Squigonometry: The Study of Imperfect Circles
Squigonometry: The Study of Imperfect Circles

Author: Robert D. Poodiack

Publisher: Springer Nature

Published: 2022-12-15

Total Pages: 292

ISBN-13: 3031137833

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This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by |x|p + |y|p = 1 where p ≥ 1. Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be. Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of π, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein’s work in design. From here, more technical pathways offer further exploration. Topics include infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor. Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.

From Natural Numbers to Quaternions
From Natural Numbers to Quaternions

Author: Jürg Kramer

Publisher: Springer

Published: 2017-11-15

Total Pages: 288

ISBN-13: 3319694294

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This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.

Improvisation as Art
Improvisation as Art

Author: Edgar Landgraf

Publisher: Bloomsbury Publishing USA

Published: 2011-05-19

Total Pages: 176

ISBN-13: 1441199322

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Improvisation as Art traces how modernity's emphasis on inventiveness has changed the meaning of improvisation; and how the ideals and laws that led improvisation to be banned from "high art" in the eighteenth century simultaneously enabled the inventive reintegration of improvisation into modernism. After an in-depth exploration of contemporary theoretical contentions surrounding improvisation, Landgraf examines how the new emphasis on inventiveness affects the understanding of improvisation in the emerging aesthetic and anthropological discourses of the late 18th and early 19th centuries. He first focuses on accounts of improvisational performances by Moritz, Goethe, and Fernow and reads them alongside the aesthetics of autonomy as it develops at the same time. In its second half, the book investigates how the problem of "planning" art receives a different treatment in German Romanticism. The final chapter focuses on the writings of Heinrich von Kleist where improvisation presents a central aesthetic principle. Kleist's figurations of improvisation recognize the anthropological predicament of the self in modern society and the social constraints that invite and often force individuals to improvise.

An Introduction to Infinite Products
An Introduction to Infinite Products

Author: Charles H. C. Little

Publisher: Springer Nature

Published: 2022-01-10

Total Pages: 258

ISBN-13: 3030906469

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This text provides a detailed presentation of the main results for infinite products, as well as several applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The book provides a detailed treatment of the main theoretical results and applications with a goal of providing the reader with a short introduction and motivation for present and future study. While the coverage does not include an exhaustive compilation of results, the reader will be armed with an understanding of infinite products within the course of more advanced studies, and, inspired by the sheer beauty of the mathematics. The book will serve as a reference for students of mathematics, physics and engineering, at the level of senior undergraduate or beginning graduate level, who want to know more about infinite products. It will also be of interest to instructors who teach courses that involve infinite products as well as mathematicians who wish to dive deeper into the subject. One could certainly design a special-topics class based on this book for undergraduates. The exercises give the reader a good opportunity to test their understanding of each section.

A Journey Through The Realm of Numbers
A Journey Through The Realm of Numbers

Author: Menny Aka

Publisher: Springer Nature

Published: 2020-10-03

Total Pages: 344

ISBN-13: 3030552330

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This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory. The journey starts with the basic idea that new number systems arise from solving different equations, leading to (abstract) algebra. Along this journey, the reader will be exposed to important ideas of mathematics, and will learn a little about how mathematics is really done. Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which numbers are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions. This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations.

Inside Interesting Integrals
Inside Interesting Integrals

Author: Paul J. Nahin

Publisher: Springer Nature

Published: 2020-06-27

Total Pages: 542

ISBN-13: 3030437884

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What’s the point of calculating definite integrals since you can’t possibly do them all? What makes doing the specific integrals in this book of value aren’t the specific answers we’ll obtain, but rather the methods we’ll use in obtaining those answers; methods you can use for evaluating the integrals you will encounter in the future. This book, now in its second edition, is written in a light-hearted manner for students who have completed the first year of college or high school AP calculus and have just a bit of exposure to the concept of a differential equation. Every result is fully derived. If you are fascinated by definite integrals, then this is a book for you. New material in the second edition includes 25 new challenge problems and solutions, 25 new worked examples, simplified derivations, and additional historical discussion.

Irresistible Integrals
Irresistible Integrals

Author: George Boros

Publisher: Cambridge University Press

Published: 2004-06-21

Total Pages: 326

ISBN-13: 9780521796361

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This book, first published in 2004, uses the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics.

Twenty-One Lectures on Complex Analysis
Twenty-One Lectures on Complex Analysis

Author: Alexander Isaev

Publisher: Springer

Published: 2017-11-29

Total Pages: 193

ISBN-13: 3319681702

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At its core, this concise textbook presents standard material for a first course in complex analysis at the advanced undergraduate level. This distinctive text will prove most rewarding for students who have a genuine passion for mathematics as well as certain mathematical maturity. Primarily aimed at undergraduates with working knowledge of real analysis and metric spaces, this book can also be used to instruct a graduate course. The text uses a conversational style with topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching. Instructors are invited to rearrange the order of topics according to their own vision. A clear and rigorous exposition is supported by engaging examples and exercises unique to each lecture; a large number of exercises contain useful calculation problems. Hints are given for a selection of the more difficult exercises. This text furnishes the reader with a means of learning complex analysis as well as a subtle introduction to careful mathematical reasoning. To guarantee a student’s progression, more advanced topics are spread out over several lectures. This text is based on a one-semester (12 week) undergraduate course in complex analysis that the author has taught at the Australian National University for over twenty years. Most of the principal facts are deduced from Cauchy’s Independence of Homotopy Theorem allowing us to obtain a clean derivation of Cauchy’s Integral Theorem and Cauchy’s Integral Formula. Setting the tone for the entire book, the material begins with a proof of the Fundamental Theorem of Algebra to demonstrate the power of complex numbers and concludes with a proof of another major milestone, the Riemann Mapping Theorem, which is rarely part of a one-semester undergraduate course.

Quadratic Number Fields
Quadratic Number Fields

Author: Franz Lemmermeyer

Publisher: Springer Nature

Published: 2021-09-18

Total Pages: 348

ISBN-13: 3030786528

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This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

Fourier Analysis
Fourier Analysis

Author: Elias M. Stein

Publisher: Princeton University Press

Published: 2011-02-11

Total Pages: 326

ISBN-13: 1400831237

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This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.