More (Almost) Impossible Integrals, Sums, and Series

A New Collection of Fiendish Problems and Surprising Solutions

verfasst von: Cornel Ioan Vălean

Verlag: Springer International Publishing

Buchreihe : Problem Books in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Integrals

Abstract

In this chapter, we encounter a rich collection of tantalizing integrals in its 60 sections. The reader is invited to evaluate a broad panel of integrals such as various integrals with parameter; generalized integrals that can be promptly turned into generalized harmonic series; curious Cauchy principal value integrals; fascinating integrals related to exponential integrals; integrals with a structure of the integrand involving Fresnel integrals; special sums of integrals with logarithms and the inverse tangent function giving surprising closed forms; unusual integrals with radicals, logarithms, and the inverse tangent function; integrals with logarithms and polylogarithms strongly related to the harmonic series; intriguing double integrals involving special functions; a few good results by Ramanujan; and a good integral involving the fractional part. A special flavor is also added by the challenging questions found in many of the sections, which the curious reader might like to consider for a more thrilling adventure.

Cornel Ioan Vălean

Chapter 2. Hints

Abstract

In this chapter, we present hints to the problems given in Chap. 1.

Cornel Ioan Vălean

Chapter 3. Solutions

In this chapter, we provide solutions to the problems given in Chap. 1.

Cornel Ioan Vălean

Chapter 4. Sums and Series

Abstract

In this chapter, we experience the problems of the second major area of the book, which is Sums and Series. Like in the first chapter, the reader will encounter a wide palette of irresistible problems, and some of the sections will also include challenging questions. As examples, we mention interesting and useful finite sums involving the central binomial coefficient; series with the central binomial coefficient; lots of (less known or not known) Cauchy products; generating functions involving generalized harmonic numbers and generalized skew-harmonic numbers; atypical harmonic series of weights 4 and 5; generalized atypical harmonic series; two (very) nice sets of atypical harmonic series involving the tail of the Riemann zeta function; the practical Fourier series expansions of the Bernoulli polynomials; sets of powerful good-to-know Fourier series of functions involving logarithms, polylogarithms, and trigonometric functions; marvelous binoharmonic series; and very useful polylogarithmic values involving a complex arguments.

Cornel Ioan Vălean

Chapter 5. Hints

Abstract

In this chapter, we present hints to the problems given in Chap. 4.

Cornel Ioan Vălean

Chapter 6. Solutions

In this chapter, we provide solutions to the problems given in Chap. 4.

Cornel Ioan Vălean

Backmatter

Titel: More (Almost) Impossible Integrals, Sums, and Series
verfasst von: Cornel Ioan Vălean
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-21262-8
Print ISBN: 978-3-031-21261-1
DOI: https://doi.org/10.1007/978-3-031-21262-8