A concrete approach to classical analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Series: | CMS books in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 433 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 0387789324 9780387789323 |
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MARC
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| 020 | |a 9780387789323 |c Gb. : ca. EUR 57.67 (freier Pr.), ca. sfr 94.00 (freier Pr.) |9 978-0-387-78932-3 | ||
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| 100 | 1 | |a Mureşan, Marian |d ca. 20./21. Jh. |e Verfasser |0 (DE-588)1127871331 |4 aut | |
| 245 | 1 | 0 | |a A concrete approach to classical analysis |c Marian Mureşan |
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XVIII, 433 S. |b graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a CMS books in mathematics | |
| 650 | 4 | |a Calculus, Integral | |
| 650 | 4 | |a Differential calculus | |
| 650 | 4 | |a Mathematical analysis | |
| 650 | 0 | 7 | |a Differentialrechnung |0 (DE-588)4012252-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Analysis |0 (DE-588)4001865-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Integralrechnung |0 (DE-588)4027232-1 |2 gnd |9 rswk-swf |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-387-78933-0 |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754134&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016754134 | ||
Record in the Search Index
| _version_ | 1804138040608686080 |
|---|---|
| adam_text | Contents
Sets and Numbers
......................................... 1
1.1
Sets
................................................... 1
1.1.1
The concept of a set
............................... 1
1.1.2
Operations on sets
................................. 3
1.1.3
Relations and functions
............................ 6
1.2
Sets of numbers
......................................... 11
1.2.1
Two examples
.................................... 11
1.2.2
The real number system
............................ 12
1.2.3
Elements of algebra
................................ 20
1.2.4
Elements of topology on
Ж
......................... 25
1.2.5
The extended real number system
................... 30
1.2.6
The complex number system
........................ 31
1.3
Exercises
............................................... 32
1.4
References and comments
................................ 40
Vector Spaces and Metric Spaces
.......................... 41
2.1
Vector spaces
........................................... 41
2.1.1
Finite-dimensional vector spaces
..................... 41
2.1.2
Vector spaces
..................................... 44
2.1.3
Normed spaces
.................................... 48
2.1.4
Hubert spaces
..................................... 49
2.1.5
Inequalities
....................................... 51
2.2
Metric spaces
........................................... 57
2.3
Compact spaces
......................................... 65
2.4
Exercises
............................................... 70
2.5
References and comments
................................ 70
Sequences and Series
...................................... 73
3.1
Numerical sequences
..................................... 73
3.1.1
Convergent sequences
.............................. 73
3.1.2
Subsequences
..................................... 76
Contents
3.1.3
Cauchy sequences
................................. 77
3.1.4 Monotonie
sequences
............................... 80
3.1.5
Upper limits and lower limits
....................... 81
3.1.6
The big Oh and small oh notations
.................. 82
3.1.7
Stolz-Cesaro theorem and some of its consequences
.... 85
3.1.8
Certain combinatorial numbers
...................... 88
3.1.9
Unimodal, log-convex, and
Pólya-frequency
sequences
. . 94
3.1.10
Some special sequences
............................. 98
3.2
Sequences of functions
...................................108
3.3
Numerical series
.........................................110
3.3.1
Series of
nonnegative
terms
.........................113
3.3.2
The root and the ratio tests
........................123
3.3.3
Partial summation
.................................125
3.3.4
Absolutely and conditionally convergent series
........126
3.3.5
The W-Z method
................................130
3.4
Series of functions
.......................................134
3.4.1
Power series
......................................137
3.4.2
Hyp
er
geometric series
..............................138
3.5
The Riemann
Zeta
function
ζ(ρ)
..........................139
3.6
Exercises
...............................................139
3.7
References and comments
................................144
Limits and Continuity
.....................................147
4.1
Limits
.................................................147
4.1.1
The limit of a function
.............................147
4.1.2
Right-hand side and left-hand side limits
.............152
4.2
Continuity
..............................................152
4.2.1
Continuity and compactness
........................155
4.2.2
Uniform continuous mappings
.......................156
4.2.3
Continuity and connectedness
.......................160
4.2.4
Discontinuities
....................................161
4.2.5
Monotonie
functions
...............................161
4.3
Periodic functions
.......................................164
4.4
Darboux functions
.......................................165
4.5
Lipschitz functions
.......................................167
4.6
Convex functions
........................................169
4.6.1
Convex functions
..................................169
4.6.2
Jensen convex functions
