A handbook of real variables: with applications to differential equations and Fourier analysis
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2004
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 201 S. graph. Darst. |
| ISBN: | 081764329X 376434329X |
Internformat
MARC
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| 100 | 1 | |a Krantz, Steven G. |d 1951- |e Verfasser |0 (DE-588)130535907 |4 aut | |
| 245 | 1 | 0 | |a A handbook of real variables |b with applications to differential equations and Fourier analysis |c Steven G. Krantz |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2004 | |
| 300 | |a XII, 201 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Baire-Kategoriesatz - Taylor-Reihe - Reelle Analysis - Weierstrass-Funktionen |2 swd | |
| 650 | 7 | |a Baire-Kategoriesatz - Weierstrass-Funktionen - Taylor-Reihe - Reelle Analysis |2 swd | |
| 650 | 7 | |a Kernfunktion - Linearer Operator |2 swd | |
| 650 | 7 | |a Linearer Operator - Kernfunktion |2 swd | |
| 650 | 7 | |a Reelle Analysis - Taylor-Reihe - Baire-Kategoriesatz - Weierstrass-Funktionen |2 swd | |
| 650 | 7 | |a Taylor-Reihe - Reelle Analysis - Baire-Kategoriesatz - Weierstrass-Funktionen |2 swd | |
| 650 | 4 | |a Functions of real variables | |
| 650 | 4 | |a Mathematical analysis | |
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Datensatz im Suchindex
| _version_ | 1804129984970752000 |
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| adam_text | STEVEN G. KRANTZ A HANDBOOK OF REAL VARIABLES WITH APPLICATIONS TO
DIFFERENTIAL EQUATIONS AND FOURIER ANALYSIS BIRKHAEUSER BOSTON BASEL
BERLIN CONTENTS PREFACE XI 1 BASICS 1 1.1 SETS 1 1.2 OPERATIONS ON SETS
1 1.3 FUNCTIONS 4 1.4 OPERATIONS ON FUNCTIONS 5 1.5 NUMBER SYSTEMS 6
1.5.1 THE REAL NUMBERS 6 1.6 COUNTABLE AND UNCOUNTABLE SETS 9 2
SEQUENCES 11 2.1 INTRODUCTION TO SEQUENCES 11 2.1.1 THE DEFINITION AND
CONVERGENCE 11 2.1.2 THE CAUCHY CRITERION 12 2.1.3 MONOTONICITY 13 2.1.4
THE PINCHING PRINCIPLE 14 2.1.5 SUBSEQUENCES 14 2.1.6 THE
BOLZANO-WEIERSTRASS THEOREM 15 2.2 LIMSUP AND LIMINF 15 2.3 SOME SPECIAL
SEQUENCES 17 3 SERIES 21 3.1 INTRODUCTION TO SERIES 21 3.1.1 THE
DEFINITION AND CONVERGENCE 21 3.1.2 PARTIAL SUMS 22 3.2 ELEMENTARY
CONVERGENCE TESTS 23 3.2.1 THE COMPARISON TEST 23 3.2.2 THE CAUCHY
CONDENSATION TEST 24 3.2.3 GEOMETRIC SERIES 24 3.2.4 THE ROOT TEST 25
3.2.5 THE RATIO TEST 25 3.2.6 ROOT AND RATIO TESTS FOR DIVERGENCE 27 3.3
ADVANCED CONVERGENCE TESTS 28 3.3.1 SUMMATION BY PARTS 28 V VI CONTENTS
3.3.2 ABEL S TEST 29 3.3.3 ABSOLUTE AND CONDITIONAL CONVERGENCE 31 3.3.4
REARRANGEMENTS OF SERIES 32 3.4 SOME PARTICULAR SERIES 33 3.4.1 THE
SERIES FOR * 33 3.4.2 OTHER REPRESENTATIONS FOR * 34 3.4.3 SUMS OF
POWERS 35 3.5 OPERATIONS ON SERIES 36 3.5.1 SUMS AND SCALAR PRODUCTS OF
SERIES 36 3.5.2 PRODUCTS OF SERIES 36 3.5.3 THE CAUCHY PRODUCT 37 4 THE
TOPOLOGY OF THE REAL LINE 39 4.1 OPEN AND CLOSED SETS 39 4.1.1 OPEN SETS
39 4.1.2 CLOSED SETS 40 4.1.3 CHARACTERIZATION OF OPEN AND CLOSED SETS
IN TERMS OF SEQUENCES 41 4.1.4 FURTHER PROPERTIES OF OPEN AND CLOSED
