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Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research

Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research

By Hongwei Chen

Catalog Code: ECA
Print ISBN: 978-0-88385-768-7
Electronic ISBN: 978-0-88385-935-3
311 pp., Hardbound, 2010
List Price: $59.95
Member Price: $49.95
Series: Classroom Resource Materials

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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

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

Excerpt: Parametric Differentiation and Integration (p. 217)

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.