Preface to the Sixth Edition(page vii) Acknowledgments(page ix) The order of presentation of the formulas(page xiii) Use of the tables(page xvii) Special functions(page xxv) Notation(page xxix) Note on the bibliographic references(page xxxiii)
0Introduction(page 1) 0.1 Finite sums (page 1) 0.11 Progressions (page 1) 0.12 Sums of powers of natural numbers (page 1) 0.13 Sums of reciprocals of natural numbers (page 2) 0.14 Sums of products of reciprocals of natural numbers (page 3) 0.15 Sums of the binomial coefficients (page 3) 0.2 Numerical series and infinite products (page 6) 0.21 The convergence of numerical series (page 6) 0.22 Convergence tests (page 6) 0.23-0.24 Examples of numerical series (page 8) 0.25 Infinite products (page 14) 0.26 Examples of infinite products (page 14) 0.3 Functional series (page 15) 0.30 Definitions and theorems (page 15) 0.31 Power series (page 16) 0.32 Fourier series (page 18) 0.33 Asymptotic series (page 20) 0.4 Certain formulas from differential calculus (page 21) 0.41 Differentiation of a definite integral with respect to a parameter (page 21) 0.42
The
nth
derivative of a product (Leibniz's rule) (page 21) 0.43 The
nth
derivative of a composite function (page 21) 0.44 Integration by substitution (page 23)
1Elementary Functions(page 25) 1.1 Power of Binomials (page 25) 1.11 Power series (page 25) 1.12 Series of rational fractions (page 26) 1.2 The Exponential Function (page 26) 1.21 Series representation (page 26) 1.22 Functional relations (page 27) 1.23 Series of exponentials (page 27) 1.3-1.4 Trigonometric and Hyperbolic Functions (page 27) 1.30 Introduction (page 28) 1.31 The basic functional relations (page 28) 1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) (page 30) 1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions (page 32) 1.34 Certain sums of trigonometric and hyperbolic functions (page 35) 1.35 Sums of powers of trigonometric functions of multiple angles (page 36) 1.36 Sums of products of trigonometric functions of multiple angles (page 37) 1.37 Sums of tangents of multiple angles (page 38) 1.38 Sums leading to hyperbolic tangents and cotangents (page 38) 1.39 The representation of cosines and sines of multiples of the angle as finite products (page 39) 1.41 The expansion of trigonometric and hyperbolic functions in power series (page 41) 1.42 Expansion in series of simple fractions (page 42) 1.43 Representation in the form of an infinite product (page 43) 1.44-1.45 Trigonometric (Fourier) series (page 44) 1.46 Series of products of exponential and trigonometric functions (page 48) 1.47 Series of hyperbolic functions (page 49) 1.48 Lobachevskiy's "Angle of parallelism"
(page 49) 1.49 The hyperbolic amplitude (the Gudermannian)
gd(x)
(page 50) 1.5 The Logarithm (page 51) 1.51 Series representation (page 51) 1.52 Series of logarithms (cf. 1.431) (page 53) 1.6 The Inverse Trigonometric and Hyperbolic Functions (page 54) 1.61 The domain of definition (page 54) 1.62-1.63 Functional relations (page 54) 1.64 Series representations (page 58)
2Indefinite Integrals of Elementary Functions(page 61) 2.0 Introduction (page 61) 2.00 General remarks (page 61) 2.01 The basic integrals (page 61) 2.02 General formulas (page 62) 2.1 Rational functions (page 64) 2.10 General integration rules (page 64) 2.11-2.13 Forms containing the binomial a+bxk(page 66) 2.14 Forms containing the binomial
(page 72) 2.15 Forms containing pairs of binomials: a+bx and
(page 76) 2.16 Forms containing the trinomial
a+bxk+cx2k(page 76) 2.17 Forms containing the quadratic trinomial a+bx+cx2 and powers of x(page 77) 2.18 Forms containing the quadratic trinomial a+bx+cx2 and the binomial
(page 79) 2.2 Algebraic functions (page 80) 2.20 Introduction (page 80) 2.21 Forms containing the binomial a+bxk and
(page 81) 2.22-2.23 Forms containing
(page 83) 2.24 Forms containing
and the binomial
(page 86) 2.25 Forms containing
(page 90) 2.26 Forms containing
and integral powers of x(page 92) 2.27 Forms containing
and integral powers of x(page 97) 2.28 Forms containing
and first-and second-degree polynomials (page 101) 2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals (page 102) 2.3 The Exponential Function (page 104) 2.31