Excursions in Classical Analysis introduces undergraduate students to advanced problem solving and undergraduate research in two ways. By offering a colorful tour of classical analysis, the book places a wide variety of problems in their historical context. Just as important, it helps students gain an understanding of mathematical discovery and proof.

In discovering a variety of possible solutions to the same sample exercise, readers come to see how connections among apparently inapplicable areas of mathematics can be exploited in problem solving.

This book can serve as excellent preparation for participation in mathematics competitions; as a valuable resource for undergraduate mathematics reading courses and seminars; and as a supplemental text in courses on analysis. It can also be used in independent study.

Table of Contents

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 of (arcsin x)2

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

Index

Excerpt

(p. 217): Parametric Differentiation and Integration

In this chapter, we present an integration method that evaluates integrals via differentiation and integration with respect to a parameter. This approach has been a favorite tool of applied mathematicians and theoretical physicists. In his autobiography, eminent physicist Richard Feynman mentioned how he frequently used this approach when confronted with difficult integrations associated with mathematics and physics problems. He referred to this approach as a "different box of tools." However, most modern texts either ignore this subject or provide only a few examples. In the following, we illustrate this method with the help of some selected examples, most of them being improper integrals. These examples will show that the parametric differentiation and integration technique requires only the mathematical maturity of calculus, and often provides a straightforward method to evaluate difficult integrals which conventionally require the more sophisticated method of contour integration.

About the Author

Hongwei Chen (Christopher Newport University, Newport News, Va.), born in China, received his Ph.D. from North Carolina State University in 1991. He has published more than 50 research articles in classical analysis and partial differential equations.