............................173
4.7
Functions of bounded variations
...........................177
4.8
Continuity of sequences of functions
.......................183
4.9
Continuity of series of functions
...........................184
4.10
Exercises
...............................................186
4.11
References and comments
................................189
Contents xi
Differential
Calculus on
К
.................................191
5.1
The derivative of a real function
...........................191
5.2
Mean value theorems
....................................197
5.2.1
Consequences of the mean value theorems
............201
5.3
The continuity and the surjectivity of derivatives
............206
5.4
L Hospital theorem
......................................207
5.5
Higher-order derivatives and the Taylor formula
.............208
5.6
Convex functions and differentiability
......................214
5.6.1
Inequalities
.......................................216
5.7
Differentiability of sequences and series of functions
..........217
5.8
Power series and Taylor series
.............................219
5.8.1
Operations with power series
........................222
5.8.2
The Taylor expansion of some elementary functions
.... 225
5.8.3
Bernoulli numbers and polynomials
..................228
5.9
Some elementary functions introduced by recurrences
........230
5.9.1
The square root function
...........................231
5.9.2
The logarithm function
.............................231
5.9.3
The exponential function
...........................235
5.9.4
The arctangent function
............................237
5.10
Functions with primitives
.................................239
5.10.1
The concept of a primitive function
..................239
5.10.2
The existence of primitives for continuous functions
.... 242
5.10.3
Operations with functions with primitives
............244
5.11
Exercises
...............................................247
5.12
References and comments
................................249
Integral Calculus on
E
....................................251
6.1
The Darboux-Stieltjes integral
............................251
6.1.1
The Darboux integral
..............................251
6.1.2
The Darboux-Stieltjes integral
......................252
6.2
Integrability of sequences and series of functions
.............262
6.3
Improper integrals
.......................................263
6.4
Euler
integrals
..........................................271
6.4.1
Gamma function
..................................271
6.4.2
Beta function
.....................................275
6.5
Polylogarithms
..........................................278
6.6
e
and
π
are transcendental
...............................280
6.7
The
Grönwall
inequality
..................................283
6.8
Exercises
...............................................284
6.9
References and comments
................................287
Contents
Differential Calculus on
Ж
................................289
7.1
Linear and bounded mappings
............................289
7.1.1
Multilinear mappings
..............................293
7.1.2
Quadratic mappings
...............................294
7.2
Differentiable functions
...................................296
7.2.1
Variations
........................................296
7.2.2
Gâteaux
differential
...............................297
7.2.3
Fréchet
differential
................................298
7.2.4
Properties of the
Fréchet
differentiable functions
......300
7.3
Partial derivatives
.......................................304
7.3.1
The inverse function theorem and the implicit function
theorem
..........................................307
7.3.2
Directional derivatives and gradients
.................312
7.4
Higher-order differentials and partial derivatives
.............312
7.4.1
The case X
=
Жп
.................................314
7.5
Taylor formula
..........................................316
7.6
Problems of local extremes
...............................317
7.6.1
First-order conditions
..............................317
7.6.2
Second-order conditions
............................318
7.6.3
Constraint local extremes
..........................319
7.7
Exercises
...............................................322
7.8
References and comments
................................322
Double Integrals, Triple Integrals, and Line Integrals
......325
8.1
Double integrals
.........................................325
8.1.1
Double integrals on rectangles
.......................325
8.1.2
Double integrals on simple domains
..................331
8.2
Triple integrals
..........................................333
8.2.1
Triple integrals on parallelepipeds
...................333
8.2.2
Triple integrals on simple domains
...................340
8.3
и
-fold integrals
.........................................341
8.3.1
Ti-fold integrals on
hyperreet
angles
..................341
8.3.2
n-fold integrals on simple domains
..................345
8.4
Line integrals
...........................................346
8.4.1
Line integrals with respect to arc length
..............346
8.4.2
Line integrals with respect to axis
...................347
8.4.3
Green formula
....................................347
8.5
Integrals depending on parameters
.........................349
8.6
Exercises
...............................................353
8.7
References and comments
................................354
Contents xiii
9
Constants
.................................................355
9.1 Pythagoras s
constant
....................................355
9.1.1
Sequences approaching
л/2 .........................