SETS 42 4.2 OTHER DISTINGUISHED POINTS 43 4.2.1 INTERIOR POINTS AND
ISOLATED POINTS 43 4.2.2 ACCUMULATION POINTS 43 4.3 BOUNDED SETS 44 4.4
COMPACT SETS 45 4.4.1 INTRODUCTION 45 4.4.2 THE HEINE-BOREL THEOREM,
PART I 45 4.4.3 THE HEINE-BOREL THEOREM, PART II 47 4.5 THE CANTOR SET
48 4.6 CONNECTED AND DISCONNECTED SETS 50 4.6.1 CONNECTIVITY 50 4.7
PERFECT SETS 51 5 LIMITS AND THE CONTINUITY OF FUNCTIONS 53 5.1
DEFINITIONS AND BASIC PROPERTIES 53 5.1.1 LIMITS 53 5.1.2 A LIMIT THAT
DOES NOT EXIST 54 5.1.3 UNIQUENESS OF LIMITS 54 5.1.4 PROPERTIES OF
LIMITS 55 5.1.5 CHARACTERIZATION OF LIMITS USING SEQUENCES 57 5.2
CONTINUOUS FUNCTIONS 57 5.2.1 CONTINUITY AT A POINT 57 5.2.2 THE
TOPOLOGICAL APPROACH TO CONTINUITY 60 5.3 TOPOLOGICAL PROPERTIES AND
CONTINUITY 62 5.3.1 THE IMAGE OF A FUNCTION 62 5.3.2 UNIFORM CONTINUITY
63 CONTENTS VII 5.3.3 CONTINUITY AND CONNECTEDNESS 65 5.3.4 THE
INTERMEDIATE VALUE PROPERTY 65 5.4 CLASSIFYING DISCONTINUITIES AND
MONOTONICITY 66 5.4.1 LEFT AND RIGHT LIMITS 66 5.4.2 TYPES OF
DISCONTINUITIES 66 5.4.3 MONOTONIE FUNCTIONS 67 6 THE DERIVATIVE 71 6.1
THE CONCEPT OF DERIVATIVE 71 6.1.1 THE DEFINITION 71 6.1.2 PROPERTIES OF
THE DERIVATIVE 72 6.1.3 THE WEIERSTRASS NOWHERE DIFFERENTIABLE FUNCTION
73 6.1.4 THE CHAIN RULE 74 6.2 THE MEAN VALUE THEOREM AND APPLICATIONS
74 6.2.1 LOCAL MAXIMA AND MINIMA 74 6.2.2 FERMAT STEST 74 6.2.3
DARBOUX S THEOREM 75 6.2.4 THE MEAN VALUE THEOREM 75 6.2.5 EXAMPLES OF
THE MEAN VALUE THEOREM 78 6.3 FURTHER RESULTS ON THE THEORY OF
DIFFERENTIATION 79 6.3.1 L HOPITAL S RULE 79 6.3.2 THE DERIVATIVE OF AN
INVERSE FUNCTION 81 6.3.3 HIGHER-ORDER DERIVATIVES 82 6.3.4 CONTINUOUS
DIFFERENTIABILITY 82 7 THE INTEGRAL 85 7.1 THE CONCEPT OF INTEGRAL 85
7.1.1 PARTITIONS 85 7.1.2 REFINEMENTS OF PARTITIONS 88 7.1.3 EXISTENCE
OF THE RIEMANN INTEGRAL 89 7.1.4 INTEGRABILITY OF CONTINUOUS FUNCTIONS
89 7.2 PROPERTIES OF THE RIEMANN INTEGRAL 89 7.2.1 EXISTENCE THEOREMS 89
7.2.2 INEQUALITIES FOR INTEGRALS 91 7.2.3 PRESERVATION OF INTEGRABLE
FUNCTIONS UNDER COMPOSITION . . 91 7.2.4 THE FUNDAMENTAL THEOREM OF
CALCULUS 92 7.3 FURTHER RESULTS ON THE RIEMANN INTEGRAL 93 7.3.1 THE
RIEMANN-STIELTJES INTEGRAL 93 7.3.2 RIEMANN S LEMMA 96 7.4 ADVANCED
RESULTS ON INTEGRATION THEORY 96 7.4.1 EXISTENCE OF THE
RIEMANN-STIELTJES INTEGRAL 96 7.4.2 INTEGRATION BY PARTS 97 7.4.3
LINEARITY PROPERTIES 98 7.4.4 BOUNDED VARIATION 98 VIU CONTENTS 8
SEQUENCES AND SERIES OF FUNCTIONS 103 8.1 PARTIAL SUMS AND POINTWISE
CONVERGENCE 103 8.1.1 SEQUENCES OF FUNCTIONS 103 8.1.2 UNIFORM
CONVERGENCE 104 8.2 MORE ON UNIFORM CONVERGENCE 106 8.2.1 COMMUTATION OF
LIMITS 106 8.2.2 THE UNIFORM CAUCHY CONDITION 107 8.2.3 LIMITS OF
DERIVATIVES 107 8.3 SERIES OF FUNCTIONS 108 8.3.1 SERIES AND PARTIAL
SUMS 108 8.3.2 UNIFORM CONVERGENCE OF A SERIES 109 8.3.3 THE WEIERSTRASS
M-TEST 110 8.4 THE WEIERSTRASS APPROXIMATION THEOREM ILL 8.4.1
WEIERSTRASS S MAIN RESULT ILL 9 SOME SPECIAL FUNCTIONS 113 9.1 POWER
SERIES 113 9.1.1 CONVERGENCE 113 9.1.2 INTERVAL OF CONVERGENCE 114 9.1.3
REAL ANALYTIC FUNCTIONS 114 9.1.4 MULTIPLICATION OF REAL ANALYTIC
