Real analysis and foundations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2005
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Studies in advanced mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XVI, 454 S. graph. Darst. |
| ISBN: | 1584884835 |
Internformat
MARC
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Datensatz im Suchindex
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|---|---|
| adam_text | Table of Contents
Preface to the Second Edition vii
Preface to the First Edition ix
Logic and Set Theory 1
1.1 Introduction......................... 1
1.2 And and Or ...................... 2
1.3 Not and If-Then .................... 4
1.4 Contrapositive, Converse, and Iff ............ 7
1.5 Quantifiers ......................... 10
1.6 Set Theory and Venn Diagrams . ............. 13
1.7 Relations and Functions.................. 18
1.8 Countable and Uncountable Sets............. 24
EXERCISES........................ 34
Number Systems 39
2.1 The Natural Numbers................... 39
2.2 Equivalence Relations and Equivalence Classes..... 42
2.3 The Integers......................... 44
2.4 The Rational Numbers................... 49
2.5 The Real Numbers..................... 58
2.6 The Complex Numbers................... 62
EXERCISES........................ 67
Sequences 75
3.1 Convergence of Sequences................. 75
3.2 Subsequences........................ 81
3.3 Limsup and Liminf.................... 85
Xlll
XIV
3.4 Some Special Sequences.................. 88
EXERCISES........................ 91
4 Series of Numbers 95
4.1 Convergence of Series ................... 95
4.2 Elementary Convergence Tests .............. 100
4.3 Advanced Convergence Tests............... 107
4.4 Some Special Series..................... 114
4.5 Operations on Series.................... 119
EXERCISES........................ 122
5 Basic Topology 129
5.1 Open and Closed Sets................... 129
5.2 Further Properties of Open and Closed Sets....... 134
5.3 Compact Sets........................ 139
5.4 The Cantor Set....................... 142
5.5 Connected and Disconnected Sets............. 145
5.6 Perfect Sets......................... 147
EXERCISES........................ 149
6 Limits and Continuity of Functions 153
6.1 Definition and Basic Properties of the Limit of a Function 153
6.2 Continuous Functions................... 159
6.3 Topological Properties and Continuity.......... 164
6.4 Classifying Discontinuities and Monotonicity...... 170
EXERCISES........................ 175
7 Differentiation of Functions 181
7.1 The Concept of Derivative................. 181
7.2 The Mean Value Theorem and Applications....... 189
7.3 More on the Theory of Differentiation.......... 197
EXERCISES........................ 201
8 The Integral 205
8.1 Partitions and The Concept of Integral.......... 205
8.2 Properties of the Riemann Integral............ 211
8.3 Another Look at the Integral............... 219
8.4 Advanced Results on Integration Theory......... 224
EXERCISES........................ 231
XV
9 Sequences and Series of Functions 237
9.1 Partial Sums and Pointwise Convergence......... 237
9.2 More on Uniform Convergence .............. 242
9.3 Series of Functions..................... 245
9.4 The Weierstrass Approximation Theorem........ 248
EXERCISES........................ 252
10 Elementary Transcendental Functions 257
10.1 Power Series......................... 257
10.2 More on Power Series: Convergence Issues........ 262
10.3 The Exponential and Trigonometric Functions ..... 267
10.4 Logarithms and Powers of Real Numbers ........ 273
10.5 The Gamma Function and Stirling s Formula...... 276
EXERCISES........................ 278
11 Applications of Analysis to Differential Equations 285
11.1 Picard s Existence and Uniqueness Theorem....... 285
11.1.1 The Form of a Differential Equation....... 285
11.1.2 Picard s Iteration Technique............ 286
11.1.3 Some Illustrative Examples............ 287
11.1.4 Estimation of the Picard Iterates......... 289
11.2 The Method of Characteristics.............. 290
11.3 Power Series Methods................... 293
EXERCISES........................ 301
12 Introduction to Harmonic Analysis 307
12.1 The Idea of Harmonic Analysis.............. 307
12.2 The Elements of Fourier Series.............. 308
12.3 An Introduction to the Fourier Transform........ 315
12.3.1 Appendix: Approximation by Smooth Functions 319
12.4 Fourier Methods in the Theory of Differential Equations 324
12.4.1 Remarks on Different Fourier Notations..... 324
12.4.2 The Dirichlet Problem on the Disc........ 325
12.4.3 The Poisson Integral................ 329
12.4.4 The Wave Equation................ 331
EXERCISES........................ 336
13 Functions of Several Variables 345
13.1 Review of Linear Algebra................. 345
13.2 A New Look at the Basic Concepts of Analysis..... 351
13.3 Properties of the Derivative................ 356
XVI
13.4 The Inverse and Implicit Function Theorems...... 361
13.5 Differential Forms ..................... 367
13.5.1 The Idea of a Differential Form.......... 368
13.5.2 Differential Forms on a Surface.......... 369
13.5.3 General Differential Forms and Stokes s Theorem 372
EXERCISES........................ 375
14 Advanced Topics 379
14.1 Metric Spaces........................ 379
14.2 Topology in a Metric Space................ 384
14.3 The Baire Category Theorem............... 387
14.4 The Ascoli-Arzela Theorem................ 391
14.5 The Lebesgue Integral................... 394
14.5.1 Measurable Sets .................. 395
14.5.2 The Lebesgue Integral............... 400
14.5.3 Calculating with the Lebesgue Integral...... 403
14.6 A Taste of Probability Theory............... 408
EXERCISES........................ 414
15 A Glimpse of Wavelet Theory 421
15.1 Localization in the Time and Space Variables...... 421
15.2 A Custom Fourier Analysis................ 424
15.3 The Haar Basis....................... 426
15.4 Some Illustrative Examples................ 432
15.5 Closing Remarks...................... 441
EXERCISES........................ 441
Bibliography 445
Index 447
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| any_adam_object | 1 |
| author | Krantz, Steven G. 1951- |
| author_GND | (DE-588)130535907 |
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| classification_tum | MAT 260f |
| ctrlnum | (OCoLC)253663513 (DE-599)BVBBV019833875 |
| dewey-full | 515.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.8 |
| dewey-search | 515.8 |
| dewey-sort | 3515.8 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Real analysis and foundations Steven G. Krantz 2. ed. Boca Raton [u.a.] Chapman & Hall/CRC 2005 XVI, 454 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Reelle Analysis (DE-588)4627581-2 gnd rswk-swf Analysis (DE-588)4001865-9 s Reelle Variable (DE-588)4202614-3 s DE-604 Reelle Analysis (DE-588)4627581-2 s 1\p DE-604 Digitalisierung UBPassau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013158913&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013158913&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Krantz, Steven G. 1951- Real analysis and foundations Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd Analysis (DE-588)4001865-9 gnd Reelle Analysis (DE-588)4627581-2 gnd |
| subject_GND | (DE-588)4202614-3 (DE-588)4001865-9 (DE-588)4627581-2 |
| title | Real analysis and foundations |
| title_auth | Real analysis and foundations |
| title_exact_search | Real analysis and foundations |
| title_full | Real analysis and foundations Steven G. Krantz |
| title_fullStr | Real analysis and foundations Steven G. Krantz |
| title_full_unstemmed | Real analysis and foundations Steven G. Krantz |
| title_short | Real analysis and foundations |
| title_sort | real analysis and foundations |
| topic | Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd Analysis (DE-588)4001865-9 gnd Reelle Analysis (DE-588)4627581-2 gnd |
| topic_facet | Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable Analysis Reelle Analysis |
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