Inside Interesting Integrals

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.

Author: Paul J. Nahin

Publisher: Springer

ISBN: 3030437876

Category: Science

Page: 503

View: 889

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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.
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Inside Interesting Integrals

This book 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.

Author: Paul J. Nahin

Publisher: Springer

ISBN: 9781493912773

Category: Science

Page: 412

View: 177

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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 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.
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Inside Interesting Integrals

The integral we evaluated was not a technically difficult one to do, ... Created by P.J.Nahin (9/12/2012) for Inside Interesting Integrals.

Author: Paul J. Nahin

Publisher: Springer Nature

ISBN: 9783030437886

Category: Science

Page: 503

View: 396

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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.
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Collected Papers

Author: George Pólya

Publisher:

ISBN: UCAL:B5159547

Category: Analytic functions

Page: 808

View: 469

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Transients for Electrical Engineers

This book offers a concise introduction to the analysis of electrical transients aimed at students who have completed introductory circuits and freshman calculus courses.

Author: Paul J. Nahin

Publisher: Springer

ISBN: 9783319775982

Category: Technology & Engineering

Page: 189

View: 743

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This book offers a concise introduction to the analysis of electrical transients aimed at students who have completed introductory circuits and freshman calculus courses. While it is written under the assumption that these students are encountering transient electrical circuits for the first time, the mathematical and physical theory is not ‘watered-down.’ That is, the analysis of both lumped and continuous (transmission line) parameter circuits is performed with the use of differential equations (both ordinary and partial) in the time domain, and the Laplace transform. The transform is fully developed in the book for readers who are not assumed to have seen it before. The use of singular time functions (unit step and impulse) is addressed and illustrated through detailed examples. The appearance of paradoxical circuit situations, often ignored in many textbooks (because they are, perhaps, considered ‘difficult’ to explain) is fully embraced as an opportunity to challenge students. In addition, historical commentary is included throughout the book, to combat the misconception that the material in engineering textbooks was found engraved on Biblical stones, rather than painstakingly discovered by people of genius who often went down many wrong paths before finding the right one. MATLAB® is used throughout the book, with simple codes to quickly and easily generate transient response curves.
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Almost Impossible Integrals Sums and Series

—Exchange between a physics graduate student and a professor of mathematics1 Shortly after my book Inside Interesting Integrals was published by Springer in ...

Author: Cornel Ioan Vălean

Publisher: Springer

ISBN: 9783030024628

Category: Mathematics

Page: 539

View: 932

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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.
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Lectures on Quantum Mechanics

answer So, what's an interesting integral ...? I suppose the honest . . . is ... Paul J. Nahin Inside Interesting Integrals, 2015. Gaussian integrals arise ...

Author: Philip L. Bowers

Publisher: Cambridge University Press

ISBN: 9781108429764

Category: Science

Page: 582

View: 439

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A leisurely but mathematically honest presentation of quantum mechanics for graduate students in mathematics with an interest in physics.
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Hot Molecules Cold Electrons

The reason I've written a partial derivative inside the integral, ... of Laplace's integral in my Inside Interesting Integrals, Springer 2015, pp. 79–82.

Author: Paul J. Nahin

Publisher: Princeton University Press

ISBN: 9780691199948

Category: Science

Page: 232

View: 282

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An entertaining mathematical exploration of the heat equation and its role in the triumphant development of the trans-Atlantic telegraph cable Heat, like gravity, shapes nearly every aspect of our world and universe, from how milk dissolves in coffee to how molten planets cool. The heat equation, a cornerstone of modern physics, demystifies such processes, painting a mathematical picture of the way heat diffuses through matter. Presenting the mathematics and history behind the heat equation, Hot Molecules, Cold Electrons tells the remarkable story of how this foundational idea brought about one of the greatest technological advancements of the modern era. Paul Nahin vividly recounts the heat equation’s tremendous influence on society, showing how French mathematical physicist Joseph Fourier discovered, derived, and solved the equation in the early nineteenth century. Nahin then follows Scottish physicist William Thomson, whose further analysis of Fourier’s explorations led to the pioneering trans-Atlantic telegraph cable. This feat of engineering reduced the time it took to send a message across the ocean from weeks to minutes. Readers also learn that Thomson used Fourier’s solutions to calculate the age of the earth, and, in a bit of colorful lore, that writer Charles Dickens relied on the trans-Atlantic cable to save himself from a career-damaging scandal. The book’s mathematical and scientific explorations can be easily understood by anyone with a basic knowledge of high school calculus and physics, and MATLAB code is included to aid readers who would like to solve the heat equation themselves. A testament to the intricate links between mathematics and physics, Hot Molecules, Cold Electrons offers a fascinating glimpse into the relationship between a formative equation and one of the most important developments in the history of human communication.
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In Praise of Simple Physics

(You can find more discussion on their appearance in physics in Inside Interesting Integrals (see note 4), pp. 212–219.) In the final chapter of this book ...

