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Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research [Kõva köide]

5.00/5 (2 hinnangut Goodreads-ist)
  • Formaat: Hardback, 301 pages, kõrgus x laius x paksus: 229x152x20 mm, kaal: 710 g, illustrations
  • Sari: Classroom Resource Materials
  • Ilmumisaeg: 30-Dec-2010
  • Kirjastus: Mathematical Association of America
  • ISBN-10: 0883857685
  • ISBN-13: 9780883857687
Teised raamatud teemal:
Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate ResearchSuurem pilt
  • Formaat: Hardback, 301 pages, kõrgus x laius x paksus: 229x152x20 mm, kaal: 710 g, illustrations
  • Sari: Classroom Resource Materials
  • Ilmumisaeg: 30-Dec-2010
  • Kirjastus: Mathematical Association of America
  • ISBN-10: 0883857685
  • ISBN-13: 9780883857687
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780883857687.html
Märksõnad:
Excursions in Classical Analysis will introduce students to advanced problem solving and undergraduate research in two ways: it will provide a tour of classical analysis, showcasing a wide variety of problems that are placed in historical context, and it will help students gain mastery of mathematical discovery and proof.The author presents a variety of solutions for the problems in the book. Some solutions reach back to the work of mathematicians like Leonhard Euler while others connect to other beautiful parts of mathematics. Readers will frequently see problems solved by using an idea that might at first glance, might not even seem to apply to that problem. Other solutions employ a specific technique that can be used to solve many different kinds of problems. Excursions emphasizes the rich and elegant interplay between continuous and discrete mathematics by applying induction, recursion, and combinatorics to traditional problems in classical analysis.The book will be useful in students' preparations for mathematics competitions, in undergraduate reading courses and seminars, and in analysis courses as a supplement. The book is also ideal for self study, since the chapters are independent of one another and may be read in any order.

Excursions in Classical Analysis will introduce students to advanced problem solving and undergraduate research in two ways: it will provide a tour of classical analysis, showcasing a wide variety of problems that are placed in historical context, and it will help students gain mastery of mathematical discovery and proof.

Arvustused

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

Muu info

A colourful introduction to classical analysis, complete with problems from past mathematics competitions, for undergraduate mathematics students.
Preface xi
1 Two Classical Inequalities
1(16)
1.1 AM-GM Inequality
1(7)
1.2 Cauchy-Schwarz Inequality
8(4)
Exercises
12(3)
References
15(2)
2 A New Approach for Proving Inequalities
17(8)
Exercises
22(1)
References
23(2)
3 Means Generated by an Integral
25(8)
Exercises
30(2)
References
32(1)
4 The L'Hopital Monotone Rule
33(6)
Exercises
37(1)
References
38(1)
5 Trigonometric Identities via Complex Numbers
39(16)
5.1 A Primer of complex numbers
39(2)
5.2 Finite Product Identities
41(2)
5.3 Finite Summation Identities
43(2)
5.4 Euler's Infinite Product
45(3)
5.5 Sums of inverse tangents
48(1)
5.6 Two Applications
49(2)
Exercises
51(2)
References
53(2)
6 Special Numbers
55(18)
6.1 Generating Functions
55(1)
6.2 Fibonacci Numbers
56(2)
6.3 Harmonic numbers
58(3)
6.4 Bernoulli Numbers
61(8)
Exercises
69(3)
References
72(1)
7 On a Sum of Cosecants
73(10)
7.1 A well-known sum and its generalization
73(1)
7.2 Rough estimates
74(2)
7.3 Tying up the loose bounds
76(3)
7.4 Final Remarks
79(1)
Exercises
79(2)
References
81(2)
8 The Gamma Products in Simple Closed Forms
83(10)
Exercises
89(2)
References
91(2)
9 On the Telescoping Sums
93(16)
9.1 The sum of products of arithmetic sequences
94(1)
9.2 The sum of products of reciprocals of arithmetic sequences
95(2)
9.3 Trigonometric sums
97(4)
9.4 Some more telescoping sums
101(5)
Exercises
106(2)
References
108(1)
10 Summation of Subseries in Closed Form
109(12)
Exercises
117(2)
References
119(2)
11 Generating Functions for Powers of Fibonacci Numbers
121(10)
Exercises
128(2)
References
130(1)
12 Identities for the Fibonacci Powers
131(12)
Exercises
140(2)
References
142(1)
13 Bernoulli Numbers via Determinants
143(10)
Exercises
149(3)
References
152(1)
14 On Some Finite Trigonometric Power Sums
153(12)
14.1 Sums involving sec2p (kπ/n)
154(2)
14.2 Sums involving csc2p (kπ/n)
156(1)
14.3 Sums involving tan2p (kπ/n)
157(2)
14.4 Sums involving cot2p (kπ/n)
159(3)
Exercises
162(2)
References
164(1)
15 Power Series of (arcsin x)2
165(10)
15.1 First Proof of the Series (15.1)
165(2)
15.2 Second Proof of the Series (15.1)
167(4)
Exercises
171(2)
References
173(2)
16 Six Ways to Sum ζ(2)
175(14)
16.1 Euler's Proof
176(1)
16.2 Proof by Double Integrals
177(4)
16.3 Proof by Trigonometric Identities
181(1)
16.4 Proof by Power Series
182(1)
16.5 Proof by Fourier Series
183(1)
16.6 Proof by Complex Variables
183(1)
Exercises
184(3)
References
187(2)
17 Evaluations of Some Variant Euler Sums
189(12)
Exercises
197(2)
References
199(2)
18 Interesting Series Involving Binomial Cofficients
201(16)
18.1 An integral representation and its applications
202(6)
18.2 Some Extensions
208(2)
18.3 Searching for new formulas for π
210(2)
Exercises
212(3)
References
215(2)
19 Parametric Differentiation and Integration
217(14)
Example 1
217(1)
Example 2
218(1)
Example 3
219(1)
Example 4
220(1)
Example 5
220(2)
Example 6
222(1)
Example 7
223(1)
Example 8
224(1)
Example 9
225(1)
Example 10
226(2)
Exercises
228(2)
References
230(1)
20 Four Ways to Evaluate the Poisson Integral
231(10)
20.1 Using Riemann Sums
232(1)
20.2 Using A Functional Equation
233(1)
20.3 Using Parametric Differentiation
234(1)
20.4 Using Infinite Series
235(1)
Exercises
235(4)
References
239(2)
21 Some Irresistible Integrals
241(14)
21.1 Monthly Problem 10611
241(2)
21.2 Monthly Problem 11206
243(1)
21.3 Monthly Problem 11275
244(1)
21.4 Monthly Problem 11277
245(1)
21.5 Monthly Problem 11322
246(1)
21.6 Monthly Problem 11329
247(2)
21.7 Monthly Problem 11331
249(101)
21.8 Monthly Problem 11418
250(2)
Exercises
252(2)
References
254(1)
Solutions to Selected Problems 255(42)
Index 297(4)
About the Author 301
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.