Limits Series And Fractional Part Integrals

Author : Ovidiu Furdui
Publisher : Springer Science & Business Media
Page : 289 pages
File Size : 42,6 Mb
Release : 2013-05-30
Category : Mathematics
ISBN : 9781461467625

Get Book

Limits, Series, and Fractional Part Integrals by Ovidiu Furdui Pdf

This book features challenging problems of classical analysis that invite the reader to explore a host of strategies and tools used for solving problems of modern topics in real analysis. This volume offers an unusual collection of problems — many of them original — specializing in three topics of mathematical analysis: limits, series, and fractional part integrals. The work is divided into three parts, each containing a chapter dealing with a particular problem type as well as a very short section of hints to select problems. The first chapter collects problems on limits of special sequences and Riemann integrals; the second chapter focuses on the calculation of fractional part integrals with a special section called ‘Quickies’ which contains problems that have had unexpected succinct solutions. The final chapter offers the reader an assortment of problems with a flavor towards the computational aspects of infinite series and special products, many of which are new to the literature. Each chapter contains a section of difficult problems which are motivated by other problems in the book. These ‘Open Problems’ may be considered research projects for students who are studying advanced calculus, and which are intended to stimulate creativity and the discovery of new and original methods for proving known results and establishing new ones. This stimulating collection of problems is intended for undergraduate students with a strong background in analysis; graduate students in mathematics, physics, and engineering; researchers; and anyone who works on topics at the crossroad between pure and applied mathematics. Moreover, the level of problems is appropriate for students involved in the Putnam competition and other high level mathematical contests.

Author : Jai Rathod
Publisher : Unknown
Page : 0 pages
File Size : 52,9 Mb
Release : 2016-04
Category : Calculus
ISBN : 1681172577

Get Book

Limits, Series, and Fractional Part Integrals by Jai Rathod Pdf

In mathematics, a limits in the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. Many times, a function can be undefined at a point, but we can think about what the function approaches as it gets closer and closer to that point (this in the limit). Other times, the function may be defined at a point, but it may approach a different limit. There are many times where the function value in the same as the limit at the point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to curve. We often attempt to find the limit at a point where the function itself in not defined. In mathematic, a series is, informally speaking, the sun of the terms if an infinite sequence. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely. The terms of the series are often produced according to a rule, such as by a formula, or by an algorithm. Fore emphasizing that there are an infinite numbers of terms, a series is often called an infinite series. The study on infinite series is a major part of mathematical analysis. Series are used in most areas of mathematical, even for studying finite structures, through generating function. The fractional part of a non-negative real number x is the excess beyond that numbers integer part. This book offers an unusual collection of problemmany of them original specializing in three topics on mathematical analysis; limits, series, and fractional part integrals. This book should be of immense valuable for undergraduate students with a strong background in analysis; graduate students in mathematical, physics, and engineering; and anyone who works on topic at the crossroad between pure and applied mathematics.

Author : Cornel Ioan Vălean
Publisher : Springer Nature
Page : 847 pages
File Size : 55,7 Mb
Release : 2023-05-24
Category : Mathematics
ISBN : 9783031212628

Get Book

More (Almost) Impossible Integrals, Sums, and Series by Cornel Ioan Vălean Pdf

This book, the much-anticipated sequel to (Almost) Impossible, Integrals, Sums, and Series, presents a whole new collection of challenging problems and solutions that are not commonly found in classical textbooks. As in the author’s previous book, these fascinating mathematical problems are shown in new and engaging ways, and illustrate the connections between integrals, sums, and series, many of which involve zeta functions, harmonic series, polylogarithms, and various other special functions and constants. Throughout the book, the reader will find both classical and new problems, with numerous original problems and solutions coming from the personal research of the author. Classical problems are shown in a fresh light, with new, surprising or unconventional ways of obtaining the desired results devised by the author. This book is accessible to readers with a good knowledge of calculus, from undergraduate students to researchers. It will appeal to all mathematical puzzlers who love a good integral or series and aren’t afraid of a challenge.

Author : Cornel Ioan Vălean
Publisher : Springer
Page : 539 pages
File Size : 54,7 Mb
Release : 2019-05-10
Category : Mathematics
ISBN : 9783030024628

Get Book

(Almost) Impossible Integrals, Sums, and Series by Cornel Ioan Vălean Pdf

This book contains a multitude of challenging problems and solutions that are not commonly found in classical textbooks. One goal of the book is to present these fascinating mathematical problems in a new and engaging way and illustrate the connections between integrals, sums, and series, many of which involve zeta functions, harmonic series, polylogarithms, and various other special functions and constants. Throughout the book, the reader will find both classical and new problems, with numerous original problems and solutions coming from the personal research of the author. Where classical problems are concerned, such as those given in Olympiads or proposed by famous mathematicians like Ramanujan, the author has come up with new, surprising or unconventional ways of obtaining the desired results. The book begins with a lively foreword by renowned author Paul Nahin and is accessible to those with a good knowledge of calculus from undergraduate students to researchers, and will appeal to all mathematical puzzlers who love a good integral or series.

