Includes research topics which can be understood by undergraduates
Author provides numerous exercises and examples
Mathematical analysis offers a solid basis for many achievements in applied mathematics and discrete mathematics. This new textbook is focused on differential and integral calculus, and includes a wealth of useful and relevant examples, exercises, and results enlightening the reader to the power of mathematical tools. The intended audience consists of advanced undergraduates studying mathematics or computer science.
The author provides excursions from the standard topics to modern and exciting topics, to illustrate the fact that even first or second year students can understand certain research problems.
The text has been divided into ten chapters and covers topics on sets and numbers, linear spaces and metric spaces, sequences and series of numbers and of functions, limits and continuity, differential and integral calculus of functions of one or several variables, constants (mainly pi) and algorithms for finding them, the W - Z method of summation, estimates of algorithms and of certain combinatorial problems. Many challenging exercises accompany the text. Most of them have been used to prepare for different mathematical competitions during the past few years. In this respect, the author has maintained a healthy balance of theory and exercises.
Content Level » Lower undergraduate
Related subjects » Analysis
Author(s): Marian Muresan
Series: CMS Books in Mathematics
Publisher: Springer
Year: 2008
Language: English
Pages: 452
Cover......Page 1
Canadian Mathematical Society......Page 2
A Concrete Approach to Classical Analysis......Page 4
DOI 10.1007/978-0-387-78933-0......Page 5
Dedication......Page 6
Contents......Page 10
List of Figures......Page 16
Preface......Page 18
1.1.1 The concept of a set......Page 20
1.1.2 Operations on sets......Page 22
1.2.1 Two examples......Page 30
1.2.2 The real number system......Page 31
1.2.3 Elements of algebra......Page 39
1.2.4 Elements of topology on R......Page 44
1.2.5 The extended real number system......Page 49
1.2.6 The complex number system......Page 50
1.3 Exercises......Page 51
1.4 References and comments......Page 59
2.1.1 Finite-dimensional vector spaces......Page 60
2.1.2 Vector spaces......Page 63
2.1.3 Normed spaces......Page 67
2.1.4 Hilbert spaces......Page 68
2.1.5 Inequalities......Page 70
2.2 Metric spaces......Page 76
2.3 Compact spaces......Page 84
2.5 References and comments......Page 89
3.1.1 Convergent sequences......Page 92
3.1.2 Subsequences......Page 95
3.1.3 Cauchy sequences......Page 96
3.1.4 Monotonic sequences......Page 99
3.1.5 Upper limits and lower limits......Page 100
3.1.6 The big Oh and small oh notations......Page 101
3.1.7 Stolz–Cesaro theorem and some of its consequences......Page 104
3.1.8 Certain combinatorial numbers......Page 107
3.1.9 Unimodal, log-convex, and P´olya-frequency sequences......Page 113
3.1.10 Some special sequences......Page 117
3.2 Sequences of functions......Page 127
3.3 Numerical series......Page 129
3.3.1 Series of nonnegative terms......Page 132
3.3.2 The root and the ratio tests......Page 142
3.3.3 Partial summation......Page 144
3.3.4 Absolutely and conditionally convergent series......Page 145
3.3.5 The W–Z method......Page 149
3.4 Series of functions......Page 153
3.4.1 Power series......Page 156
3.4.2 Hypergeometric series......Page 157
3.6 Exercises......Page 158
3.7 References and comments......Page 163
4.1.1 The limit of a function......Page 166
4.2 Continuity......Page 171
4.2.1 Continuity and compactness......Page 174
4.2.2 Uniform continuous mappings......Page 175
4.2.3 Continuity and connectedness......Page 179
4.2.5 Monotonic functions......Page 180
4.3 Periodic functions......Page 183
4.4 Darboux functions......Page 184
4.5 Lipschitz functions......Page 186
4.6.1 Convex functions......Page 188
4.6.2 Jensen convex functions......Page 192