355
9.2
Archimedes constant
....................................356
9.2.1
Recurrence relation
................................356
9.2.2
Buffon needle problem
.............................358
9.3
Arithmetic-geometric mean
...............................358
9.4
BBP formulas
...........................................363
9.4.1
Computing the nth binary or hexadecimal digit of
π
. . 363
9.4.2
BBP formulas by binomial sums
.....................368
9.5
Ramanujan formulas
.....................................372
9.6
Several natural ways to introduce number
e
................374
9.7
Optimal stopping problem
................................377
9.8
References and comments
................................378
10
Asymptotic and Combinatorial Estimates
..................381
10.1
Asymptotic estimates
....................................381
10.2
Algorithm analysis
.......................................384
10.3
Combinatorial estimates
..................................390
10.3.1
Counting relations, topologies, and partial orders
......394
10.3.2
Generalized Fubini numbers
........................396
10.3.3
The Catalan numbers and binary trees
...............401
10.4
References and comments
................................409
References
.....................................................411
List of Symbols
................................................419
Author Index
..................................................423
Subject Index
.................................................425
xvi
List of Figures
10.3
Monotonie
paths of length 2n through an n-by-n grid
........404
10.4
Binary trees
.............................................405
|
| adam_txt |
Contents
Sets and Numbers
. 1
1.1
Sets
. 1
1.1.1
The concept of a set
. 1
1.1.2
Operations on sets
. 3
1.1.3
Relations and functions
. 6
1.2
Sets of numbers
. 11
1.2.1
Two examples
. 11
1.2.2
The real number system
. 12
1.2.3
Elements of algebra
. 20
1.2.4
Elements of topology on
Ж
. 25
1.2.5
The extended real number system
. 30
1.2.6
The complex number system
. 31
1.3
Exercises
. 32
1.4
References and comments
. 40
Vector Spaces and Metric Spaces
. 41
2.1
Vector spaces
. 41
2.1.1
Finite-dimensional vector spaces
. 41
2.1.2
Vector spaces
. 44
2.1.3
Normed spaces
. 48
2.1.4
Hubert spaces
. 49
2.1.5
Inequalities
. 51
2.2
Metric spaces
. 57
2.3
Compact spaces
. 65
2.4
Exercises
. 70
2.5
References and comments
. 70
Sequences and Series
. 73
3.1
Numerical sequences
. 73
3.1.1
Convergent sequences
. 73
3.1.2
Subsequences
. 76
Contents
3.1.3
Cauchy sequences
. 77
3.1.4 Monotonie
sequences
. 80
3.1.5
Upper limits and lower limits
. 81
3.1.6
The big Oh and small oh notations
. 82
3.1.7
Stolz-Cesaro theorem and some of its consequences
. 85
3.1.8
Certain combinatorial numbers
. 88
3.1.9
Unimodal, log-convex, and
Pólya-frequency
sequences
. . 94
3.1.10
Some special sequences
. 98
3.2
Sequences of functions
.108
3.3
Numerical series
.110
3.3.1
Series of
nonnegative
terms
.113
3.3.2
The root and the ratio tests
.123
3.3.3
Partial summation
.125
3.3.4
Absolutely and conditionally convergent series
.126
3.3.5
The W-Z method
.130
3.4
Series of functions
.134
3.4.1
Power series
.137
3.4.2
Hyp
er
geometric series
.138
3.5
The Riemann
Zeta
function
ζ(ρ)
.139
3.6
Exercises
.139
3.7
References and comments
.144
Limits and Continuity
.147
4.1
Limits
.147
4.1.1
The limit of a function
.147
4.1.2
Right-hand side and left-hand side limits
.152
4.2
Continuity
.152
4.2.1
Continuity and compactness
.155
4.2.2
Uniform continuous mappings
.156
4.2.3
Continuity and connectedness
.160
4.2.4
Discontinuities
.161
4.2.5
Monotonie
functions
.161
4.3
Periodic functions
.164
4.4
Darboux functions
.165
4.5
Lipschitz functions
.167
4.6
Convex functions
.169
4.6.1
Convex functions
.169
4.6.2
Jensen convex functions
.173
4.7
Functions of bounded variations
.177
4.8
Continuity of sequences of functions
.183
4.9
Continuity of series of functions
.184
4.10
Exercises
.186
4.11
References and comments
.189
Contents xi
Differential
Calculus on
К
.191
5.1
The derivative of a real function
.191
5.2
Mean value theorems
.197
5.2.1
Consequences of the mean value theorems
.201
5.3
The continuity and the surjectivity of derivatives
.206
5.4
L'Hospital theorem
.207
5.5
Higher-order derivatives and the Taylor formula