FUNCTIONS 115 9.1.5 DIVISION OF REAL ANALYTIC FUNCTIONS 115 9.2 MORE ON
POWER SERIES: CONVERGENCE ISSUES 117 9.2.1 THE HADAMARD FORMULA 117
9.2.2 THE DERIVED SERIES 118 9.2.3 FORMULA FOR THE COEFFICIENTS OF A
POWER SERIES 119 9.2.4 TAYLOR S EXPANSION 121 9.3 THE EXPONENTIAL AND
TRIGONOMETRIC FUNCTIONS 123 9.3.1 THE SERIES DEFINITION 123 9.3.2 THE
TRIGONOMETRIC FUNCTIONS 126 9.3.3 EULER S FORMULA 126 9.3.4 THE
TRIGONOMETRIC FUNCTIONS 127 9.4 LOGARITHMS AND POWERS OF REAL NUMBERS
130 9.4.1 THE LOGARITHMIC FUNCTION 130 9.4.2 CHARACTERIZATION OF THE
LOGARITHM 132 9.5 THE GAMMA FUNCTION AND STIRLING S FORMULA 132 9.5.1
STIRLING S FORMULA 133 9.6 AN INTRODUCTION TO FOURIER SERIES 133 9.6.1
TRIGONOMETRIC SERIES 134 9.6.2 FORMULA FOR THE FOURIER COEFFICIENTS 135
9.6.3 BESSEL S INEQUALITY 135 9.6.4 THE DIRICHLET KERNEL 136 CONTENTS IX
10 ADVANCED TOPICS 139 10.1 METRIE SPACES 139 10.1.1 THE CONCEPT OF A
METRIE 139 10.1.2 EXAMPLES OF METRIE SPACES 139 10.1.3 CONVERGENCE IN A
METRIE SPACE 140 10.1.4 THE CAUCHY CRITERION 141 10.1.5 COMPLETENESS 142
10.1.6 ISOLATED POINTS 142 10.2 TOPOLOGY IN A METRIE SPACE 144 10.2.1
BALLS IN A METRIE SPACE 144 10.2.2 ACCUMULATION POINTS 145 10.2.3
COMPACTNESS 146 10.3 THE BAIRE CATEGORY THEOREM 147 10.3.1 DENSITY 147
10.3.2 CLOSURE 148 10.3.3 BAIRE S THEOREM 149 10.4 THE ASCOLI-ARZELA
THEOREM 150 10.4.1 EQUICONTINUITY 150 10.4.2 EQUIBOUNDEDNESS 151 10.4.3
THE ASCOLI-ARZELA THEOREM 151 11 DIFFERENTIAL EQUATIONS 153 11.1
PICARD S EXISTENCE AND UNIQUENESS THEOREM 153 11.1.1 THE FORM OF A
DIFFERENTIAL EQUATION 153 11.1.2 PICARD S ITERATION TECHNIQUE 154 11.1.3
SOME ILLUSTRATIVE EXAMPLES 155 11.1.4 ESTIMATION OF THE PICARD ITERATES
156 11.2 THE METHOD OF CHARACTERISTICS 158 11.3 POWER SERIES METHODS 160
11.4 FOURIER ANALYTIC METHODS 168 11.4.1 REMARKS ON DIFFERENT FOURIER
NOTATIONS 168 11.4.2 THE DIRICHLET PROBLEM ON THE DISC 169 11.4.3 THE
POISSON INTEGRAL 172 11.4.4 THE WAVE EQUATION 174 GLOSSARY OF TERMS FROM
REAL VARIABLE THEORY 177 LIST OF NOTATION 189 GUIDE TO THE LITERATURE
193 BIBLIOGRAPHY 197 INDEX 199 *}
|
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| author | Krantz, Steven G. 1951- |
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| dewey-ones | 515 - Analysis |
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| dewey-sort | 3515 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV017109853 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:13:50Z |
| institution | BVB |
| isbn | 081764329X 376434329X |
| language | English |
| lccn | 2003050248 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010317142 |
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| owner_facet | DE-824 DE-91G DE-BY-TUM DE-11 |
| physical | XII, 201 S. graph. Darst. |
| publishDate | 2004 |
