Ce vol. III expose la thA(c)orie classique de Cauchy dans un esprit orientA(c) bien davantage vers ses innombrables utilisations que vers une thA(c)orie plus ou moins complA]te des fonctions analytiques. On montre ensuite comment les intA(c)grales curvilignes A la Cauchy se gA(c)nA(c)ralisent A un nombre quelconque de variables rA(c)elles (formes diffA(c)rentielles, formules de type Stokes). Les bases de la thA(c)orie des variA(c)tA(c)s sont ensuite exposA(c)es, principalement pour fournir au lecteur le langage "canonique" et quelques thA(c)orA]mes importants (changement de variables dans les intA(c)grales, A(c)quations diffA(c)rentielles). Un dernier chapitre montre comment on peut utiliser ces thA(c)ories pour construire la surface de Riemann compacte d'une fonction algA(c)brique, sujet rarement traitA(c) dans la littA(c)rature non spA(c)cialisA(c)e bien que n'A(c)xigeant que des techniques A(c)lA(c)mentaires. Un volume IV exposera, outre, l'intA(c)grale de Lebesgue, un bloc de mathA(c)matiques spA(c)cialisA(c)es vers lequel convergera tout le contenu des volumes prA(c)cA(c)dents: sA(c)ries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, thA(c)orie classique des fonctions modulaires et la version moderne utilisant la structure de groupe de Lie de SL(2, R).

Les deux premiers volumes de cet ouvrage sont consacrA(c)s aux fonctions dans R ou C, y compris la thA(c)orie A(c)lA(c)mentaire des sA(c)ries et intA(c)grales de Fourier et une partie de celle des fonctions holomorphes. L'exposA(c), non strictement linA(c)aire, combine indications historiques et raisonnements rigoureux. Il montre la diversitA(c) des voies d'accA]s aux principaux rA(c)sultats afin de familiariser le lecteur avec les mA(c)thodes de raisonnement et idA(c)es fondamentales plutAt qu'avec les techniques de calcul, point de vue utile aussi aux personnes travaillant seules.
Les volumes 3 et 4 traiteront principalement des fonctions analytiques (thA(c)orie de Cauchy, thA(c)orie analytique des nombres et fonctions modulaires), ainsi que du calcul diffA(c)rentiel sur les variA(c)tA(c)s, avec un court exposA(c) de l'intA(c)grale de Lebesgue, en suivant d'assez prA]s le cA(c)lA]bre cours donnA(c) longtemps par l'auteur A l'UniversitA(c) Paris 7.
On reconnaA(R)tra dans ce nouvel ouvrage le style inimitable de l'auteur, et pas seulement par son refus de l'A(c)criture condensA(c)e en usage dans de nombreux manuels.

This course in real analysis is directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and nonspecialists alike. The text covers three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal, down-to-earth style, the author gives motivation and overview of new ideas, while still supplying full details and complete proofs. He provides a great many exercises and suggestions for further study.

This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.

Super-real fields are a class of large totally ordered fields. These fields are larger than the real line. They arise from quotients of the algebra of continuous functions on a compact space by a prime ideal, and generalize the well-known class of ultrapowers, and indeed the continuous ultrapowers. These fields are of interest in their own right and have many surprising applications, both in analysis and logic. The authors introduce some exciting new fields, including a natural generalization of the real line R, and resolve a number of open problems. The book is intended to be accessible to analysts and logicians. After an exposition of the general theory of ordered fields and a careful proof of some classic theorems, including Kaplansky's embedding theorems , the authors establish important new results in Banach algebra theory, non-standard analysis, an model theory.

Gute Kenntnisse in MaA- und Integrationstheorie sind unerlAAlich fA1/4r fast alle Bereiche der hAheren Analysis, Wahrscheinlichkeitstheorie, Statistik und Physik. In dem vorliegenden Lehrbuch wird diese Theorie von den allerersten AnfAngen - Was soll eine Inhaltsmessung eigentlich leisten? - systematisch bis zur Theorie der RadonmaAe entwickelt. Besonderer Wert ist auf ausfA1/4hrliche Motivationen der neu eingefA1/4hrten Begriffe gelegt. Dem Zugang von L. Schwartz folgend werden RadonmaAe auf beliebigen topologischen RAumen behandelt, wodurch der Rieszsche Darstellungssatz sehr allgemein und A1/4bersichtlich bewiesen werden kann. Den BedA1/4rfnissen der Wahrscheinlichkeitstheorie wird durch die Behandlung von MaAen auf unendlichen Produkten (ProduktmaAe, Satz von Kolmogoroff) angemessen Rechnung getragen.

