Publisher's Synopsis
This outstanding and unusual introduction to functions of one complex variable is now being reissued after 60 years by Blue Collar Scholar/Createspace as part of our long term goal of unearthing long-out-print quality textbooks in mathematics and making them available again to all students, teachers and autodidacts of mathematics in inexpensive editions. This book by legendary analyst/teacher Phillip Franklin gives a complete and above all, balanced first course in analytic function theory. Where most introductory modern books on complex analysis usually take either a geometric approach (via complex differentiability and conformal mappings) or an analytic approach (via local power series expansions of complex functions), Franklin gives neither precedence. Both aspects of the subject are woven together in equal measure in his presentation, emphasizing the connections between them. Chapter 1 describes the algebra and topology of the complex plane. Chapter 2 introduces functions of one complex variable and their calculus: limits, differentiation and some applications to geometry and analysis such as the solution of the Laplace equation. Chapter 3 extends power series approximations and their convergence properties to complex valued functions. Chapter 4 discusses the so-called elementary functions that form the building blocks of complex analysis; the exponential, the trigonometric functions, hyperbolic functions and power functions. Chapters 5 and 6 discuss the conformal mappings, which completely characterize the deep interplay of plane geometry and analysis with the complex numbers. Chapter 7 discusses integration in the complex plane and the Cauchy-Goursat theorem which is the complex analogue of the fundamental theorem of calculus for path integrals in C. Chapter 8 discusses the Taylor series expansion in the complex plane and its many applications to analytic functions including Louisville's and Morera's theorems. Chapter 9 discusses singularities, the Laurent series and the coefficients of the local expansions, which are the residues. The final chapter describes many of the applications of residues in complex analysis. The book builds incrementally in intricacy-it begins with the simplest and most obvious properties of the complex plane and gradually increases in difficulty as one proceeds. This is a very wise organizational approach, born of many years teaching the subject to strong students in both pure and applied mathematics. There are a host of excellent exercises which include many side lemmas and theorems that are not covered explicitly in the text and are quite diverse in level. The result is the problem sets not only provide much needed practice in both calculation and proof, they expand the actual coverage of the book significantly. This text can be used for either an intensive undergraduate course or a beginning graduate course for either mathematics students or students of the physical sciences. Prerequisites for the book are minimal: A working knowledge of calculus up to and including partial derivatives, multiple and line integrals and a one semester rigorous analysis course on the real line where limits are defined and used carefully. In addition, BCS editor/ founder Karo Maestro has written a personal and entertaining new preface for the beginning student and instructor on the remarkable field this book is an introduction to. He has also added a new recommended reading section guiding the student to further study. Literately and clearly written with many pictures, this is a superior, inspiring and affordable first course on complex analysis that not only will be treasured by pure and applied mathematics students, it will provide a rock-solid foundation for further study.
