Real Analysis and Foundations (Hardcover, 4)

Number Systems

The Real Numbers

Appendix: Construction of the Real Numbers

The Complex Numbers

Sequences

Convergence of Sequences

Subsequences

Limsup and Liminf

Some Special Sequences

Series of Numbers

Convergence of Series

Elementary Convergence Tests

Advanced Convergence Tests

Some Special Series

Operations on Series

Basic Topology

Open and Closed Sets

Further Properties of Open and Closed Sets

Compact Sets

The Cantor Set

Connected and Disconnected Sets

Perfect Sets

Limits and Continuity of Functions

Basic Properties of the Limit of a Function

Continuous Functions

Topological Properties and Continuity

Classifying Discontinuities and Monotonicity

Differentiation of Functions

The Concept of Derivative

The Mean Value Theorem and Applications

More on the Theory of Differentiation

The Integral

Partitions and the Concept of Integral

Properties of the Riemann Integral

Change of Variable and Related Ideas

Another Look at the Integral

Advanced Results on Integration Theory

Sequences and Series of Functions

Partial Sums and Pointwise Convergence

More on Uniform Convergence

Series of Functions

The Weierstrass Approximation Theorem

Elementary Transcendental Functions

Power Series .

More on Power Series: Convergence Issues

The Exponential and Trigonometric Functions

Logarithms and Powers of Real Numbers

Differential Equations

Picard’s Existence and Uniqueness Theorem

The Form of a Differential Equation

Picard’s Iteration Technique

Some Illustrative Examples

Estimation of the Picard Iterates

Power Series Methods

Introduction to Harmonic Analysis

The Idea of Harmonic Analysis

The Elements of Fourier Series

An Introduction to the Fourier Transform

Appendix: Approximation by Smooth Functions

Fourier Methods and Differential Equations

Remarks on Different Fourier Notations

The Dirichlet Problem on the Disc

Introduction to the Heat and Wave Equations

Boundary Value Problems

Derivation of the Wave Equation

Solution of the Wave Equation

The Heat Equation

Functions of Several Variables

A New Look at the Basic Concepts of Analysis

Properties of the Derivative

The Inverse and Implicit Function Theorems

Appendix I: Elementary Number Systems

Appendix II: Logic and Set Theory

Appendix III: Review of Linear Algebra