Introduction to Analysis with Complex Numbers

✍ Scribed by Irena Swanson

• Year: 2020
• Tongue: English
• Leaves: 388
• Category: Library

✦ Synopsis

"These notes were written expressly for Mathematics 112 at Reed College, with first use in the spring of 2013. The title of the course is “Introduction to Analysis”. The prerequisite is calculus. I maintain two versions of these notes, one in which the natural, rational and real numbers are constructed and the Least upper bound theorem is proved for the ordered field of real numbers, and one version in which the Least upper bound property is assumed for the ordered field of real numbers. You are reading the longer, former version."

✦ Table of Contents

Table of Contents
Preface
The briefest overview, motivation, notation
Chapter 1: How we will do mathematics
1.1 Statements and proof methods
1.2 Statements with quantifiers
1.3 More proof methods
1.4 Logical negation
1.5 Summation
1.6 Proofs by (mathematical) induction
1.7 Pascal's triangle
Chapter 2: Concepts with which we will do mathematics
2.1 Sets
2.2 Cartesian product
2.3 Relations, equivalence relations
2.4 Functions
2.5 Binary operations
2.6 Fields
2.7 Order on sets, ordered fields
2.8 What are the integers and the rational numbers?
2.9 Increasing and decreasing functions
2.10 Absolute values
Chapter 3: Construction of the number systems
3.1 Inductive sets, a construction of natural numbers
3.2 Arithmetic on ℕ₀
3.3 Order on ℕ₀
3.4 Cancellation in ℕ₀
3.5 Construction of ℤ, arithmetic, and order on ℤ
3.6 Construction of the ordered field ℚ
3.7 Construction of the field ℝ of real numbers
3.8 Order on ℝ, the Least upper bound theorem
3.9 Complex numbers
3.10 Functions related to complex numbers
3.11 Absolute value in ℂ
3.12 Polar coordinates
3.13 Topology on the fields of real and complex numbers
3.14 The Heine–Borel theorem
Chapter 4: Limits of functions
4.1 Limit of a function
4.2 When a number is not a limit
4.3 More on the definition of a limit
4.4 Limit theorems
4.5 Infinite limits (for real-valued functions)
4.6 Limits at infinity
Chapter 5: Continuity
5.1 Continuous functions
5.2 Topology and the Extreme value theorem
5.3 Intermediate value theorem
5.4 Radical functions
5.5 Uniform continuity
Chapter 6: Differentiation
6.1 Definition of derivatives
6.2 Basic properties of derivatives
6.3 The Mean value theorem
6.4 L'Hôpital's rule
6.5 Higher-order derivatives, Taylor polynomials
Chapter 7: Integration
7.1 Approximating areas
7.2 Computing integrals from upper and lower sums
7.3 What functions are integrable?
7.4 The Fundamental theorem of calculus
7.5 Integration of complex-valued functions
7.6 Natural logarithm and the exponential functions
7.7 Applications of integration
Chapter 8: Sequences
8.1 Introduction to sequences
8.2 Convergence of infinite sequences
8.3 Divergence of infnite sequences and infinite limits
8.4 Convergence theorems via epsilon-N proofs
8.5 Convergence theorems via functions
8.6 Bounded sequences, monotone sequences, ratio test
8.7 Cauchy sequences, completeness of ℝ, ℂ
8.8 Subsequences
8.9 Liminf, limsup for real-valued sequences
Chapter 9: Infinite series and power series
9.1 Infinite series
9.2 Convergence and divergence theorems for series
9.3 Power series
9.4 Differentiation of power series
9.5 Numerical evaluations of some series
9.6 Some technical aspects of power series
9.7 Taylor series
Chapter 10: Exponential and trigonometric functions
10.1 The exponential function
10.2 The exponential function, continued
10.3 Trigonometry
10.4 Examples of L'Hôpital's rule
10.5 Trigonometry for the computation of some series
Appendix A: Advice on writing mathematics
Appendix B: What one should never forget
Index