Infinite series in a history of analysis: stages up to the verge of summability
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
De Gruyter Oldenbourg
2015
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| Series: | De Gruyter textbook
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Angekündigt u.d.T.: Körle, Hans-Heinrich: A Concise History of Infinite Series |
| Physical Description: | X, 132 S. graph. Darst. 170 mm x 240 mm |
| ISBN: | 311034372X 9783110343724 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
FOREWORD VII
1 THE GREEK ERA: SPECULATION, IDEAS AND THE COMET OF RIGOR 1
1.1 INFINITY, CHALLENGING GREEK PHILOSOPHERS 1
1.2 INFINITY, MASTERED BY A GREEK GEOMETER : EUDOXUS 2
1.3 ARCHIMEDES AND THE GEOMETRIC SERIES TO BE 5
1.4 MEMORIZING A JOINT VENTURE 9
2 SUCCESSFUL EXPERIMENTS ON INFINITE SUMMATION 11
2.1 ORESMUS AND THE GEOMETRIC SERIES 11
2.2 FREESTYLE: CAVALIERI SUMMING INDIVISIBLES 13
2.3 LEIBNIZ TELESCOPES 16
3 SERIES OF FUNCTIONS: THE PROTOTYPES 19
3.1 PRELIMINARY: ASPECTS 19
3.1.1 SERIES VERSUS SEQUENCES 19
3.1.2 SERIES: EVALUATION AND EXPANSION 20
3.2 THE POWER OF POWER SERIES 23
3.2.1 NTLAKANTHA AND THE ARCUSTANGENT SERIES 23
3.2.2 THE HYPERBOLIC LOGARITHM AND ITS SERIES 26
3.2.3 NEWTON AND THE BINOMIAL SERIES 29
3.2.4 TAYLOR SERIES AND EXPANSION; ORDINARY POWER SERIES 32
3.2.5 THE BINOMIAL SERIES IN THE HANDS OF EULER 36
3.2.6 EULER S ALGEBRAICAL ANALYSIS: EVALUATION AND VALUATION 42
3.3 THE POWER OF TRIGONOMETRIC SERIES 46
3.3.1 FACING ANTIPODES 46
3.3.2 EULER EVALUATING A TRIGONOMETRIC SERIES 46
3.3.3 PRIMING PROBLEMS 49
3.3.4 THE MYSTERIOUS . .. FOURIER SERIES 50
3.4 CAUCHY 55
3.4.1 SERIES OF POWERS IN THE HANDS OF CAUCHY AND LAURENT 55
3.4.2 THE CAUCHY PRODUCT 61
HTTP://D-NB.INFO/1041645287
X CONTENTS
4 SERIES SERIOUSLY: THE GREEK COMET REAPPEARS 67
4.1 RIGOR IN RETROSPECT AND PROSPECT 67
4.2 FAREWELL TO THE INFINITELY SMALL; THE EPSILONTICS 69
4.2.1 CONVERGENCE, CONTINUITY 69
4.2.2 DIRICHLET, HEINE: UNIFORM CONTINUITY 71
4.2.3 FROM ABEL TO WEIERSTRAB: UNIFORM CONVERGENCE 72
4.3 WELCOME TO IRRATIONALS: THE COMPLETE SPACE OF REAL NUMBERS 78
4.4 ANOTHER COMPLETE SPACE: THE HOME OF CONVERGENT SEQUENCES 84
5 ON THE VERGE OF SUMMABILITY 89
5.1 DIVERGENT SERIES: SUSPECTED AND RESPECTED 89
5.2 THE INITIATION OF CESARO AND ABEL SUMMATION 90
5.2.1 GRANDI S SERIES, RECURRENT SERIES 90
5.2.2 FROBENIUS THEOREM AND THE LIMIT THEOREMS OF CAUCHY AND ABEL 92
5.2.3 THE CAUCHY PRODUCT REVISITED BY ABEL AND CESARO 94
5.2.4 INTRODUCING THE METHODS OF BASIC CESARO AND OF ABEL MEANS 95
5.3 FEATURES OF CESARO AND ABEL MEANS 98
5.3.1 INCLUSION, LIMITATION, EFFICIENCY 98
5.3.2 THE Q LIMIT: A CONTINUOUS FUNCTIONAL ON A BANACH SPACE 101
5.3.3 AN EARLY TRIUMPH OF ARITHMETIC MEANS: FEJER S THEOREM 103
5.3.4 ON THE SCALE OF CESARO MEANS 103
5.3.5 CESARO S CAUCHY PRODUCT 106
5.3.6 INVERSE THEOREMS, TAUBER S THEOREM 107
APPENDIXES (OFNOTES, PROOFS, EXERCISES) 111
LIFE DATA 119
REFERENCES 121
INDEX 127
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| any_adam_object | 1 |
| author | Körle, Hans-Heinrich 1934- |
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| author_facet | Körle, Hans-Heinrich 1934- |
| author_role | aut |
| author_sort | Körle, Hans-Heinrich 1934- |
| author_variant | h h k hhk |
| building | Verbundindex |
| bvnumber | BV041589805 |
| classification_rvk | SG 500 SG 590 |
| ctrlnum | (OCoLC)923000530 (DE-599)DNB1041645287 |
| dewey-full | 515.243 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.243 |
| dewey-search | 515.243 |
| dewey-sort | 3515.243 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| era | Geschichte gnd |
| era_facet | Geschichte |
| format | Book |
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| indexdate | 2024-07-10T01:00:18Z |
| institution | BVB |
| isbn | 311034372X 9783110343724 |
| language | German |
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| physical | X, 132 S. graph. Darst. 170 mm x 240 mm |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | De Gruyter Oldenbourg |
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| series2 | De Gruyter textbook |
| spelling | Körle, Hans-Heinrich 1934- Verfasser (DE-588)128511699 aut Infinite series in a history of analysis stages up to the verge of summability Hans-Heinrich Körle A Concise History of Infinite Series Berlin [u.a.] De Gruyter Oldenbourg 2015 X, 132 S. graph. Darst. 170 mm x 240 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter textbook Angekündigt u.d.T.: Körle, Hans-Heinrich: A Concise History of Infinite Series Geschichte gnd rswk-swf Reihe (DE-588)4049197-3 gnd rswk-swf Reihe (DE-588)4049197-3 s Geschichte z DE-604 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-039916-5 Erscheint auch als Online-Ausgabe, PDF 978-3-11-035983-1 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027034895&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Körle, Hans-Heinrich 1934- Infinite series in a history of analysis stages up to the verge of summability Reihe (DE-588)4049197-3 gnd |
| subject_GND | (DE-588)4049197-3 |
| title | Infinite series in a history of analysis stages up to the verge of summability |
| title_alt | A Concise History of Infinite Series |
| title_auth | Infinite series in a history of analysis stages up to the verge of summability |
| title_exact_search | Infinite series in a history of analysis stages up to the verge of summability |
| title_full | Infinite series in a history of analysis stages up to the verge of summability Hans-Heinrich Körle |
| title_fullStr | Infinite series in a history of analysis stages up to the verge of summability Hans-Heinrich Körle |
| title_full_unstemmed | Infinite series in a history of analysis stages up to the verge of summability Hans-Heinrich Körle |
| title_short | Infinite series in a history of analysis |
| title_sort | infinite series in a history of analysis stages up to the verge of summability |
| title_sub | stages up to the verge of summability |
| topic | Reihe (DE-588)4049197-3 gnd |
| topic_facet | Reihe |
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