Numbers and functions : from a classical-experimental mathematician's point of view
著者
書誌事項
Numbers and functions : from a classical-experimental mathematician's point of view
（Student mathematical library, v. 65）
American Mathematical Society, c2012
- : pbk
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注記
Includes bibliographical references (p. 473-491) and index
内容説明・目次
内容説明
New mathematics often comes about by probing what is already known. Mathematicians will change the parameters in a familiar calculation or explore the essential ingredients of a classic proof. Almost magically, new ideas emerge from this process. This book examines elementary functions, such as those encountered in calculus courses, from this point of view of experimental mathematics. The focus is on exploring the connections between these functions and topics in number theory and combinatorics. There is also an emphasis throughout the book on how current mathematical software can be used to discover and prove interesting properties of these functions. The book provides a transition between elementary mathematics and more advanced topics, trying to make this transition as smooth as possible. Many topics occur in the book, but they are all part of a bigger picture of mathematics. By delving into a variety of them, the reader will develop this broad view. The large collection of problems is an essential part of the book. The problems vary from routine verifications of facts used in the text to the exploration of open questions.
目次
Preface
The number systems
Factorials and binomial coefficients
The Fibonacci numbers
Polynomials
Binomial sums
Catalan numbers
The Stirling numbers of the second kind
Rational functions
Wallis' formula
Farey fractions
The exponential function
Trigonometric functions
Bernoulli polynomials
A sample of classical polynomials: Legendre, Chebyshev, and Hermite
Landen transformations
Three special functions: ,𝜓, and 𝜁
Bibliography
Index
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