7 Best Books on Theory of Analytic Functions

“Analytic Function Theory” Book Review: The objective of this book is to provide an understanding of the major aspects of analytic function theory. It begins with a detailed exploration of the key concepts and historical background of continuity in analytic function theory. The book thoroughly explains canonical topics such as elliptic functions, entire and meromorphic functions, and conformal mapping. The chapters highlight the evolving frontiers and significant applications of analytic function theory. The final section of the book covers major aspects and topics, including majorization and functions holomorphic in a half-plane. By discussing into these subjects, this book offers readers a deep insight into the rich and diverse field of analytic function theory.

“Elementary Theory of Analytic Functions of One or Several Complex Variables” Book Review: This book provides a foundational theory of analytic functions in a complex variable, extending to analytic functions in multiple real or complex variables. It introduces an existence theorem for solving differential systems when the data is analytic. The chapters of this book are structured around key topics, including power series in one variable, Taylor and Laurent expansions, analytic functions in several variables, harmonic functions, convergence of sequences and series of functions, holomorphic transformations, and holomorphic systems of differential equations. The book presents a comprehensive theory of abstract complex manifolds of one complex dimension, holomorphic functions, and Cauchy’s integral. Each chapter concludes with a set of exercises to reinforce understanding and proficiency in the subject matter.

“Introduction to Analytic Number Theory” Book Review: This book is dedicated to exploring the fundamental techniques and principles of number theory. It discusses the principles of analytic number theory, providing a comprehensive overview of the subject. The book starts by presenting clear explanations of basic concepts, such as divisibility, congruence, and arithmetical functions. It then progresses to more advanced and intricate topics, including Dirichlet’s theorem on primes in progressions, Gauss sums, quadratic residues, Dirichlet series, and Euler products. A chapter on partitions is also included. Throughout the book, basic calculus, complex integration, and residue calculus are utilized as tools to handle featured theorems and methods. Numerous exercises are provided at the end of each chapter. With its well-structured, precise, and clear chapters, this book is an ideal choice for undergraduate students seeking an understanding of number theory.

“Introduction to Circle Packing: The Theory of Discrete Analytic Functions” Book Review: This book is an updated and revised edition that encompasses the latest theory, computations, and applications of circle packing. Each chapter of the book offers an exploration of the elegance of circle geometry, providing clear-cut theory, classical topics, and emerging applications of circle packing. Beginning with the fundamentals of circle packing, the book gradually progresses to more advanced topics, including complex analysis and Riemann surfaces. The book is richly illustrated with various images.

“Banach Spaces of Analytic Functions” Book Review: This book provides an exploration of the role of functional analysis and analytic function theory in mathematics. Each chapter covers essential topics related to Fourier series, analytic and harmonic functions in the unit disc, H1 space, factorization of Hp functions, analytic functions with continuous boundary values, the shift operator, Hp space in the half-plane, and Hp as a Banach space. To enhance the reader’s understanding, the book includes a dedicated section of systematic notes. Each chapter concludes with a set of exercises to reinforce the concepts and principles discussed. This book serves as a valuable resource for those seeking a thorough understanding of functional analysis and analytic function theory, offering clear explanations and practical exercises for readers at various levels of expertise.

“Modular Functions in Analytic Number Theory” Book Review: This book is dedicated to showcasing the applications of modular functions in number-theoretic functions. The chapters are carefully organized and self-contained, providing an exploration of the subject matter. The book extensively covers the general theory of automorphic functions, a theory of far-reaching significance, as well as Riemann surface theory. Special emphasis is placed on recent developments in the theory of automorphic functions, ensuring that readers are up-to-date with the latest advancements. Designed to benefit graduating students interested in analytic number theory, this book serves as a valuable resource for gaining a deeper understanding of the applications and significance of modular functions in this field.