Author: Paul J. Nahin

Publisher: Princeton University Press

ISBN: 9781400880515

Category: Science

Page: 272

View: 967

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Fun puzzles that use physics to explore the wonders of everyday life Physics can explain many of the things that we commonly encounter. It can tell us why the night is dark, what causes the tides, and even how best to catch a baseball. With In Praise of Simple Physics, popular math and science writer Paul Nahin presents a plethora of situations that explore the science and math behind the wonders of everyday life. Roaming through a diverse range of puzzles, he illustrates how physics shows us ways to wring more energy from renewable sources, to measure the gravity in our car garages, to figure out which of three light switches in the basement controls the light bulb in the attic, and much, much more. How fast can you travel from London to Paris? How do scientists calculate the energy of an atomic bomb explosion? How do you kick a football so it stays in the air and goes a long way downfield? Nahin begins with simpler problems and progresses to more challenging questions, and his entertaining, accessible, and scientifically and mathematically informed explanations are all punctuated by his trademark humor. Readers are presumed to have some background in beginning differential and integral calculus. Whether you simply have a personal interest in physics' influence in the world or you're an engineering and science student who wants to gain more physics know-how, this book has an intriguing scenario for you. In Praise of Simple Physics proves that if we look carefully at the world around us, physics has answers for the most astonishing day-to-day occurrences.
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Complex Variables and Analytic Functions An Illustrated Introduction

292) P.J. Nahin, Inside Interesting Integrals, Springer, 2015. (Cited on p. 143) A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, 1993.

Author: Bengt Fornberg

Publisher: SIAM

ISBN: 9781611975987

Category: Mathematics

Page: 371

View: 717

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At almost all academic institutions worldwide, complex variables and analytic functions are utilized in courses on applied mathematics, physics, engineering, and other related subjects. For most students, formulas alone do not provide a sufficient introduction to this widely taught material, yet illustrations of functions are sparse in current books on the topic. This is the first primary introductory textbook on complex variables and analytic functions to make extensive use of functional illustrations. Aiming to reach undergraduate students entering the world of complex variables and analytic functions, this book utilizes graphics to visually build on familiar cases and illustrate how these same functions extend beyond the real axis. It covers several important topics that are omitted in nearly all recent texts, including techniques for analytic continuation and discussions of elliptic functions and of Wiener–Hopf methods. It also presents current advances in research, highlighting the subject’s active and fascinating frontier. The primary audience for this textbook is undergraduate students taking an introductory course on complex variables and analytic functions. It is also geared toward graduate students taking a second semester course on these topics, engineers and physicists who use complex variables in their work, and students and researchers at any level who want a reference book on the subject.
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Mathematical Methods in the Earth and Environmental Sciences

Inside Interesting Integrals. Springer, New York. Niklas, Karl J. 1994. Plant Allometry: The Scaling of Form and Process. University of Chicago Press, ...

Author: Adrian Burd

Publisher: Cambridge University Press

ISBN: 9781107117488

Category: Mathematics

Page: 550

View: 339

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An accessible introduction to the mathematical methods essential for understanding processes in the Earth and environmental sciences.
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Scalar Vector and Matrix Mathematics

P. Nahin, Inside Interesting Integrals. Princeton: Princeton University Press, 2015. (Cited on pp. 996, 1102, 1104, 1106, 1107, 1109, 1114, 1116, 1121, ...

Author: Dennis S. Bernstein

Publisher: Princeton University Press

ISBN: 9780691176536

Category: Mathematics

Page: 1600

View: 296

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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
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Theories of Integration

The Henstock-Kurzweil integral studied in Chapter 4 is obtained by using ... however, the McShane integral has some very interesting properties and it is ...

Author: Douglas S Kurtz

Publisher: World Scientific Publishing Company

ISBN: 9789813108264

Category: Mathematics

Page: 312

View: 700

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The book uses classical problems to motivate a historical development of the integration theories of Riemann, Lebesgue, Henstock–Kurzweil and McShane, showing how new theories of integration were developed to solve problems that earlier integration theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this book useful as a textbook.
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Integral Methods in Science and Engineering Volume 1

Table 12.5 The first four real-valued interior transmission eigenvalues for a circle ... It would also be interesting to investigate whether one is able to ...