Author : Bernard Candelpergher
Publisher : Springer
Page : 195 pages
File Size : 54,8 Mb
Release : 2017-09-12
Category : Mathematics
ISBN : 9783319636306

Get Book

Ramanujan Summation of Divergent Series by Bernard Candelpergher Pdf

The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.

Author : Ioannis Roussos
Publisher : CRC Press
Page : 464 pages
File Size : 52,5 Mb
Release : 2023-09-27
Category : Mathematics
ISBN : 9781000958140

Get Book

Improper Riemann Integrals by Ioannis Roussos Pdf

The scope of this book is the improper or generalized Riemann integral and infinite sum (series). The reader will study its convergence, principal value, evaluation and application to science and engineering. Improper Riemann integrals and infinite sums are interconnected. In the new edition, the author has involved infinite sums more than he did in the first edition. Apart from having computed and listed a large number of improper integrals and infinite sums, we have also developed the necessary theory and various ways of evaluating them or proving their divergence. Questions, problems and applications involving various improper integrals and infinite sums (series) of numbers emerge in science and application very often. Their complete presentations and all rigorous proofs would require taking the graduate-level courses on these subjects. Here their statements are adjusted to a level students of all levels can understand and use them efficiently as powerful tools in a large list of problems and applications.

Author : Anatoly Kochubei,Yuri Luchko
Publisher : Walter de Gruyter GmbH & Co KG
Page : 489 pages
File Size : 51,5 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9783110571622

Get Book

Basic Theory by Anatoly Kochubei,Yuri Luchko Pdf

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

Author : Hongwei Chen
Publisher : CRC Press
Page : 443 pages
File Size : 49,6 Mb
Release : 2022-12-16
Category : Mathematics
ISBN : 9781000786545

Get Book

Classical Analysis by Hongwei Chen Pdf

A conceptually clear induction to fundamental analysis theorems, a tutorial for creative approaches for solving problems, a collection of modern challenging problems, a pathway to undergraduate research—all these desires gave life to the pages here. This book exposes students to stimulating and enlightening proofs and hard problems of classical analysis mainly published in The American Mathematical Monthly. The author presents proofs as a form of exploration rather than just a manipulation of symbols. Drawing on the papers from the Mathematical Association of America's journals, numerous conceptually clear proofs are offered. Each proof provides either a novel presentation of a familiar theorem or a lively discussion of a single issue, sometimes with multiple derivations. The book collects and presents problems to promote creative techniques for problem-solving and undergraduate research and offers instructors an opportunity to assign these problems as projects. This book provides a wealth of opportunities for these projects. Each problem is selected for its natural charm—the connection with an authentic mathematical experience, its origination from the ingenious work of professionals, develops well-shaped results of broader interest.

Author : Hongwei Chen
Publisher : CRC Press
Page : 324 pages
File Size : 50,9 Mb
Release : 2021-07-06
Category : Mathematics
ISBN : 9781000402285

Get Book

Monthly Problem Gems by Hongwei Chen Pdf

This book is an outgrowth of a collection of sixty-two problems offered in the The American Mathematical Monthly (AMM) the author has worked over the last two decades. Each selected problem has a central theme, contains gems of sophisticated ideas connected to important current research, and opens new vistas in the understanding of mathematics. The AMM problem section provides one of the most challenging and interesting problem sections among the various journals and online sources currently available. The published problems and solutions have become a treasure trove rife with mathematical gems. The author presents either his published solution in the AMM or an alternative solution to the published one to present and develop problem-solving techniques. A rich glossary of important theorems and formulas is included for easy reference. The reader may regard this book as a starter set for AMM problems, providing a jumping of point to new ideas, and extending their personal lexicon of problems and solutions. This collection is intended to encourage the reader to move away from routine exercises toward creative solutions, as well as offering the reader a systematic illustration of how to organize the transition from problem solving to exploring, investigating and discovering new results.

