文件名称:
A Concrete Approach to Classical Analysis (Springer 2009)
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详细说明： Preface This book reflects the conclusions of the author to some simple questions: “What should an easy comprehensible introduction to classical mathematical analysis look like? Can we avoid the basic results on differential and integral calculus to jump into a bstract results? Actually, which results are considered as basic? Is the book a bridge to some new topics of research?” The influence of functional analysis and Bourbakism has been clear for a long time. At the same time, numerical methods emerged from analysis. It is hard to imagine discrete mathematics without analysis. New and even unexpected tendencies appeared. It is enough to mention some of them, experimental mathematics and scientific computing. These two topics are illustrated by two remarkable books [22] and [17]. Our answer to all these questions consists of our somehow taking all these fields into account. We mean, on the solid ground of classical results (sets, functions, metric spaces, sequences, series, limits, continuity, differentiability, and integrability) that we have to introduce newer results. Why introduce new results? They forcefully appear every day. Moreover, new and incredible methods appear. We mention only two of them presently considered as belonging to “experimental mathematics,” namely the fast computation of the π number based on BBP methods, Ramanujan methods. Other methods explore strange functions by computers, that is, the nowhere differentiable functions. The latter topic has been considered as one belonging to “pure mathematics.” Presently it came down into the laboratory of mathematical experiments. This means that by experimental methods we catch a result and then prove it rigorously. We are pressed to take into account some parts from mathematics and to neglect many others. The present book is focused on differential and integral calculus. Mathematical analysis offers a solid ground to many achievements in applied and discrete mathematics. In spite of the fact that this book concerns part of what is customarily called mathematical analysis, we have tried to include useful and relevant examples, exercises, and results enlightening the xviii Preface reader on the power of mathematical analysis tools. In this respect the topics covered by our book are quite “concrete.” The strong interplay between so-called theoretical mathematics and scientific computing has been emphasized by D. H. Bailey as “To this day I live in two worlds, theoretical math and scientific computing. I’m trying to marry these two by applying advanced computing to problems in pure mathematics. Experimental mathematics is the outcome.” We continuously had in front of our eyes a generic student wishing to know more about mathematical analysis at the beginning of his or her student life. We tried to offer paths from the standard knowledge of a student to modern and exciting topics in this way showing that a student from the first or second year is able to understand certain research problems. The book has been divided into ten chapters and covers topics on sets and numbers, linear and metric spaces, sequences and series of numbers and functions, limits and continuity, differential and integral calculus of functions of one or several variables, constants (mainly π, but not only) and algorithms for finding them, the W –Z method of summation, and estimates of algorithms and of certain combinatorial problems. Many challenging exercises accompany the text. Most of them have been the subjects of different mathematical competitions during the last few years. In this respect we consider that there is an appropriate balance between what is traditionally called theory and exercises. The topics of the last two chapters bring the student closer to topics belonging also to computer science. In this way it is shown that the frontier between “pure” mathematics and other related topics is more or less a matter of taste. It is the proper moment and place to express our sincere gratitude to Professor Heiner Gonska of the University of Duisburg-Essen, Germany, giving us, among others, the opportunity of using all the facilities of his department and library. Thanks are due to Professor Jonathan M. Borwein of Dalhousie University, Halifax, Nova Scotia, Canada, for his constant and warm friendship along the years and to Professor Karl Dilcher of Dalhousie University, Halifax, Nova Scotia, Canada for his firm support in the publication of this book. The author is also grateful to the editors of Springer-Verlag, New York, for very strong and constant support offered to us. Above all I express my deep gratitude to my wife Viorica for her unbroken encouragement, strong moral support, and constant understanding during the many days of work on this book. Cluj-Napoca, December 2007 Marian Mure¸san Babe¸s-Bolyai University ...展开收缩
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