Real and Complex Analysis

The Spaces R, R^k, and C

The Real Numbers R

The Real Spaces R^k

The Complex Numbers C

Point-Set Topology

Bounded Sets

Classification of Points

Open and Closed Sets

Nested Intervals and the Bolzano–Weierstrass Theorem

Compactness and Connectedness

Limits and Convergence

Definitions and First Properties

Convergence Results for Sequences

Topological Results for Sequences

Properties of Infinite Series

Manipulations of Series in R

Functions: Definitions and Limits

Definitions

Functions as Mappings

Some Elementary Complex Functions

Limits of Functions

Functions: Continuity and Convergence

Continuity

Uniform Continuity

Sequences and Series of Functions

The Derivative

The Derivative for f: D¹ ? R

The Derivative for f: D^k ? R

The Derivative for f: D^k ? R^p

The Derivative for f: D ? C

The Inverse and Implicit Function Theorems

Real Integration

The Integral of f: [a, b] ? R

Properties of the Riemann Integral

Further Development of Integration Theory

Vector-Valued and Line Integrals

Complex Integration

Introduction to Complex Integrals

Further Development of Complex Line Integrals

Cauchy’s Integral Theorem and Its Consequences

Cauchy’s Integral Formula

Further Properties of Complex Differentiable Functions

Appendices: Winding Numbers Revisited

Taylor Series, Laurent Series, and the Residue Calculus

Power Series

Taylor Series

Analytic Functions

Laurent’s Theorem for Complex Functions

Singularities

The Residue Calculus

Complex Functions as Mappings

The Extended Complex Plane

Lineal Fractional Transformations

Conformal Mappings

Bibliography

Index