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Examples and Theorems in Analysis

Examples and Theorems in Analysis takes a unique and very practical approach to mathematical analysis. It makes the subject more accessible by giving the examples equal status with the theorems. The results are introduced and motivated by reference to examples which illustrate their use, and further...

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Bibliographic Details
Main Author: Walker, Peter
Format: eBook
Language:English
Published: London Springer London 2004, 2004
Edition:1st ed. 2004
Subjects:
Fourier Analysis
Mathematical Analysis
Functions Of Real Variables
Analysis
Analysis (mathematics)
Real Functions
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1. Sequences
  • 1.1 Examples, Formulae and Recursion
  • 1.2 Monotone and Bounded Sequences
  • 1.3 Convergence
  • 1.4 Subsequences
  • 1.5 Cauchy Sequences
  • Exercises
  • 2. Functions and Continuity
  • 2.1 Examples
  • 2.2 Monotone and Bounded Functions
  • 2.3 Limits and Continuity
  • 2.4 Bounds and Intermediate Values
  • 2.5 Inverse Functions
  • 2.6 Recursive Limits and Iteration
  • 2.7 One-Sided and Infinite Limits. Regulated Functions
  • 2.8 Countability
  • Exercises
  • 3. Differentiation
  • 3.1 Differentiable Functions
  • 3.2 The Significance of the Derivative
  • 3.3 Rules for Differentiation
  • 3.4 Mean Value Theorems and Estimation
  • 3.5 More on Iteration
  • 3.6 Optimisation
  • Exercises
  • 4. Constructive Integration
  • 4.1 Step Functions
  • 4.2 The Integral of a Regulated function
  • 4.3 Integration and Differentiation
  • 4.4 Applications
  • 4.5 Further Mean Value Theorems
  • Exercises
  • 5. Improper Integrals
  • 5.1 Improper Integrals on an Interval
  • 5.2 Improper Integrals at Infinity
  • 5.3 The Gamma function
  • Exercises
  • 6. Series
  • 6.1 Convergence
  • 6.2 Series with Positive Terms
  • 6.3 Series with Arbitrary Terms
  • 6.4 Power Series
  • 6.5 Exponential and Trigonometric Functions
  • 6.6 Sequences and Series of Functions
  • 6.7 Infinite Products
  • Exercises
  • 7. Applications
  • 7.1 Fourier Series
  • 7.2 Fourier Integrals
  • 7.3 Distributions
  • 7.4 Asymptotics
  • 7.5 Exercises
  • A. Fubini’s Theorem
  • B. Hints and Solutions for Exercises

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