4.7 Functions of bounded variations......Page 196
4.8 Continuity of sequences of functions......Page 202
4.9 Continuity of series of functions......Page 203
4.10 Exercises......Page 205
4.11 References and comments......Page 208
5.1 The derivative of a real function......Page 210
5.2 Mean value theorems......Page 216
5.3 The continuity and the surjectivity of derivatives......Page 225
5.4 L’Hospital theorem......Page 226
5.5 Higher-order derivatives and the Taylor formula......Page 227
5.6 Convex functions and differentiability......Page 233
5.6.1 Inequalities......Page 235
5.7 Differentiability of sequences and series of functions......Page 236
5.8 Power series and Taylor series......Page 238
5.8.1 Operations with power series......Page 241
5.8.2 The Taylor expansion of some elementary functions......Page 244
5.8.3 Bernoulli numbers and polynomials......Page 247
5.9 Some elementary functions introduced by recurrences......Page 249
5.9.2 The logarithm function......Page 250
5.9.3 The exponential function......Page 254
5.9.4 The arctangent function......Page 256
5.10.1 The concept of a primitive function......Page 258
5.10.2 The existence of primitives for continuous functions......Page 261
5.10.3 Operations with functions with primitives......Page 263
5.11 Exercises......Page 266
5.12 References and comments......Page 268
6.1.1 The Darboux integral......Page 270
6.1.2 The Darboux–Stieltjes integral......Page 271
6.2 Integrability of sequences and series of functions......Page 281
6.3 Improper integrals......Page 282
6.4.1 Gamma function......Page 290
6.4.2 Beta function......Page 294
6.5 Polylogarithms......Page 297
6.6 e and π are transcendental......Page 299
6.7 The Gr¨onwall inequality......Page 302
6.8 Exercises......Page 303
6.9 References and comments......Page 306
7.1 Linear and bounded mappings......Page 308
7.1.1 Multilinear mappings......Page 312
7.1.2 Quadratic mappings......Page 313
7.2.1 Variations......Page 315
7.2.2 Gateaux differential......Page 316
7.2.3 Fr´echet differential......Page 317
7.3 Partial derivatives......Page 323
7.3.1 The inverse function theorem and the implicit function theorem......Page 326
7.4 Higher-order differentials and partial derivatives......Page 331
7.4.1 The case X = R^n......Page 333
7.5 Taylor formula......Page 335
7.6.1 First-order conditions......Page 336
7.6.2 Second-order conditions......Page 337
7.6.3 Constraint local extremes......Page 338
7.8 References and comments......Page 341
8.1.1 Double integrals on rectangles......Page 344
8.1.2 Double integrals on simple domains......Page 350
8.2.1 Triple integrals on parallelepipeds......Page 352
8.2.2 Triple integrals on simple domains......Page 359
8.3.1 n-fold integrals on hyperrectangles......Page 360
8.3.2 n-fold integrals on simple domains......Page 364
8.4.1 Line integrals with respect to arc length......Page 365
8.4.3 Green formula......Page 366
8.5 Integrals depending on parameters......Page 368
8.6 Exercises......Page 372
8.7 References and comments......Page 373
9.1.1 Sequences approaching √2......Page 374
9.2.1 Recurrence relation......Page 375
9.3 Arithmetic–geometric mean......Page 377
9.4.1 Computing the nth binary or hexadecimal digit of π......Page 382
9.4.2 BBP formulas by binomial sums......Page 387
9.5 Ramanujan formulas......Page 391
9.6 Several natural ways to introduce number e......Page 393
9.7 Optimal stopping problem......Page 396
9.8 References and comments......Page 397
10.1 Asymptotic estimates......Page 400
10.2 Algorithm analysis......Page 403
10.3 Combinatorial estimates......Page 409
10.3.1 Counting relations, topologies, and partial orders......Page 413
10.3.2 Generalized Fubini numbers......Page 415
10.3.3 The Catalan numbers and binary trees......Page 420
10.4 References and comments......Page 428
References......Page 430
List of Symbols......Page 438
Author Index......Page 442
Subject Index......Page 444
