Invitation to classical analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2012
|
| Schriftenreihe: | Pure and applied undergraduate texts
17 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XIII, 392 S. Ill., graph. Darst. 26 cm |
| ISBN: | 9781470463212 9780821869321 0821869329 |
Internformat
MARC
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| 245 | 1 | 0 | |a Invitation to classical analysis |c Peter Duren |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2012 | |
| 300 | |a XIII, 392 S. |b Ill., graph. Darst. |c 26 cm | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: Invitation to classical analysis
Autor: Duren, Peter L
Jahr: 2012
Contents
Preface
xi
Chapter 1. Basic Principles 1
1.1. Mathematical induction 1
1.2. Real numbers 2
1.3. Completeness principles 5
1.4. Numerical sequences 12
1.5. Infinite series 18
1.6. Continuous functions and derivatives 22
1.7. The Riemann integral 27
1.8. Uniform convergence 33
1.9. Historical remarks 39
1.10. Metrie spaces 41
1.11. Complex numbers 42
Exercises 46
Chapter 2. Special Sequences 51
2.1. The number e 51
2.2. Irrationality of tr 55
2.3. Euler s constant 56
2.4. Vieta s produet formula 60
2.5. Wallis produet formula 61
2.6. Stirling s formula 63
Exercises 66
vi Contents
Chapter 3. Power Series and Related Topics 73
3.1. General properties of power series 73
3.2. Abel s theorem 76
3.3. Cauchy products and Mertens theorem 81
3.4. Taylor s formula with remainder 83
3.5. Newton s binomial series 87
3.6. Composition of power series 89
3.7. Euler s sum 92
3.8. Continuous nowhere differentiable functions 98
Exercises 102
Chapter 4. Inequalities 109
4.1. Elementary inequalities 109
4.2. Cauchy s inequality 112
4.3. Arithmetic-geometric mean inequality 117
4.4. Integral analogues 118
4.5. Jensen s inequality 119
4.6. Hilbert s inequality 122
Exercises 127
Chapter 5. Infinite Products 131
5.1. Basic concepts 131
5.2. Absolute convergence 135
5.3. Logarithmic series 136
5.4. Uniform convergence 138
Exercises 141
Chapter 6. Approximation by Polynomials 145
6.1. Interpolation 145
6.2. Weierstrass approximation theorem 151
6.3. Landau s proof 153
6.4. Bernstein polynomials 157
6.5. Best approximation 160
6.6. Stone-Weierstrass theorem 164
6.7. Refinements of Weierstrass theorem 168
Exercises 171
Contents vii
Chapter 7. Tauberian Theorems 179
7.1. Summation of divergent series 179
7.2. Tauber s theorem 182
7.3. Theorems of Hardy and Littlewood 183
7.4. Karamata s proof 185
7.5. Hardy s power series 190
Exercises 193
Chapter 8. Fourier Series 197
8.1. Physical origins 197
8.2. Orthogonality relations 199
8.3. Mean-square approximation 200
8.4. Convergence of Fourier series 203
8.5. Examples 207
8.6. Gibbs phenomenon 212
8.7. Arithmetic means of partial sums 215
8.8. Continuous functions with divergent Fourier series 219
8.9. Fourier transforms 221
8.10. Inversion of Fourier transforms 228
8.11. Poisson summation formula 232
Exercises 236
Chapter 9. The Gamma Function 247
9.1. Probability integral 247
9.2. Gamma function 249
9.3. Beta function 251
9.4. Legendre s duplication formula 252
9.5. Euler s reflection formula 253
9.6. Infinite produet representation 255
9.7. Generalization of Stirling s formula 257
9.8. Bohr-Mollerup theorem 257
9.9. A special integral 261
Exercises 262
vüi Contents
Chapter 10. Two Topics in Number Theory 269
10.1. Equidistributed sequences 269
10.2. Weyl s criterion 271
10.3. The Riemann zeta function 276
10.4. Connection with the gamma function 280
10.5. Fünctional equation 282
Exercises 286
Chapter 11. Bernoulli Numbers 291
11.1. Calculation of Bernoulli numbers 291
11.2. Sums of positive powers 294
11.3. Euler s sums 295
11.4. Bernoulli polynomials 297
11.5. Euler-Maclaurin summation formula 300
11.6. Applications of Euler-Maclaurin formula 302
Exercises 305
Chapter 12. The Cantor Set 309
12.1. Cardinal numbers 309
12.2. Lebesgue measure 313
12.3. The Cantor set 315
12.4. The Cantor-Scheeffer function 317
12.5. Space-füling curves 320
Exercises 323
Chapter 13. Differential Equations 327
13.1. Existence and uniqueness of Solutions 327
13.2. Wronskians 333
13.3. Power series Solutions 336
13.4. Bessel functions 343
13.5. Hypergeometric functions 348
13.6. Oscillation and comparison theorems 354
13.7. Refinements of Sturm s theory 358
Exercises 360
Contents ix
Chapter 14. Elliptic Integrals 369
14.1. Standard forms 369
14.2. Fagnano s duplication formula 371
14.3. The arithmetic-geometric mean 373
14.4. The Legendre relation 381
Exercises 384
Index of Names 387
Subject Index 389
|
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| indexdate | 2024-07-10T00:22:43Z |
| institution | BVB |
| isbn | 9781470463212 9780821869321 0821869329 |
| language | English |
| lccn | 2011045853 |
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| series | Pure and applied undergraduate texts |
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| spelling | Duren, Peter L. 1935-2020 Verfasser (DE-588)135675502 aut Invitation to classical analysis Peter Duren Providence, RI American Math. Soc. 2012 XIII, 392 S. Ill., graph. Darst. 26 cm txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 17 The Sally series Includes bibliographical references and index Functional analysis Textbooks Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Pure and applied undergraduate texts 17 (DE-604)BV035489189 17 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025225420&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Duren, Peter L. 1935-2020 Invitation to classical analysis Pure and applied undergraduate texts Functional analysis Textbooks Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Invitation to classical analysis |
| title_auth | Invitation to classical analysis |
| title_exact_search | Invitation to classical analysis |
| title_full | Invitation to classical analysis Peter Duren |
| title_fullStr | Invitation to classical analysis Peter Duren |
| title_full_unstemmed | Invitation to classical analysis Peter Duren |
| title_short | Invitation to classical analysis |
| title_sort | invitation to classical analysis |
| topic | Functional analysis Textbooks Analysis (DE-588)4001865-9 gnd |
| topic_facet | Functional analysis Textbooks Analysis |
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