This is a text that develops calculus 'from scratch', with complete rigorous arguments. Its aim is to introduce the reader not only to the basic facts about calculus but, as importantly, to mathematical reasoning. It covers in great detail calculus of one variable and multivariable calculus. Additionally it offers a basic introduction to the topology of Euclidean space. It is intended to more advanced or highly motivated undergraduates.

Schon seit geraumer Zeit hat die Lehre von den reellen Funk tionen aufgehort, eine blosse Sammlung von Merkwurdigkeiten zu sein: sie ist zu einer Theorie der reellen Funktionen geworden, die eine grosse Anzahl bedeutungsvoller und weittragender Gesetze aufgedeckt hat; nicht mehr das Suchen nach Ausnahmen ist ihre Absicht, sondern das Suchen nach Regeln. Und da immer haufiger Frage stellungen aus den verschiedensten Gebieten der Mathematik bei grundlicher Behandlung auf Fragen aus der Theorie der reellen Funk tionen fiihrten, so hat diese Theorie auch aufgehort, Alleinbesitr. einiger Spezialisten zu sein und in immer steigendem Masse das Interesse der mathematischen Allgemeinheit gefunden. Von vielen Seiten wurde daher der Mangel einer zusammenfassenden, systemati schen Darstellung dieser Theorie schmerzlich empfunden. Ich habe es deshalb mit Freuden begrusst, als vor einer Reihe von Jahren Herr A. Schoenflies an mich mit der Aufforderung herantrat, an einer Neuauflage seines Berichtes "Die Entwickelung der Lehre von den Punktmiumigfaltigkeiten" mitzuarbeiten, und zwar insbesondere die Anwendungen der Mengenlehre auf die Theorie der reellen Funktionen zu behandeln. Im Jahre 1914 war diese Darstellung nahezu beendet. Der Ausbruch des Krieges, der mich von meinem damaligen 'Wohnsitze Czernowitz trennte, sodann meine Einberufung zur osterreichischen Armee, eine schwere Verwundung, schliesslich meine Ubersiedlung nach Bann verzogerten die endgultige Fertigstellung, und. als diese endlich erfolgt war, machten die mittlerweile einge tretenen traurigen Verhaltnisse die Drucklegung unmoglich. Ich war schon darauf g fasst, das Manuskript in meinem Schreibtische begraben zu mussen, ."

Designed for courses in advanced calculus and introductory real analysis, the second edition of Elementary Classical Analysis strikes a careful and thoughtful balance between pure and applied mathematics, with the emphasis on techniques important to classical analysis, without vector calculus or complex analysis. As such, it's a perfect teaching and learning resource for mathematics undergraduate courses in classical analysis. The book includes detailed coverage of the foundations of the real number system and focuses primarily on analysis in Euclidean space with a view towards application. As well as being suitable for students taking pure mathematics, it can also be used by students taking engineering and physical science courses. There's now even more material on variable calculus, expanding the textbook's already considerable coverage of the subject.

While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory-uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis-the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor-Schroeder-Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

SageMath, or Sage for short, is an open-source mathematical software system based on the Python language and developed by an international community comprising hundreds of teachers and researchers, whose aim is to provide an alternative to the commercial products Magma, Maple, Mathematica, and MATLAB (R). To achieve this, Sage relies on many open-source programs, including GAP, Maxima, PARI, and various scientific libraries for Python, to which thousands of new functions have been added. Sage is freely available and is supported by all modern operating systems. Sage provides a wonderful scientific and graphical calculator for high school students, and it efficiently supports undergraduates in their computations in analysis, linear algebra, calculus, etc. For graduate students, researchers, and engineers in various mathematical specialties, Sage provides the most recent algorithms and tools, which is why several universities around the world already use Sage at the undergraduate level. Computational Mathematics with SageMath, written by researchers and by teachers at the high school, undergraduate, and graduate levels, focuses on the underlying mathematics necessary to use Sage efficiently and is illustrated with concrete examples. Part I is accessible to high school and undergraduate students and Parts II, III, and IV are suitable for graduate students, teachers, and researchers. This book is available under a Creative Commons license at sagebook.gforge.inria.fr.

Real analysis provides the fundamental underpinnings for calculus, arguably the most useful and influential mathematical idea ever invented. It is a core subject in any mathematics degree, and also one which many students find challenging. A Sequential Introduction to Real Analysis gives a fresh take on real analysis by formulating all the underlying concepts in terms of convergence of sequences. The result is a coherent, mathematically rigorous, but conceptually simple development of the standard theory of differential and integral calculus ideally suited to undergraduate students learning real analysis for the first time.This book can be used as the basis of an undergraduate real analysis course, or used as further reading material to give an alternative perspective within a conventional real analysis course.