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| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut A handbook of real variables with applications to differential equations and Fourier analysis Steven G. Krantz Boston [u.a.] Birkhäuser 2004 XII, 201 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Baire-Kategoriesatz - Taylor-Reihe - Reelle Analysis - Weierstrass-Funktionen swd Baire-Kategoriesatz - Weierstrass-Funktionen - Taylor-Reihe - Reelle Analysis swd Kernfunktion - Linearer Operator swd Linearer Operator - Kernfunktion swd Reelle Analysis - Taylor-Reihe - Baire-Kategoriesatz - Weierstrass-Funktionen swd Taylor-Reihe - Reelle Analysis - Baire-Kategoriesatz - Weierstrass-Funktionen swd Functions of real variables Mathematical analysis GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010317142&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- A handbook of real variables with applications to differential equations and Fourier analysis Baire-Kategoriesatz - Taylor-Reihe - Reelle Analysis - Weierstrass-Funktionen swd Baire-Kategoriesatz - Weierstrass-Funktionen - Taylor-Reihe - Reelle Analysis swd Kernfunktion - Linearer Operator swd Linearer Operator - Kernfunktion swd Reelle Analysis - Taylor-Reihe - Baire-Kategoriesatz - Weierstrass-Funktionen swd Taylor-Reihe - Reelle Analysis - Baire-Kategoriesatz - Weierstrass-Funktionen swd Functions of real variables Mathematical analysis |
| title | A handbook of real variables with applications to differential equations and Fourier analysis |
| title_auth | A handbook of real variables with applications to differential equations and Fourier analysis |
| title_exact_search | A handbook of real variables with applications to differential equations and Fourier analysis |
| title_full | A handbook of real variables with applications to differential equations and Fourier analysis Steven G. Krantz |
| title_fullStr | A handbook of real variables with applications to differential equations and Fourier analysis Steven G. Krantz |
| title_full_unstemmed | A handbook of real variables with applications to differential equations and Fourier analysis Steven G. Krantz |
| title_short | A handbook of real variables |
| title_sort | a handbook of real variables with applications to differential equations and fourier analysis |
| title_sub | with applications to differential equations and Fourier analysis |
| topic | Baire-Kategoriesatz - Taylor-Reihe - Reelle Analysis - Weierstrass-Funktionen swd Baire-Kategoriesatz - Weierstrass-Funktionen - Taylor-Reihe - Reelle Analysis swd Kernfunktion - Linearer Operator swd Linearer Operator - Kernfunktion swd Reelle Analysis - Taylor-Reihe - Baire-Kategoriesatz - Weierstrass-Funktionen swd Taylor-Reihe - Reelle Analysis - Baire-Kategoriesatz - Weierstrass-Funktionen swd Functions of real variables Mathematical analysis |
| topic_facet | Baire-Kategoriesatz - Taylor-Reihe - Reelle Analysis - Weierstrass-Funktionen Baire-Kategoriesatz - Weierstrass-Funktionen - Taylor-Reihe - Reelle Analysis Kernfunktion - Linearer Operator Linearer Operator - Kernfunktion Reelle Analysis - Taylor-Reihe - Baire-Kategoriesatz - Weierstrass-Funktionen Taylor-Reihe - Reelle Analysis - Baire-Kategoriesatz - Weierstrass-Funktionen Functions of real variables Mathematical analysis |
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