface
Part III. Analytic Tools
9. Bernoulli Polynomials and the Gamma Function
9.1 Bernoulli Numbers and Polynomials
9.1.1 Generating Functions for Bernoulli Polynomials
9.1.2 Further Recurrences for Bernoulli Polynomials
9.1.3 Computing a Single Bernoulli Number
9.1.4 Bernoulli Polynomials and Fourier Series
9.2 Analytic Applications of Bernoulli Polynomials
9.2.1 Asymptotic Expansions
9.2.2 The Euler-MacLaurin Summation Formula
9.2.3 The Remainder Term and the Constant Term
9.2.4 Euler-MacLaurin and the Laplace Transform
9.2.5 Basic Applications of the Euler-MacLaurin Formula
9.3 Applications to Numerical Integration
9.3.1 Standard Euler-MacLaurin Numerical Integration
9.3.2 The Basic Tanh-Sinh Numerical Integration Method
9.3.3 General Doubly Exponential Numerical Integration
9.4 x-Bernoulli Numbers, Polynomials, and Functions
9.4.1 x-Bernoulli Numbers and Polynomials
9.4.2 x-Bernoulli Functions
9.4.3 The x-Euler-MacLaurin Summation Formula
9.5 Arithmetic Properties of Bernoulli Numbers
9.5.1 x-Power Sums
9.5.2 The Generalized Clausen-von Staudt Congruence
9.5.3 The Voronoi Congruence
9.5.4 The Kummer Congruences
9.5.5 The Almkvist-Meurman Theorem
9.6 The Real and Complex Gamma Functions
9.6.1 The Hurwitz Zeta Function
9.6.2 Definition of the Gamma Function
9.6.3 Preliminary Results for the Study of r(s)
9.6.4 Properties of the Gamma Function
9.6.5 Specific Properties of the Function w(s)
9.6.6 Fourier Expansions of S(s,x) and log(F(x))
9.7 Integral Transforms
9.7.1 Generalities on Integral Transforms
9.7.2 The Fourier Transform
9.7.3 The Mellin Transform
9.7.4 The Laplace Transform
9.8 Bessel Functions
9.8.1 Definitions
9.8.2 Integral Representations and Applications
9.9 Exercises for Chapter 9
10. Dirichlet Series and L-Functions
10.1 Arithmetic Functions and Dirichlet Series
10.1.1 Operations on Arithmetic Functions
10.1.2 Multiplicative Functions
10.1.3 Some Classical Arithmetical Functions
10.1.4 Numerical Dirichlet Series