Part 1 begins with an overview of properties of the real numbers and starts to introduce the notions of set theory. The absolute value and in particular inequalities are considered in great detail before functions and their basic properties are handled. From this the authors move to differential and integral calculus. Many examples are discussed. Proofs not depending on a deeper understanding of the completeness of the real numbers are provided. As a typical calculus module, this part is thought as an interface from school to university analysis.Part 2 returns to the structure of the real numbers, most of all to the problem of their completeness which is discussed in great depth. Once the completeness of the real line is settled the authors revisit the main results of Part 1 and provide complete proofs. Moreover they develop differential and integral calculus on a rigorous basis much further by discussing uniform convergence and the interchanging of limits, infinite series (including Taylor series) and infinite products, improper integrals and the gamma function. In addition they discussed in more detail as usual monotone and convex functions.Finally, the authors supply a number of Appendices, among them Appendices on basic mathematical logic, more on set theory, the Peano axioms and mathematical induction, and on further discussions of the completeness of the real numbers. Remarkably, Volume I contains ca. 360 problems with complete, detailed solutions.

Optimal control theory has numerous applications in both science and engineering. This book presents basic concepts and principles of mathematical programming in terms of set-valued analysis and develops a comprehensive optimality theory of problems described by ordinary and partial differential inclusions.

This textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland's path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics. A pool of 200 'Dialing In' problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way. Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.

Elementary Real Analysis is a core course in nearly all mathematics departments throughout the world. It enables students to develop a deep understanding of the key concepts of calculus from a mature perspective. Elements of Real Analysis is a student-friendly guide to learning all the important ideas of elementary real analysis, based on the author's many years of experience teaching the subject to typical undergraduate mathematics majors. It avoids the compact style of professional mathematics writing, in favor of a style that feels more comfortable to students encountering the subject for the first time. It presents topics in ways that are most easily understood, yet does not sacrifice rigor or coverage. In using this book, students discover that real analysis is completely deducible from the axioms of the real number system. They learn the powerful techniques of limits of sequences as the primary entry to the concepts of analysis, and see the ubiquitous role sequences play in virtually all later topics. They become comfortable with topological ideas, and see how these concepts help unify the subject. Students encounter many interesting examples, including "pathological" ones, that motivate the subject and help fix the concepts. They develop a unified understanding of limits, continuity, differentiability, Riemann integrability, and infinite series of numbers and functions.

Ideal For The One-Semester Undergraduate Course, Basic Real Analysis Is Intended For Students Who Have Recently Completed A Traditional Calculus Course And Proves The Basic Theorems Of Single Variable Calculus In A Simple And Accessible Manner. It Gradually Builds Upon Key Material As To Not Overwhelm Students Beginning The Course And Becomes More Rigorous As They Progresses. Optional Appendices On Sets And Functions, Countable And Uncountable Sets, And Point Set Topology Are Included For Those Instructors Who Wish Include These Topics In Their Course. The Author Includes Hints Throughout The Text To Help Students Solve Challenging Problems. An Online Instructor's Solutions Manual Is Also Available.

This book is written by award-winning author, Frank Morgan. It offers a simple and sophisticated point of view, reflecting Morgan's insightful teaching, lecturing, and writing style. Intended for undergraduates studying real analysis, this book builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as 'closed and bounded', via sequences, and via open covers.Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem. This concise text not only provides efficient proofs, but also shows students how to derive them. The excellent exercises are accompanied by select solutions. Ideally suited as an undergraduate textbook, this complete book on real analysis will fit comfortably into one semester. Frank Morgan received the first Haimo Award for distinguished college teaching from the Mathematical Association of America. He has also garnered top teaching awards from Rice University (Houston, TX) and MIT (Cambridge, MA).

Diese Einfuhrung besticht durch zwei ungewohnliche Aspekte: Sie gibt einen Einblick in die Mathematik als Bestandteil unserer Kultur, und sie vermittelt die Hintergrunde der Mathematik vom Schulstoff ausgehend bis zum Niveau von Mathematikvorlesungen im ersten Studienjahr. Die Stoffdarstellung geht vom Aufbau der naturlichen Zahlen aus; der Schwerpunkt liegt aber in den exakten Begrundungen der Zahlenbegriffe, der Geometrie der Ebene und der Funktionen einer Veranderlichen. Dabei werden alle Satze bis hin zum Hauptsatz der Algebra vollstandig bewiesen. Der klare Aufbau des Buches mit Stichwortregister wichtiger Begriffe erleichtert das systematische Lernen und Nachschlagen. Die zweite Auflage enthalt teilweise ausfuhrliche Darstellungen fur die Losungen der zahlreichen Ubungsaufgaben.
Da viele Aspekte zur Sprache kommen, die so weder im Unterricht noch im Studium behandelt werden, erganzt die Einfuhrung ideal den Vorlesungsstoff fur Lehramtskandidaten und Diplomstudenten."