(Almost) Impossible Integrals, Sums, and Series

1. Front Matter

   Pages i-xxxviii

   PDF

2. Integrals

   • Cornel Ioan Vălean

   Pages 1-37

3. Hints

   • Cornel Ioan Vălean

   Pages 39-53

4. Solutions

   • Cornel Ioan Vălean

   Pages 55-277

5. Sums and Series

   • Cornel Ioan Vălean

   Pages 279-314

6. Hints

   • Cornel Ioan Vălean

   Pages 315-328

7. Solutions

   • Cornel Ioan Vălean

   Pages 329-536

8. Back Matter

   Pages 537-539

   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.

• Differentiation Under the Integral
• Integration Methods
• Integration Techniques
• Integrals involving zeta functions
• logarithmic integrals
• methods for evaluating series
• generalized harmonic numbers
• antidifferentiation
• gamma, beta, and polygamma functions
• higher logarithm functions

“This is a collection of 120 challenging integrals and series which can be evaluated by (more or less) elementary means but require a high level of technical proficiency. … The author gives hints and solutions … . This volume can be used to hone one’s technical skills … . Mainly, however, it will appeal to afficionados who enjoy cracking the puzzles presented.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)

• Teremia Mare, Timiş County, Romania

  Cornel Ioan Vălean

Cornel Ioan Valean lives in Romania. While his background is in accounting and business informatics, he is also an independent researcher and self-educated in the area of the calculation of integrals, series and limits. He has published his work in the Journal of Classical Analysis and the Mediterranean Journal of Mathematics, and has made several contributions to the Problems section of the American Mathematical Monthly.

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