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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 the 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. Volume 1 focuses on the analysis of real-valued functions of a real variable. Volume 2 goes on to consider metric and topological spaces. This third volume develops the classical theory of functions of a complex variable. It carefully establishes the properties of the complex plane, including a proof of the Jordan curve theorem. Lebesgue measure is introduced, and is used as a model for other measure spaces, where the theory of integration is developed. The RadonÐNikodym theorem is proved, and the differentiation of measures discussed.

Table of Contents

  1. Cover
  2. Half-title page
  3. Title
  4. Copyright
  5. Contents
  6. Introduction
  7. Part Five Complex analysis
    1. 20 Holomorphic functions and analytic functions
      1. 20.1 Holomorphic functions
      2. 20.2 The Cauchy-Riemann equations
      3. 20.3 Analytic functions
      4. 20.4 The exponential, logarithmic and circular functions
      5. 20.5 Infinite products
      6. 20.6 The maximum modulus principle
    2. 21 The topology of the complex plane
      1. 21.1 Winding numbers
      2. 21.2 Homotopic closed paths
      3. 21.3 The Jordan curve theorem
      4. 21.4 Surrounding a compact connected set
      5. 21.5 Simply connected sets
    3. 22 Complex integration
      1. 22.1 Integration along a path
      2. 22.2 Approximating path integrals
      3. 22.3 Cauchy’s theorem
      4. 22.4 The Cauchy kernel
      5. 22.5 The winding number as an integral
      6. 22.6 Cauchy’s integral formula for circular and square paths
      7. 22.7 Simply connected domains
      8. 22.8 Liouville’s theorem
      9. 22.9 Cauchy’s theorem revisited
      10. 22.10 Cycles; Cauchy’s integral formula revisited
      11. 22.11 Functions defined inside a contour
      12. 22.12 The Schwarz reflection principle
    4. 23 Zeros and singularities
      1. 23.1 Zeros
      2. 23.2 Laurent series
      3. 23.3 Isolated singularities
      4. 23.4 Meromorphic functions and the complex sphere
      5. 23.5 The residue theorem
      6. 23.6 The principle of the argument
      7. 23.7 Locating zeros
    5. 24 The calculus of residues
      1. 24.1 Calculating residues
      2. 24.2 Integrals of the form
      3. 24.3 Integrals of the form
      4. 24.4 Integrals of the form
      5. 24.5 Integrals of the form
    6. 25 Conformal transformations
      1. 25.1 Introduction
      2. 25.2 Univalent functions on C
      3. 25.3 Univalent functions on the punctured plane C*
      4. 25.4 The Möbius group
      5. 25.5 The conformal automorphisms of D
      6. 25.6 Some more conformal transformations
      7. 25.7 The space H (U) of holomorphic functions on a domain U
      8. 25.8 The Riemann mapping theorem
    7. 26 Applications
      1. 26.1 Jensen’s formula
      2. 26.2 The function π cot πz
      3. 26.3 The functions πcosec πz
      4. 26.4 Infinite products
      5. 26.5 *Euler’s product formula*
      6. 26.6 Weierstrass products
      7. 26.7 The gamma function revisited
      8. 26.8 Bernoulli numbers, and the evaluation of ζ(2k)
      9. 26.9 The Riemann zeta function revisited
  8. Part Six Measure and Integration
    1. 27 Lebesgue measure on R
      1. 27.1 Introduction
      2. 27.2 The size of open sets, and of closed sets
      3. 27.3 Inner and outer measure
      4. 27.4 Lebesgue measurable sets
      5. 27.5 Lebesgue measure on R
      6. 27.6 A non-measurable set
    2. 28 Measurable spaces and measurable functions
      1. 28.1 Some collections of sets
      2. 28.2 Borel sets
      3. 28.3 Measurable real-valued functions
      4. 28.4 Measure spaces
      5. 28.5 Null sets and Borel sets
      6. 28.6 Almost sure convergence
    3. 29 Integration
      1. 29.1 Integrating non-negative functions
      2. 29.2 Integrable functions
      3. 29.3 Changing measures and changing variables
      4. 29.4 Convergence in measure
      5. 29.5 The spaces and
      6. 29.6 The spaces and , for 0 < p < ∞
      7. 29.7 The spaces and
    4. 30 Constructing measures
      1. 30.1 Outer measures
      2. 30.2 Caratheodory’s extension theorem
      3. 30.3 Uniqueness
      4. 30.4 Product measures
      5. 30.5 Borel measures on R, I
    5. 31 Signed measures and complex measures
      1. 31.1 Signed measures
      2. 31.2 Complex measures
      3. 31.3 Functions of bounded variation
    6. 32 Measures on metric spaces
      1. 32.1 Borel measures on metric spaces
      2. 32.2 Tight measures
      3. 32.3 Radon measures
    7. 33 Differentiation
      1. 33.1 The Lebesgue decomposition theorem
      2. 33.2 Sublinear mappings
      3. 33.3 The Lebesgue differentiation theorem
      4. 33.4 Borel measures on R, II
    8. 34 Applications
      1. 34.1 Bernstein polynomials
      2. 34.2 The dual space of , for 1 ≤ p < oo
      3. 34.3 Convolution
      4. 34.4 Fourier series revisited
      5. 34.5 The Poisson kernel
      6. 34.6 Boundary behaviour of harmonic functions
  9. Index
  10. Contents for Volume I
  11. Contents for Volume II