Sharpening Mathematical Analysis Skills

ISBN-13: 9783030771416 / Angielski / Miękka / 2022 / 536 str.

Alina Sîntămărian; Ovidiu Furdui

Opis

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. 

Recenzje

"It should be of use to a variety of readers including, those teaching Calculus sequence, researchers who routinely deal with series, and mathematical laypeople who wish to enjoy some punchy, beautiful identities and revel at some mathematical jokes. ... the solutions are frequently high-quality step-by-step derivations. The book is more than a collection of problems." (Russel Jay, Hendel, MAA Reviews, May 23, 2022)

"Think of competition level problems like those of the Putnam competition or the Mathematical Olympaids like SEEMOUS. This book can be used to prepare for such contests both by potential participants as well as by those that have to set up the questions. ... This is a challenging, surprising, and interesting way to explore the classical topics in analysis clearly intended for the those who want to excel in mathematics." (Adhemar Bultheel, zbMATH 1482.00001, 2022)

Spis treści

Preface.- Notations.- Sequences of Real Numbers.- Series of Real Numbers.- Power Series.- Derivatives and Applications.- Partial Derivatives and Applications.- Implicit Functions.- Challenges, Gems, and Mathematical Beauties.- An Artistry of Quadratic Series. Two New Proofs of Sandham-Yeung Series.- Solutions.- References.- Index.

O autorze

Alina Sîntămărian is a Professor of Mathematics at Technical University of Cluj-Napoca, Romania. She holds a PhD in Mathematics from Babeș-Bolyai University, Cluj-Napoca, Romania. Her research interests focus on non-linear operators, fixed point theory, multivalued operators, mathematical analysis, constants evaluation, asymptotic expansions.

Ovidiu Furdui is an Associate Professor of Mathematics at Technical University of Cluj-Napoca, Romania. He has published more than 500 problems in journals, with a problem column, from all over the world and he is the author of Limits, Series, and Fractional Part Integrals and has co-authored Square Matrices of Order 2, both published by Springer.