Mathematical Analysis

1. Real Numbers and Natural Numbers. - 1. 1 Introduction. - 1. 2 The Axiomatic Approach to Real Numbers. - 1. 3 Natural Numbers. - 1. 4 Summing Up. - 1. 5 Exercises. - 2. Sequences of Real Numbers. - 2. 1 Sequences. - 2. 2 Equivalent Formulations of the Continuity Axiom. - 2. 3 Limits of Sequences and Continuity. - 2. 4 Some Special Sequences. - 2. 5 An Alternative Definition of Exponentials and Logarithms. - 2. 6 Summing Up. - 2. 7 Exercises. - 3. Integer Numbers: Congruences Counting and Infinity. - 3. 1 Congruences. - 3. 2 Combinatorics. - 3. 3 Infinity. - 3. 4 Summing Up. - 3. 5 Exercises. - 4. Complex Numbers. - 4. 1 Complex Numbers. - 4. 2 Sequences of Complex Numbers. - 4. 3 Some Elementary Applications. - 4. 4 Summing Up. - 4. 5 Exercises. - 5. Polynomials Rational Functions and Trigonometric Polynomials. - 5. 1 Polynomials. - 5. 2 Solutions of Polynomial Equations. - 5. 3 Rational Functions. - 5. 4 Sinusoidal Functions and Their Sums. - 5. 5 Summing Up. - 5. 6 Exercises. - 6. Series. - 6. 1 Basic Facts. - 6. 2 Taylor Series e and ?. - 6. 3 Series of Nonnegative Terms. - 6. 4 Series of Terms of Arbitrary Sign. - 6. 5 Series of Products. - 6. 6 Products of Series. - 6. 7 Rearrangements. - 6. 8 Summing Up. - 6. 9 Exercises. - 7. Power Series. - 7. 1 Basic Theory. - 7. 2 Further Results. - 7. 3 Some Applications. - 7. 4 Further Applications. - 7. 5 Summing Up. - 7. 6 Exercises. - 8. Discrete Processes. - 8. 1 Recurrences. - 8. 2 One-Dimensional Dynamical Systems. - 8. 3 Two-Dimensional Dynamical Systems. - 8. 4 Exercises. - A. Mathematicians and Other Scientists. - B. Bibliographical Notes. - C. Index. Language: English