Excursions in Classical Analysis

Hongwei Chen was born in China, and received his PhD from North Carolina State University in 1991. He is currently a Professor of Mathematics at Christopher Newport University. He has published more than fifty research articles in classical analysis and partial differential equations.

• Preface; 1. Two classical inequalities; 2. A new approach for proving inequalities; 3. Means generated by an integral; 4. The L'Hôpital monotone rule; 5. Trigonometric identities via complex numbers; 6. Special numbers; 7. On a sum of cosecants; 8. The gamma products in simple closed forms; 9. On the telescoping sums; 10. Summation of subseries in closed form; 11. Generating functions for powers of Fibonacci numbers; 12. Identities for the Fibonacci powers; 13. Bernoulli numbers via determinants; 14. On some finite trigonometric power sums; 15. Power series; 16. Six ways to sum ζ(2); 17. Evaluations of some variant Euler sums; 18. Interesting series involving binomial coefficients; 19. Parametric differentiation and integration; 20. Four ways to evaluate the Poisson integral; 21. Some irresistible integrals; Solutions to selected problems.

Based on his obviously very rich and far-reaching experience in this didactic realm, the author offers a colorful panorama of various topics in calculus, both elementary and advanced, as well as a wide variety of typical problems placed in their respective historical contexts. Generally, starting from analyzing simple cases, the present book illustrates creative problem solving techniques along selected case studies, on the one hand, and helps students grasp the art of mathematical experimentation, guessing, discovery, and proof, on the other. In fact, this text often presents different approaches and solutions to a particular problem, thereby illuminating the fascinating interplay between original classical ideas, related more recent viewpoints, and various methods of proof likewise. ...No doubt, this fine book will be of great use and value for students preparing for mathematics competitions, participating in undergraduate analysis courses, seminars, and research projects, or conducting any kind of self-study in the field."" - Zentrallblatt Math""Chen (Christopher Newport Univ) offers many enjoyable trips through classical analysis, providing wonderful insights and remarkable connections. The focus is problem solving, while modeling various aspects of discovery, proof, and multiple solutions. The chapters are organized around specific themes (e.g., classical inequalities or trigonometric identities via complex numbers or evaluating the Poisson integral), each designed to be experienced independently. The chapters consistently include historical perspectives, connections to modern research, and powerful interactions between continuous and discrete mathematics. ...The ""excursions"" are designed for upper-level mathematics students with strong backgrounds in calculus."" - CHOICE Magazine""Chen's book is a wonderful tour of classical analysis and would serve as an excellent source of undergraduate enrichment/research problems. It recalls the type of gems in classical analysis, number theory, and combinatorics I first encountered in the books of Polya and Szego as an undergraduate many years ago. Peruse the table of contents and see if some of the topics and subtopics don't grab you."" - Henry Ricardo, MAA Reviews