Author: Christian Constanda

Publisher: Birkhäuser

ISBN: 9783319593845

Category: Mathematics

Page: 340

View: 614

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This contributed volume contains a collection of articles on the most recent advances in integral methods. The first of two volumes, this work focuses on the construction of theoretical integral methods. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Fourteenth International Conference on Integral Methods in Science and Engineering, held July 25-29, 2016, in Padova, Italy. A broad range of topics is addressed, such as:• Integral equations• Homogenization• Duality methods• Optimal design• Conformal techniques This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines, and to other professionals who use integration as an essential tool in their work.
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Linear Integral Equations

Incidentally, the function f(t) occurring in the integral (8.3.9) is called the ... from inside or outside the contour L, to the point t on this contour; ...

Author: Ram P. Kanwal

Publisher: Springer Science & Business Media

ISBN: 9781461207658

Category: Mathematics

Page: 318

View: 655

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This second edition of Linear Integral Equations continues the emphasis that the first edition placed on applications. Indeed, many more examples have been added throughout the text. Significant new material has been added in Chapters 6 and 8. For instance, in Chapter 8 we have included the solutions of the Cauchy type integral equations on the real line. Also, there is a section on integral equations with a logarithmic kernel. The bibliography at the end of the book has been exteded and brought up to date. I wish to thank Professor B.K. Sachdeva who has checked the revised man uscript and has suggested many improvements. Last but not least, I am grateful to the editor and staff of Birkhauser for inviting me to prepare this new edition and for their support in preparing it for publication. RamP Kanwal CHAYfERl Introduction 1.1. Definition An integral equation is an equation in which an unknown function appears under one or more integral signs Naturally, in such an equation there can occur other terms as well. For example, for a ~ s ~ b; a :( t :( b, the equations (1.1.1) f(s) = ib K(s, t)g(t)dt, g(s) = f(s) + ib K(s, t)g(t)dt, (1.1.2) g(s) = ib K(s, t)[g(t)fdt, (1.1.3) where the function g(s) is the unknown function and all the other functions are known, are integral equations. These functions may be complex-valued functions of the real variables s and t.
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Path Integrals in Quantum Mechanics

6.5.3 Discussion Operator ordering and ambiguities in the path integral. ... in most interesting examples, interactions contain monomials involving both a ...

Author: Jean Zinn-Justin

Publisher: Oxford University Press

ISBN: 9780198566748

Category: Science

Page: 318

View: 853

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Quantum field theory is hardly comprehensible without path integrals: the goal of this book is to introduce students to this topic within the context of ordinary quantum mechanics and non-relativistic many-body theory, before facing the problems associated with the more involved quantum field theory formalism.
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Path Integrals In Quantum Mechanics Statistics Polymer Physics And Financial Markets 5th Edition

... a monopole can be brought in from infinity inside an infinitely thin tube of flux q = nopo (n ... leading to many new and interesting insights.

Author: Kleinert Hagen

Publisher: World Scientific

ISBN: 9789814365260

Category: Science

Page: 1624

View: 637

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This is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.The relevance of path integrals to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions.
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Open Quantum Systems and Feynman Integrals

... a particle which is confined initially inside a potential barrier . ... e.g. in Avron ( 1982 ) ; the model is interesting in connection with the so ...

Author: Pavel Exner

Publisher: Springer Science & Business Media

ISBN: 9027716781

Category: Science

Page: 356

View: 228

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Every part of physics offers examples of non-stability phenomena, but probably nowhere are they so plentiful and worthy of study as in the realm of quantum theory. The present volume is devoted to this problem: we shall be concerned with open quantum systems, i.e. those that cannot be regarded as isolated from the rest of the physical universe. It is a natural framework in which non-stationary processes can be investigated. There are two main approaches to the treatment of open systems in quantum theory. In both the system under consideration is viewed as part of a larger system, assumed to be isolated in a reasonable approximation. They are differentiated mainly by the way in which the state Hilbert space of the open system is related to that of the isolated system - either by orthogonal sum or by tensor product. Though often applicable simultaneously to the same physical situation, these approaches are complementary in a sense and are adapted to different purposes. Here we shall be concerned with the first approach, which is suitable primarily for a description of decay processes, absorption, etc. The second approach is used mostly for the treatment of various relaxation phenomena. It is comparably better examined at present; in particular, the reader may consult a monograph by E. B. Davies.
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The Subregular Germ of Orbital Integrals

We work inside B2. ... exp(8 K-2) exp(x(m)xn) exp(-& X-2) € Ng. The only interesting case occurs when m = Ga(6). By SUBREGULAR GERM OF ORBITAL INTEGRALS, V 91.

Author: Thomas Callister Hales

Publisher: American Mathematical Soc.

ISBN: 9780821825396

Category: Mathematics

Page: 142

View: 593

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Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on $p$-adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety $Y$ to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes.This monograph constructs the variety $Y$ and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over $p$-adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy.
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