1 Sets and Numbers

1.1 Sets

1.1.1 The concept of a set

1.1.2 Operations on sets

1.1.3 Relations and functions

1.2 Sets of numbers

1.2.1 Two examples

1.2.2 The real number system

1.2.3 Elements of algebra

1.2.4 Elements of topology on **R**

1.2.5 The extended real number system

1.2.6 The complex number system

1.3 Exercises

1.4 References and comments

2 Vector Spaces and Metric Spaces

2.1 Vector spaces

2.1.1 Finite-dimensional vector spaces

2.1.2 Vector spaces

2.1.3 Normed spaces

2.1.4 Hilbert spaces

2.1.5 Inequalities

2.2 Metric spaces

2.3 Compact spaces

2.4 Exercises

2.5 References and comments

3 Sequences and Series

3.1 Numerical sequences

3.1.1 Convergent sequences

3.1.2 Subsequences

3.1.3 Cauchy sequences

3.1.4 Monotonic sequences

3.1.5 Upper limits and lower limits

3.1.6 The big Oh and small oh notations

3.1.7 Stolz-Cesaro theorem and some of its consequences

3.1.8 Certain combinatorial numbers

3.1.9 Unimodal, log-convex, and Pó1ya-frequency sequences

3.1.10 Some special sequences

3.2 Sequences of functions

3.3 Numerical series

3.3.1 Series of nonnegative terms

3.3.2 The root and the ratio tests

3.3.3 Partial summation

3.3.4 Absolutely and conditionally convergent series

3.3.5 The W – Z method

3.4 Series of functions

3.4.1 Power series

3.4.2 Hypergeometric series

3.5 The Riemann Zeta function ζ(p)

3.6 Exercises

3.7 References and comments

4 Limits and Continuity

4.1 Limits

4.1.1 The limit of a function

4.1.2 Right-hand side and left-hand side limits

4.2 Continuity

4.2.1 Continuity and compactness

4.2.2 Uniform continuous mappings

4.2.3 Continuity and connectedness

4.2.4 Discontinuities

4.2.5 Monotonic functions

4.3 Periodic functions

4.4 Darboux functions

4.5 Lipschitz functions

4.6 Convex functions

4.6.1 Convex functions

4.6.2 Jensen convex functions

4.7 Functions of bounded variations

4.8 Continuity of sequences of functions

4.9 Continuity of series of functions

4.10 Exercises

4.11 References and comments

5 Differential Calculus on **R**

5.1 The derivative of a real function

5.2 Mean value theorems

5.2.1 Consequences of the mean value theorems

5.3 The continuity and the surjectivity of derivatives

5.4 L’Hospital theorem

5.5 Higher-order derivatives and the Taylor formula

5.6 Convex functions and differentiability

5.6.1 Inequalities

5.7 Differentiability of sequences and series of functions

5.8 Power series and Taylor series

5.8.1 Operations with power series

5.8.2 The Taylor expansion of some elementary functions

5.8.3 Bernoulli numbers and polynomials

5.9 Some elementary functions introduced by recurrences

5.9.1 The square root function

5.9.2 The logarithm function

5.9.3 The exponential function

5.9.4 The arctangent function

5.10 Functions with primitives

5.10.1 The concept of a primitive function

5.10.2 The existence of primitives for continuous functions

5.11 Exercises

5.12 References and comments

6 Integral Calculus on **R**

6.1 The Darboux-Stieltjes integral

6.1.1 The Darboux integral

6.1.2 The Darboux-Stieltjes integral

6.2 Integrability of sequences and series of functions

6.3 Improper integrals

6.4 Euler integrals

6.4.1 Gamma function

6.4.2 Beta function

6.5 Polylogarithms

6.6 *e* and π are transcendental

6.7 The Gronwall inequality

6.8 Exercises

6.9 References and comments

7 Differential Calculus on **R**^{n}

7.1 Linear and bounded mappings

7.1.1 Multilinear mappings

7.1.2 Quadratic mappings

7.2 Differentiable functions

7.2.1 Variations

7.2.2 Gateaux differential

7.2.3 Frechet differential

7.2.4 Properties of the Frechet differentiable functions

7.3 Partial derivatives

7.3.1 The inverse function theorem and the implicit function theorem

7.3.2 Directional derivatives and gradients

7.4 Higher-order differentials and partial derivatives

7.4.1 The case X = **R**^{n}

7.5 Taylor formula

7.6 Problems of local extremes

7.6.1 First-order conditions

7.6.2 Second-order conditions

7.6.3 Constraint local extremes

7.7 Exercises

7.8 References and comments

8 Double Integrals, Triple Integrals, and Line Integrals

8.1 Double integrals

8.1.1 Double integrals on rectangles

8.1.2 Double integrals on simple domains

8.2 Triple integrals

8.2.1 Triple integrals on parallelepipeds

8.2.2 Triple integrals on simple domains

8.3 n-fold integrals

8.3.1 n-fold integrals on hyperrectangles

8.3.2 n-fold integrals on simple domains

8.4 Line integrals

8.4.1 Line integrals with respect to arc length

8.4.2 Line integrals with respect to axis

8.4.3 Green formula

8.5 Integrals depending on parameters

8.6 Exercises

8.7 References and comments

9 Constants

9.1 Pythagoras's constant

9.1.1 Sequences approaching √2

9.2 Archimedes' constant

9.2.1 Recurrence relation

9.2.2 Buffon needle problem

9.3 Arithmetic-geometric mean

9.4 BBP formulas

9.4.1 Computing the nth binary or hexadecimal digit of π

9.4.2 BBP formulas by binomial sums

9.5 Ramanujan formulas

9.6 Several natural ways to introduce number *e*

9.7 Optimal stopping problem

9.8 References and comments

10 Asymptotic and Combinatorial Estimates

10.1 Asymptotic estimates

10.2 Algorithm analysis

10.3 Combinatorial estimates

10.3.1 Counting relations, topologies, and partial orders

10.3.2 Generalized Fubini numbers

10.3.3 The Catalan numbers and binary trees

10.4 References and comments

References

List of Symbols

Author Index

Subject Index