Author : Vasile Pop,Ovidiu Furdui
Publisher : Springer
Page : 370 pages
File Size : 53,7 Mb
Release : 2017-04-04
Category : Mathematics
ISBN : 9783319549392

Get Book

Square Matrices of Order 2 by Vasile Pop,Ovidiu Furdui Pdf

This unique and innovative book presents an exciting and complete detail of all the important topics related to the theory of square matrices of order 2. The readers exploring every detailed aspect of matrix theory are gently led toward understanding advanced topics. They will follow every notion of matrix theory with ease, accumulating a thorough understanding of algebraic and geometric aspects of matrices of order 2. The prime jewel of this book is its offering of an unusual collection of problems, theoretically motivated, most of which are new, original, and seeing the light of publication for the first time in the literature. Nearly all of the exercises are presented with detailed solutions and vary in difficulty from easy to more advanced. Many problems are particularly challenging. These, and not only these, invite the reader to unleash their creativity and research capabilities and to discover their own methods of attacking a problem. Matrices have a vast practical importance to mathematics, science, and engineering; therefore the readership of this book is intended to be broad: high school students wishing to learn the fundamentals of matrix theory, first year students who like to participate in mathematical competitions, graduate students who want to learn more about an application of a certain technique, doctoral students who are preparing for their prelim exams in linear algebra, and linear algebra instructors. Chapters 1–3 complement a standard linear algebra course. Pure and applied mathematicians who use matrix theory for their applications will find this book useful as a refresher. In fact, anyone who is willing to explore the methodologies discussed in this book and work through a collection of problems involving matrices of order 2 will be enriched.

Author : Dennis S. Bernstein
Publisher : Princeton University Press
Page : 1593 pages
File Size : 46,8 Mb
Release : 2018-02-27
Category : Mathematics
ISBN : 9780691176536

Get Book

Scalar, Vector, and Matrix Mathematics by Dennis S. Bernstein Pdf

The essential reference book on matrices—now fully updated and expanded, with new material on scalar and vector mathematics Since its initial publication, this book has become the essential reference for users of matrices in all branches of engineering, science, and applied mathematics. In this revised and expanded edition, Dennis Bernstein combines extensive material on scalar and vector mathematics with the latest results in matrix theory to make this the most comprehensive, current, and easy-to-use book on the subject. Each chapter describes relevant theoretical background followed by specialized results. Hundreds of identities, inequalities, and facts are stated clearly and rigorously, with cross-references, citations to the literature, and helpful comments. Beginning with preliminaries on sets, logic, relations, and functions, this unique compendium covers all the major topics in matrix theory, such as transformations and decompositions, polynomial matrices, generalized inverses, and norms. Additional topics include graphs, groups, convex functions, polynomials, and linear systems. The book also features a wealth of new material on scalar inequalities, geometry, combinatorics, series, integrals, and more. Now more comprehensive than ever, Scalar, Vector, and Matrix Mathematics includes a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. Fully updated and expanded with new material on scalar and vector mathematics Covers the latest results in matrix theory Provides a list of symbols and a summary of conventions for easy and precise use Includes an extensive bibliography with back-referencing plus an author index

Author : Alina Sîntămărian,Ovidiu Furdui
Publisher : Springer Nature
Page : 543 pages
File Size : 43,7 Mb
Release : 2021-10-25
Category : Mathematics
ISBN : 9783030771393

Get Book

Sharpening Mathematical Analysis Skills by Alina Sîntămărian,Ovidiu Furdui Pdf

This book gathers together a novel collection of problems in mathematical analysis that are challenging and worth studying. They cover most of the classical topics of a course in mathematical analysis, and include challenges presented with an increasing level of difficulty. Problems are designed to encourage creativity, and some of them were especially crafted to lead to open problems which might be of interest for students seeking motivation to get a start in research. The sets of problems are comprised in Part I. The exercises are arranged on topics, many of them being preceded by supporting theory. Content starts with limits, series of real numbers and power series, extending to derivatives and their applications, partial derivatives and implicit functions. Difficult problems have been structured in parts, helping the reader to find a solution. Challenges and open problems are scattered throughout the text, being an invitation to discover new original methods for proving known results and establishing new ones. The final two chapters offer ambitious readers splendid problems and two new proofs of a famous quadratic series involving harmonic numbers. In Part II, the reader will find solutions to the proposed exercises. Undergraduate students in mathematics, physics and engineering, seeking to strengthen their skills in analysis, will most benefit from this work, along with instructors involved in math contests, individuals who want to enrich and test their knowledge in analysis, and anyone willing to explore the standard topics of mathematical analysis in ways that aren’t commonly seen in regular textbooks.

Author : Tian-Xiao He
Publisher : Springer Science & Business Media
Page : 234 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461221005

Get Book

Dimensionality Reducing Expansion of Multivariate Integration by Tian-Xiao He Pdf

This book focuses primarily on a powerful tool: dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics.

Author : David Applebaum
Publisher : OUP Oxford
Page : 224 pages
File Size : 42,8 Mb
Release : 2012-03-01
Category : Mathematics
ISBN : 9780191627873

Get Book

Limits, Limits Everywhere by David Applebaum Pdf

A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series. Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and π, continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject. A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.

Author : Stanley Rabinowitz,Mark Bowron
Publisher : MathPro Press
Page : 548 pages
File Size : 46,8 Mb
Release : 1999
Category : Mathematics
ISBN : 0962640123

Get Book

Index to Mathematical Problems, 1975-1979 by Stanley Rabinowitz,Mark Bowron Pdf