An introduction to analysis:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2017]
|
| Edition: | revised edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes index. - Die Information, dass es sich um eine 'revised edition' handelt, ist dem Vorwort entnommen; die vorherige Ausgabe ist 1993 unter gleichem Titel ebenfalls bei Wiley erschienen. |
| Physical Description: | xiv, 303 Seiten Diagramme |
| ISBN: | 9789813202610 |
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| 240 | 1 | 0 | |a Wstęp do analizy matematycznej |
| 245 | 1 | 0 | |a An introduction to analysis |c Jan Mikusiński, Piotr Mikusiński (University of Central Florida, USA) |
| 250 | |a revised edition | ||
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Record in the Search Index
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| adam_text | Contents
Preface vii
1. REAL NUMBERS 1
1.1 Addition........................................................ 1
1.2 Multiplication.................................................. 4
1.3 Distributivity................................................ 8
1.4 Inequalities................................................... 10
1.5 Bounded sets................................................... 13
1.6 Countable and uncountable sets.............................. 16
2. LIMITS OF FUNCTIONS 21
2.1 Functions...................................................... 21
2.2 Geometric interpretations...................................... 24
2.3 The absolute value............................................ 27
2.4 Limit of a function............................................ 30
2.5 Algebraic operations on limits................................. 37
2.6 Limits and inequalities........................................ 41
2.7 Limit at infinity.............................................. 43
2.8 Monotone functions............................................. 45
3. CONTINUOUS FUNCTIONS 49
3.1 Continuity of a function at a point............................ 49
3.2 Functions continuous on an interval............................ 54
3.3 Bounded functions.............................................. 58
3.4 Minimum and maximum of a function.......................... 60
3.5 Composition of functions....................................... 62
3.6 Inverse functions.............................................. 64
4. DERIVATIVES 67
4.1 Ascent and velocity............................................ 67
7
xi
xii
An Introduction to Analysis
4.2 Derivative..................................................... 69
4.3 Differentiability and continuity............................... 70
4.4 Derivative of a product........................................ 71
4.5 Derivative of a sum and a difference .......................... 73
4.6 Derivative of a quotient ...................................... 74
4.7 Derivative of composition of functions......................... 75
4.8 Derivative of the inverse function............................. 77
4.9 Differentials.................................................. 78
4.10 Minimum and maximum of a function revisited.................... 79
4.11 The Mean Value Theorem......................................... 82
4.12 Derivatives of monotone functions.............................. 84
5. SEQUENCES AND SERIES 87
5.1 Convergent sequences . ........................................ 87
5.2 Monotone sequences and subsequences............................ 90
5.3 Geometric sequence and series.................................. 93
5.4 Convergent series.............................................. 95
5.5 Absolutely convergent series................................... 99
5.6 Rearrangements and partitions..................................104
5.7 Decimal expansions.............................................110
6. SEQUENCES AND SERIES OF FUNCTIONS 115
6.1 Power series...................................................115
6.2 Sequences of functions.........................................118
6.3 Uniform convergence of power series............................122
6.4 Differentiation of sequences of functions......................123
7. ELEMENTARY FUNCTIONS 127
7.1 The exponential function ex....................................127
7.2 General exponential functions..................................132
7.3 The number ....................................................134
7.4 Natural logarithm..............................................136
7.5 Arbitrary logarithms...........................................138
7.6 Derivatives of power functions.................................139
7.7 Natural logarithms of natural numbers..........................141
7.8 Common logarithms..............................................144
7.9 Sine and cosine.............................................. 146
7.10 Tangent........................................................151
7.11 Arctangent ....................................................153
7.12 Machines formula...............................................155
7.13 Newton’s binomial..............................................156
7.14 Arccosine and arcsine........................................ 160
Contents
xiii
8. LIMITS REVISITED 163
8.1 Improper limits...............................................163
8.2 The easy PHospitaTs theorem....................................164
8.3 The difficult PHospitaPs theorem..............................166
8.4 Some applications of PHospitaPs rule..........................169
8.5 A version of PHospitaPs rule for sequences.....................171
