Introduction to classical real analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Belmont, California
Wadsworth
1981
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 575 S. |
| ISBN: | 0534980120 |
Staff View
MARC
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| 245 | 1 | 0 | |a Introduction to classical real analysis |c Karl R. Stromberg, Kansas State University |
| 264 | 1 | |a Belmont, California |b Wadsworth |c 1981 | |
| 300 | |a IX, 575 S. | ||
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Record in the Search Index
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| adam_text | CONTENTS
0 PRELIMINARIES 1
Sets and Subsets 1
Operations on Sets 2
Ordered Pairs and Relations 3
Equivalence Relations 3
Functions 4
Products of Sets 5
1 NUMBERS 7
Axioms for U 8
The Supremum Principle 12
The Natural Numbers 13
Integers 16
Decimal Representation of Natural Numbers 17
Roots 20
Rational and Irrational Numbers 21
Complex Numbers 22
Some Inequalities 24
Extended Real Numbers 27
Finite and Infinite Sets 28
Newton s Binomial Theorem 33
Exercises 35
v
Vi CONTENTS
2 SEQUENCES AND SERIES 39
Sequences in C 39
Sequences in E? 43
Cauchy Sequences 51
Subsequences 52
Series of Complex Terms 53
Series of Nonnegative Terms 59
Decimal Expansions 65
The Number e 67
The Root and Ratio Tests 69
Power Series 71
Multiplication of Series 72
Lebesgue Outer Measure 76
Cantor Sets 80
Exercises 84
3 LIMITS AND CONTINUITY 91
Metric Spaces 91
Topological Spaces 95
Compactness 102 !
Connectedness 107
Completeness 107
Baire Category 109
Exercises 111
Limits of Functions at a Point 114
Exercises 119
Compactness, Connectedness, and Continuity 122
Exercises 124
Simple Discontinuities and Monotone Functions 128
Exercises 131
Expand Log 134
Powers 136
Exercises 138
Uniform Convergence 139
Exercises 144
Stone Weierstrass Theorems 146
Exercises 156
Total Variation 159
Absolute Continuity 162
Exercises 163
Equicontinuity 164 ;:
Exercises 168 1
CONTENTS vii
4 DIFFERENTIATION 170
Dini Derivates 170
??A Nowhere Differentiable, Everywhere Continuous, Function 174
Some Elementary Formulas 175
Local Extrema 177
Mean Value Theorems 178
L Hospital s Rule 179
Exercises 182
Higher Order Derivatives 188
Taylor Polynomials 192
Exercises 197
?Convex Functions 199
?Exercises 203
Differentiability Almost Everywhere 206
Exercises 211
?Termwise Differentiation of Sequences 213
?Exercises 215
? Complex Derivatives 219
?Exercises 223
5 THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 226
The Exponential Function 226
The Trigonometric Functions 227
The Argument 231
Exercises 232
?Complex Logarithms and Powers 236
?Exercises 240
??n is Irrational 240
??Exercises 242
?Log Series and the Inverse Tangent 242
??Rational Approximation to n 245
?Exercises 246
??The Sine Product and Related pansions 247
??Stirling s Formula 253
??Exercises 255
6 INTEGRATION 257
Step Functions 257
The First Extension 263
Integrable Functions 264
Vlii CONTENTS
Two Limit Theorems 266
The Riemann Integral 269
Exercises 274
Measureable Functions 285
Complex Valued Functions 289
Measurable Sets 293
Structure of Measurable Functions 300
Integration Over Measurable Sets 305
Exercises 306
The Fundamental Theorem of Calculus 318
Integration by Parts 323
Integration Substitution 323
Two Mean Value Theorems 327
?Arc Length 329
Exercises 331
Holder s and Minkowski s Inequalities 339 j
The Lp Spaces 341 j
Exercises 343 !
Integration on R 345
Iteration of Integrals 349
Exercises 356
Some Differential Calculus in Higher Dimensions 364
Exercises 376
Transformations of Integrals on R 385
Exercises 393
i
i
7 INFINITE SERIES AND INFINITE PRODUCTS 398
Series Having Monotone Terms 398
Limit Comparison Tests 401
?*Two Log Tests 404
•?Other Ratio Tests 406
?Exercises 409
??Infinite Products 411
??Exercises 418
Some Theorems of Abel 420
Exercises 426
? ?Another Ratio Test and the Binomial Series 434
??Exercises 440
Rearrangements and Double Series 443 j
Exercises 454 j
??The Gamma Function 460 j
CONTENTS ix
??Exercises 470
Divergent Series 473
Exercises 484
Tauberian Theorems 494
Exercises 500
8 TRIGONOMETRIC SERIES 502
Trigonometric Series and Fourier Series 503
Which Trigonometric Series are Fourier Series? 510
Exercises 518
?Divergent Fourier Series 526
?Exercises 530
Summability of Fourier Series 534
Riemann Localization and Convergence Criteria 541
Growth Rate of Partial Sums 551
Exercises 554
BIBLIOGRAPHY 567
INDEX 569
* Sections marked with a single asterisk are not actually needed in conjunction with
any of the unmarked sections. These sections may be safely omitted if time permits only a
short course.
**Sections marked with a double asterisk are not actually needed in conjunction with
any other section. They are included as interesting and useful applications of the theory.
|
| any_adam_object | 1 |
| author | Stromberg, Karl R. 1931-1994 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
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| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:43:27Z |
| institution | BVB |
| isbn | 0534980120 |
| language | English |
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| physical | IX, 575 S. |
| publishDate | 1981 |
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| spelling | Stromberg, Karl R. 1931-1994 Verfasser (DE-588)105811041 aut Introduction to classical real analysis Karl R. Stromberg, Kansas State University Belmont, California Wadsworth 1981 IX, 575 S. txt rdacontent n rdamedia nc rdacarrier Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001504827&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stromberg, Karl R. 1931-1994 Introduction to classical real analysis Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Introduction to classical real analysis |
| title_auth | Introduction to classical real analysis |
| title_exact_search | Introduction to classical real analysis |
| title_full | Introduction to classical real analysis Karl R. Stromberg, Kansas State University |
| title_fullStr | Introduction to classical real analysis Karl R. Stromberg, Kansas State University |
| title_full_unstemmed | Introduction to classical real analysis Karl R. Stromberg, Kansas State University |
| title_short | Introduction to classical real analysis |
| title_sort | introduction to classical real analysis |
| topic | Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analyse mathématique Mathematical analysis Analysis |
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