Table of Contents
Preface ix
1 Number systems 1
1.1 Sets 1
1.2 Domains 2
1.3 Assumptions in Mathematics® 5
1.4 Quantifiers 7
1.5 Complex numbers 10
1.6 Real numbers 11
1.7 Infinities 13
1.8 Integers and the Principle of Mathematical Induction 15
1.8.1 Example 16
1.8.2 Example 16
1.9 Algebraic equations and algebraic numbers 18
1.10 Non-algebraic equations 21
1.11 Sequences of real numbers and their limits 22
1.11.1 Example 25
1.11.2 Example: the number e 25
1.12 Supremumandinfimum 27
1.12.1 Example 29
2 Recursive sequences, discrete dynamical systems and their limits 33
2.1 Example 34
2.2 Example: the Fibonacci sequence 41
3 Series 47
3.1 Sequences and series 47
3.2 The functions Sum and NSum 49
3.3 Absolute convergence 51
3.4 Convergence of series with terms of constant signs 52
3.4.1 Example 53
3.5 Convergence of series with terms of non-constant signs 55
3.5.1 Grouping of terms 55
3.5.2 Example 56
3.5.3 Abel's summation formula 57
3.5.4 Dirichlet's and Abel's tests 58
3.5.5 Example 60
3.6 The function SumConvergence 62
3.6.1 Example 63
3.7 Riemann's theorem on conditionally convergent series 65
3.8 The Cauchy product of series 67
3.9 Divergent series 68
3.10 Power series 71
4 Limits of functions and continuity 75
4.1 Limits of functions 75
4.2 One-sided limits 76
4.3 Continuous functions 78
4.4 Discontinuous functions 79
4.4.1 Example: the Dirichlet function 82
4.5 The main theorems on continuous functions 84
4.5.1 Example 85
4.6 Inverse functions and their continuity 86
4.7 Example: recursive sequences and continuity 88
4.8 Uniform continuity and the Lipschitz property 90
4.8.1 Example 92
5 Differentiation 95
5.1 Difference quotient and derivative of a function 95
5.2 Differentiation in Mathematica® 102
5.2.1 Differentiation of expressions using D 102
5.2.2 Differentiation of functions using Derivative 105
5.2.3 Algebraic rules of differentiation 107
5.2.4 Example: user-defined derivative 108
5.3 Main properties of differentiable functions 109
5.3.1 Example: global and local extrema 111
5.3.2 Example 114
5.3.3 Example: the inverse function of the hyperbolic sine 117
5.4 Convex functions 119
5.4.1 Jensen's inequality 123
6 Sequences and series of functions 125
6.1 Power series continued 125
6.1.1 Example 126
6.1.2 Example 127
6.2 Taylor polynomials and Taylor series 129
6.2.1 Example 135
6.2.2 Example 136
6.2.3 Approximating functions by Taylor polynomials 137
6.2.4 Example: rational approximation of $$$ 139
6.2.5 Example: illustration of approximation of functions with Taylor polynomials 140
6.3 Convergence of sequences and series of functions 142
6.3.1 Examples: pointwise, uniform and almost uniform convergence of function sequences 145
6.3.2 Continuity and differentiability of limits and sums 149
6.3.3 Examples: pointwise, uniform and almost uniform convergence of function series 153
6.3.4 Example 156
7 Integration 159
7.1 Indefinite integrals 159
7.2 The Risch algorithm 163
7.2.1 Differential algebras 163
7.2.2 Example 1: integration of rational functions 165
7.2.3 Example 2: the Risch algorithm for an exponential extension 167
7.2.4 Limitations of Mathematica®'s integration 168
7.3 The Riemann integral 169
7.3.1 Using Integrate and NIntegrate with definite integrals 175
7.3.2 Riemann sums 176
7.4 Improper integrals 178
7.4.1 Integrals over infinite intervals (improper integrals of the first type) 178
7.4.2 Improper integrals of the first type and infinite sums 180
7.4.3 Integrals of unbounded functions (improper integrals of the second type) 184
Bibliography 187
Index 189
