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A Course in Mathematical Analysis

Book Description

The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. This first volume focuses on the analysis of real-valued functions of a real variable. Besides developing the basic theory it describes many applications, including a chapter on Fourier series. It also includes a Prologue in which the author introduces the axioms of set theory and uses them to construct the real number system. Volume II goes on to consider metric and topological spaces and functions of several variables. Volume III covers complex analysis and the theory of measure and integration.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Introduction
  6. Part One Prologue: The foundations of analysis
    1. 1 The axioms of set theory
      1. 1.1 The need for axiomatic set theory
      2. 1.2 The first few axioms of set theory
      3. 1.3 Relations and partial orders
      4. 1.4 Functions
      5. 1.5 Equivalence relations
      6. 1.6 Some theorems of set theory
      7. 1.7 The foundation axiom and the axiom of infinity
      8. 1.8 Sequences, and recursion
      9. 1.9 The axiom of choice
      10. 1.10 Concluding remarks
    2. 2 Number systems
      1. 2.1 The non-negative integers and the natural numbers
      2. 2.2 Finite and infinite sets
      3. 2.3 Countable sets
      4. 2.4 Sequences and subsequences
      5. 2.5 The integers
      6. 2.6 Divisibility and factorization
      7. 2.7 The field of rational numbers
      8. 2.8 Ordered fields
      9. 2.9 Dedekind cuts
      10. 2.10 The real number field
  7. Part Two Functions of a real variable
    1. 3 Convergent sequences
      1. 3.1 The real numbers
      2. 3.2 Convergent sequences
      3. 3.3 The uniqueness of the real number system
      4. 3.4 The Bolzano–Weierstrass theorem
      5. 3.5 Upper and lower limits
      6. 3.6 The general principle of convergence
      7. 3.7 Complex numbers
      8. 3.8 The convergence of complex sequences
    2. 4 Infinite series
      1. 4.1 Infinite series
      2. 4.2 Series with non-negative terms
      3. 4.3 Absolute and conditional convergence
      4. 4.4 Iterated limits and iterated sums
      5. 4.5 Rearranging series
      6. 4.6 Convolution, or Cauchy, products
      7. 4.7 Power series
    3. 5 The topology of R
      1. 5.1 Closed sets
      2. 5.2 Open sets
      3. 5.3 Connectedness
      4. 5.4 Compact sets
      5. 5.5 Perfect sets, and Cantor’s ternary set
    4. 6 Continuity
      1. 6.1 Limits and convergence of functions
      2. 6.2 Orders of magnitude
      3. 6.3 Continuity
      4. 6.4 The intermediate value theorem
      5. 6.5 Point-wise convergence and uniform convergence
      6. 6.6 More on power series
    5. 7 Differentiation
      1. 7.1 Differentiation at a point
      2. 7.2 Convex functions
      3. 7.3 Differentiable functions on an interval
      4. 7.4 The exponential and logarithmic functions; powers
      5. 7.5 The circular functions
      6. 7.6 Higher derivatives, and Taylor’s theorem
    6. 8 Integration
      1. 8.1 Elementary integrals
      2. 8.2 Upper and lower Riemann integrals
      3. 8.3 Riemann integrable functions
      4. 8.4 Algebraic properties of the Riemann integral
      5. 8.5 The fundamental theorem of calculus
      6. 8.6 Some mean-value theorems
      7. 8.7 Integration by parts
      8. 8.8 Improper integrals and singular integrals
    7. 9 Introduction to Fourier series
      1. 9.1 Introduction
      2. 9.2 Complex Fourier series
      3. 9.3 Uniqueness
      4. 9.4 Convolutions, and Parseval’s equation
      5. 9.5 An example
      6. 9.6 The Dirichlet kernel
      7. 9.7 The Fejér kernel and the Poisson kernel
    8. 10 Some applications
      1. 10.1 Infinite products
      2. 10.2 The Taylor series of logarithmic functions
      3. 10.3 The beta function
      4. 10.4 Stirling’s formula
      5. 10.5 The gamma function
      6. 10.6 Riemann’s zeta function
      7. 10.7 Chebyshev’s prime number theorem
      8. 10.8 Evaluating ζ(2)
      9. 10.9 The irrationality of er
      10. 10.10 Theirrationalityof π
  8. Appendix A Zorn’s lemma and the well-ordering principle
    1. A.1 Zorn’s lemma
    2. A.2 The well-ordering principle
  9. Index