The idea of complex numbers dates back at least 300 years-to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers. This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications. Features: This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysis This book has an exceptionally large number of examples and a large number of figures. The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking. Incisive applications appear throughout the book. Partial differential equations are used as a unifying theme.

This new edition of an indispensable text provides a clear treatment of Fourier Series, Fourier Transforms, and FFTs. The unique software, included with the book and newly updated for this edition, allows the reader to generate, firsthand, images of all aspects of Fourier analysis described in the text. Topics covered include applications to vibrating strings, heat conduction, removal of noise and frequency detection, filtering of Fourier Series and improvement of convergence, and much more.

This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book."

The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings. This book is based on Steven G. Krantz's notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974. This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.

During the past decade, mathematics education has changed rapidly, giving rise to a polarization of opinions among the community of research mathematicians. What is the appropriate balance among theory, technique, and applications? What is the role of technology? How do we fulfill the needs of students entering other fields? The purpose of this volume, the proceedings of a conference held at the Mathematical Sciences Research Institute in Berkeley in 1996, is to present a serious discussion of these educational issues, with a balanced representation of opposing ideas. Part I deals with general issues in university mathematics education; Part II presents case studies on particular projects; Part III presents a range of opinions on mathematics education in elementary and secondary schools; and Part IV presents the reports of the working groups.