# Real and Complex Analysis

## 1st Edition

Chapman and Hall/CRC

567 pages | 67 B/W Illus.

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### Description

Presents Real & Complex Analysis Together Using a Unified Approach

A two-semester course in analysis at the advanced undergraduate or first-year graduate level

Unlike other undergraduate-level texts, Real and Complex Analysis develops both the real and complex theory together. It takes a unified, elegant approach to the theory that is consistent with the recommendations of the MAA’s 2004 Curriculum Guide.

By presenting real and complex analysis together, the authors illustrate the connections and differences between these two branches of analysis right from the beginning. This combined development also allows for a more streamlined approach to real and complex function theory. Enhanced by more than 1,000 exercises, the text covers all the essential topics usually found in separate treatments of real analysis and complex analysis. Ancillary materials are available on the book’s website.

This book offers a unique, comprehensive presentation of both real and complex analysis. Consequently, students will no longer have to use two separate textbooks—one for real function theory and one for complex function theory.

The Spaces R, Rk, and C

The Real Numbers R

The Real Spaces Rk

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: Dk → R

The Derivative for f: Dk → Rp

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

Exercises appear at the end of each chapter.

### About the Authors/Editors

Christopher Apelian is an associate professor and chair of the Department of Mathematics and Computer Science at Drew University. Dr. Apelian has published papers on the application of probability and stochastic processes to the modeling of turbulent transport.

Steve Surace is an associate professor in the Department of Mathematics and Computer Science at Drew University. Dr. Surace is also the Associate Director of the New Jersey Governor’s School in the Sciences held at Drew University every summer.

### Subject Categories

##### BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis