Functions of A Complex Variable

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.
About the Author: Phillip Franklin was one of the most famous American-born mathematicians of the first half of the 20th century. He was one of the first American mathematicians with a truly modern training; getting his doctorate at Princeton University under Oswald Velben in 1922 and beginning his career by spreading knowledge of the frontiers of mathematics to the other top programs. Marshall Stone famously commented that the entire Harvard University department-students and faculty alike- learned the elements of topology from the seminar Franklin taught when he arrived there as a new PhD. He then moved on to the Massachusetts Institute of Technology where we spent the bulk of his career, both teaching and doing research. Franklin became well known as an analyst, but the truth is he published in a number of fields, including geometry, number theory, graph theory and topology. He's best known for authoring some of the most well-received and influential textbooks of his generation. Among his most famous texts were Differential equations for electrical engineers (1933), the exhaustive undergraduate analysis text A Treatise on advanced calculus (1940), the standard introductory calculus text, Differential and integral calculus (1953) and the honors calculus text Compact calculus (1963). About Publisher/ Editor: Karo Maestro and The Mathemagician are both nome-de-plume of a former graduate student in mathematics. Some things this enigma can reveal: He was a distinguished undergraduate student as a double major in mathematics and biochemistry whose poor health & personal tragedies prevented completing graduate studies. Unbowed and undaunted, he plans to return to ultimately obtain a PhD in pure mathematics before dying. Partially to that end, he is building Blue Collar Scholar, a publishing company committed to making high quality sources of mathematics-both original works and reprints-available widely and inexpensively to students of all backgrounds. He has 3 bachelors' degrees, in philosophy, physical chemistry and mathematics as well as minors in biochemistry and psychology. He is also a past reviewer of textbooks for the Mathematical Association Of America.He's painfully blunt, opinionated and has a comment on just about everything in creation. He's painfully blunt, opinionated and has a comment on just about everything in creation-from math textbooks to progressive politics to how to make a perfect cup of organic green tea.