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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.
Essentials of Mathematical Thinking addresses the growing need to
better comprehend mathematics today. Increasingly, our world is
driven by mathematics in all aspects of life. The book is an
excellent introduction to the world of mathematics for students not
majoring in mathematical studies. The author has written this book
in an enticing, rich manner that will engage students and introduce
new paradigms of thought. Careful readers will develop critical
thinking skills which will help them compete in today's world. The
book explains: What goes behind a Google search algorithm How to
calculate the odds in a lottery The value of Big Data How the
nefarious Ponzi scheme operates Instructors will treasure the book
for its ability to make the field of mathematics more accessible
and alluring with relevant topics and helpful graphics. The author
also encourages readers to see the beauty of mathematics and how it
relates to their lives in meaningful ways.
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.
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