Impossible Integrals, Sums, and Series

Introduction

For students and enthusiasts of advanced mathematics in India, few challenges are as exhilarating as tackling the most formidable integrals and series. Impossible Integrals, Sums, and Series by Cornel Ioan Valean is a masterful guide that transforms the seemingly impossible into the achievable. Published by Springer, this hardcover volume is an essential companion for anyone seeking to deepen their understanding of mathematical analysis, problem-solving, and the elegant beauty hidden within complex expressions.

Book Overview

This book is part of the renowned Problem Books in Mathematics series and is a treasure trove of advanced problems and solutions. It focuses on integrals, sums, and series that are often considered too difficult for standard coursework. Valean systematically presents a wide array of challenging problems, each accompanied by detailed, step-by-step solutions. The book bridges the gap between classical calculus and modern research-level techniques, making it a unique resource for Indian students preparing for competitive exams like the IIT JAM, GATE, or the NBHM, as well as for researchers in pure and applied mathematics.

Key Highlights

• Over 400 carefully selected problems ranging from difficult to research-level
• Complete, rigorous solutions provided for every problem
• Focus on special functions, polylogarithms, Euler sums, and harmonic series
• Innovative techniques such as contour integration, series manipulations, and symmetry
• Designed to build intuition and problem-solving stamina
• Ideal for self-study or as a supplement to advanced calculus courses

Inside the Book

The book is structured into thematic chapters that gradually increase in complexity. It begins with foundational integrals and sums, then moves into more specialized topics like logarithmic integrals, polylogarithms, and multiple zeta values. Each chapter includes a rich collection of problems, followed by solutions that reveal clever substitutions, unconventional approaches, and deep connections between seemingly unrelated topics. The author's clear exposition and attention to detail make even the most intricate derivations accessible to dedicated readers.

Key Topics

• Definite and indefinite integrals involving elementary and special functions
• Infinite sums and series, including alternating and harmonic series
• Euler sums and their generalizations
• Polylogarithms, Clausen functions, and Dirichlet beta functions
• Contour integration and residue theorem applications
• Symmetry and transformation techniques in integration

Reader Benefits

By working through this book, readers will develop a robust toolkit for tackling integrals and series that appear in research papers, advanced textbooks, and competitions. The problem-solution format encourages active learning, while the diversity of methods ensures a well-rounded skill set. Indian students will find the material particularly valuable for cracking the toughest problems in exams like ISI, CMI, and the International Mathematical Olympiad (IMO) training. Researchers will discover new perspectives on classical problems and may even find inspiration for their own work.

Learning Outcomes

• Master advanced integration techniques beyond standard calculus courses
• Gain proficiency in manipulating infinite series and products
• Understand the theory and application of special functions
• Learn to recognize patterns and symmetries in complex expressions
• Develop the confidence to tackle unsolved or research-level problems

Who Should Read

This book is intended for advanced undergraduate and graduate students in mathematics, physics, and engineering who have a solid foundation in calculus and real analysis. It is also an excellent resource for mathematics educators, competitive exam aspirants, and professional mathematicians looking to expand their problem-solving repertoire. If you have a passion for mathematical challenges and are ready to push your limits, this book is for you.

About the Author

Cornel Ioan Valean is a distinguished mathematician known for his deep contributions to the field of integrals and series. His work has appeared in leading mathematical journals, and he is celebrated for his ability to solve problems that have stumped researchers for decades. Valean's writing combines rigorous mathematics with pedagogical clarity, making his books invaluable for self-learners and academics alike.

About the Publisher

Springer is a globally respected academic publisher with a long tradition of producing high-quality mathematics books. The Problem Books in Mathematics series, in particular, is known for its rigorous content and excellent production standards. This hardcover edition is built to last through years of use, making it a worthy addition to any serious mathematician's library in India.

Conclusion

Impossible Integrals, Sums, and Series is more than a problem book—it is a journey into the heart of mathematical beauty and ingenuity. For Indian readers who aspire to excel in advanced mathematics, this volume offers both a challenge and a reward: the satisfaction of mastering the impossible. Add it to your collection today and unlock the secrets of the most captivating integrals and series ever compiled.