Real and Complex Analysis

Volume 1This unique book develops the subject of analysis organically, by presenting techniques and concepts that apply more generally to functions of various types. By considering these various function types together (real-valued functions of a single or several real variables, vector-valued functions of a single or several real variables, and complex functions), the student can better appreciate what is common to all of them, and what is distinctive to each. For the instructor, this approach also provides for certain pedagogical advantages. Like the first, the aim of this second edition of Real and Complex Analysis, Volume 1, is to benefit the student of analysis as best we can. Changes were made with the student in mind. The most evident change is one that we hope provides more flexibility to instructors, and more affordability to students. The new edition splits the text into two volumes, available individually or as a set. Other changes include clarifications and improvements suggested by readers. Exercises within each chapter have been thoroughly reviewed and reorganized, and partial solutions provided. Volume one has over 800 exercises in total, with more than 400 embedded exercises, and nearly the same number of end-of-chapter supplementary exercises. Solutions to embedded and supplementary exercises are located on the Instructor Resources Download Hub, which can be accessed from the Instructor & Student Resources tab further down this page. Volume 2This text presents real and complex integration theory, as well as mapping properties of complex functions from the complex plane to itself. Comprising the second volume to the second edition of Real and Complex Analysis, this work completes the development, begun in Volume 1, of real and complex functions and their properties. The two volumes together present a unique, yet elegant and approachable treatment of analysis. Like the first volume, Volume 2 was written with the student in mind. Containing more examples and a more thorough treatment of mappings than the first edition, the text offers over 300 exercises. It provides hints and solutions to all odd-numbered embedded problems, and continues the authors’ philosophy of exploring real and complex functions side-by-side, as in Volume 1. Chapter 1 rigorously develops the Riemann integral and its properties, while Chapter 2 is devoted to the rich and rewarding theory of complex integration. Chapter 3 introduces power series, Taylor series, and Laurent series, including the fundamentals of the residue calculus for computing both real and complex integrals. Finally, the fourth chapter explores complex functions as mappings of the complex plane. Intended for advanced undergraduates who have completed a college-level calculus sequence and a first course in proof techniques, it is also well-suited for a first-year graduate course in analysis.