8.6 D’Alembert’s test............................................ 173
8.7 Theorem of Abel................................................175
9. ANTIDERIVATIVES 179
9.1 Antiderivatives................................................179
9.2 Integration by substitution....................................182
9.3 Integration by parts ..........................................184
9.4 Integration of rational functions..............................185
9.5 Integration of some irrational expressions ....................190
9.6 Integration of trigonometric functions ........................193
10. THE LEBESGUE INTEGRAL 197
10.1 Introduction................................................. 197
10.2 Integrable functions...........................................198
10.3 Examples of expansions of integrable functions ................200
10.4 Step functions.................................................202
10.5 Basic properties of the integral...............................209
10.6 The absolute value of an integrable function...................214
10.7 Series of integrable functions.................................217
10.8 Null functions and null sets...................................219
10.9 Norm convergence ..............................................221
10.10 Convergence almost everywhere..................................224
10.11 Theorems of Riesz .............................................227
10.12 The Monotone Convergence Theorem and the Dominated
Convergence Theorem.............................................229
10.13 Integral over an interval......................................231
11. MISCELLANEOUS TOPICS 237
11.1 Integration and differentiation................................237
11.2 The change of variables theorem ...............................243
11.3 Cauchy’s formula...............................................246
11.4 Taylor’s formula...............................................248
11.5 The integral test for convergence of infinite series...........250
11.6 Examples of trigonometric series ..............................251
11.7 General Fourier series ........................................255
11.8 Euler’s constant...............................................257
XIV
An Introduction to Analysis
11.9 Iterated integrals...............................................259
11.10 Euler’s gamma and beta functions ...............................266
11.11 The Riemann integral ...........................................268
12. CONSTRUCTION OF REAL NUMBERS 273
12.1 Introduction.....................................................273
12.2 Axioms of natural numbers........................................274
12.3 Some simple consequences of Axioms I-HI..........................276
12.4 The Infinity Principle...........................................278
12.5 Addition in 91...................................................279
12.6 Multiplication in 91.............................................282
12.7 Addition table and multiplication table..........................284
12.8 Equivalence classes..............................................289
12.9 Defining integers by natural numbers.............................290
12.10 Defining rational numbers by integers ..........................292
12.11 Defining real numbers by rational numbers.......................294
Index
301
|
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| author | Mikusiński, Jan 1913-1987 Mikusiński, Piotr |
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| discipline | Mathematik |
| edition | revised edition |
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| isbn | 9789813202610 |
| language | English |
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| spelling | Mikusiński, Jan 1913-1987 (DE-588)1035621681 aut Wstęp do analizy matematycznej An introduction to analysis Jan Mikusiński, Piotr Mikusiński (University of Central Florida, USA) revised edition New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2017] © 2017 xiv, 303 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Includes index. - Die Information, dass es sich um eine 'revised edition' handelt, ist dem Vorwort entnommen; die vorherige Ausgabe ist 1993 unter gleichem Titel ebenfalls bei Wiley erschienen. Mathematical analysis Calculus Functions of real variables Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Mikusiński, Piotr (DE-588)12359944X aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029637467&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mikusiński, Jan 1913-1987 Mikusiński, Piotr An introduction to analysis Mathematical analysis Calculus Functions of real variables Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4151278-9 |
| title | An introduction to analysis |
| title_alt | Wstęp do analizy matematycznej |
| title_auth | An introduction to analysis |
| title_exact_search | An introduction to analysis |
| title_full | An introduction to analysis Jan Mikusiński, Piotr Mikusiński (University of Central Florida, USA) |
| title_fullStr | An introduction to analysis Jan Mikusiński, Piotr Mikusiński (University of Central Florida, USA) |
| title_full_unstemmed | An introduction to analysis Jan Mikusiński, Piotr Mikusiński (University of Central Florida, USA) |
| title_short | An introduction to analysis |
| title_sort | an introduction to analysis |
| topic | Mathematical analysis Calculus Functions of real variables Analysis (DE-588)4001865-9 gnd |
| topic_facet | Mathematical analysis Calculus Functions of real variables Analysis Einführung |
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