.208
5.6
Convex functions and differentiability
.214
5.6.1
Inequalities
.216
5.7
Differentiability of sequences and series of functions
.217
5.8
Power series and Taylor series
.219
5.8.1
Operations with power series
.222
5.8.2
The Taylor expansion of some elementary functions
. 225
5.8.3
Bernoulli numbers and polynomials
.228
5.9
Some elementary functions introduced by recurrences
.230
5.9.1
The square root function
.231
5.9.2
The logarithm function
.231
5.9.3
The exponential function
.235
5.9.4
The arctangent function
.237
5.10
Functions with primitives
.239
5.10.1
The concept of a primitive function
.239
5.10.2
The existence of primitives for continuous functions
. 242
5.10.3
Operations with functions with primitives
.244
5.11
Exercises
.247
5.12
References and comments
.249
Integral Calculus on
E
.251
6.1
The Darboux-Stieltjes integral
.251
6.1.1
The Darboux integral
.251
6.1.2
The Darboux-Stieltjes integral
.252
6.2
Integrability of sequences and series of functions
.262
6.3
Improper integrals
.263
6.4
Euler
integrals
.271
6.4.1
Gamma function
.271
6.4.2
Beta function
.275
6.5
Polylogarithms
.278
6.6
e
and
π
are transcendental
.280
6.7
The
Grönwall
inequality
.283
6.8
Exercises
.284
6.9
References and comments
.287
Contents
Differential Calculus on
Ж"
.289
7.1
Linear and bounded mappings
.289
7.1.1
Multilinear mappings
.293
7.1.2
Quadratic mappings
.294
7.2
Differentiable functions
.296
7.2.1
Variations
.296
7.2.2
Gâteaux
differential
.297
7.2.3
Fréchet
differential
.298
7.2.4
Properties of the
Fréchet
differentiable functions
.300
7.3
Partial derivatives
.304
7.3.1
The inverse function theorem and the implicit function
theorem
.307
7.3.2
Directional derivatives and gradients
.312
7.4
Higher-order differentials and partial derivatives
.312
7.4.1
The case X
=
Жп
.314
7.5
Taylor formula
.316
7.6
Problems of local extremes
.317
7.6.1
First-order conditions
.317
7.6.2
Second-order conditions
.318
7.6.3
Constraint local extremes
.319
7.7
Exercises
.322
7.8
References and comments
.322
Double Integrals, Triple Integrals, and Line Integrals
.325
8.1
Double integrals
.325
8.1.1
Double integrals on rectangles
.325
8.1.2
Double integrals on simple domains
.331
8.2
Triple integrals
.333
8.2.1
Triple integrals on parallelepipeds
.333
8.2.2
Triple integrals on simple domains
.340
8.3
и
-fold integrals
.341
8.3.1
Ti-fold integrals on
hyperreet
angles
.341
8.3.2
n-fold integrals on simple domains
.345
8.4
Line integrals
.346
8.4.1
Line integrals with respect to arc length
.346
8.4.2
Line integrals with respect to axis
.347
8.4.3
Green formula
.347
8.5
Integrals depending on parameters
.349
8.6
Exercises
.353
8.7
References and comments
.354
Contents xiii
9
Constants
.355
9.1 Pythagoras's
constant
.355
9.1.1
Sequences approaching
л/2 .
355
9.2
Archimedes' constant
.356
9.2.1
Recurrence relation
.356
9.2.2
Buffon needle problem
.358
9.3
Arithmetic-geometric mean
.358
9.4
BBP formulas
.363
9.4.1
Computing the nth binary or hexadecimal digit of
π
. . 363
9.4.2
BBP formulas by binomial sums
.368
9.5
Ramanujan formulas
.372
9.6
Several natural ways to introduce number
e
.374
9.7
Optimal stopping problem
.377
9.8
References and comments
.378
10
Asymptotic and Combinatorial Estimates
.381
10.1
Asymptotic estimates
.381
10.2
Algorithm analysis
.384
10.3
Combinatorial estimates
.390
10.3.1
Counting relations, topologies, and partial orders
.394
10.3.2
Generalized Fubini numbers
.396
10.3.3
The Catalan numbers and binary trees
.401
10.4
References and comments
.409
References
.411
List of Symbols
.419
Author Index
.423
Subject Index
.425
xvi
List of Figures
10.3
Monotonie
paths of length 2n through an n-by-n grid
.404
10.4
Binary trees
.405 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Mureşan, Marian ca. 20./21. Jh |
| author_GND | (DE-588)1127871331 |
| author_facet | Mureşan, Marian ca. 20./21. Jh |
| author_role | aut |
| author_sort | Mureşan, Marian ca. 20./21. Jh |
| author_variant | m m mm |
| building | Verbundindex |
| bvnumber | BV035085953 |
| callnumber-first | Q - Science |
| callnumber-label | QA300 |
| callnumber-raw | QA300 |
| callnumber-search | QA300 |
| callnumber-sort | QA 3300 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 400 |
| ctrlnum | (OCoLC)233933476 (DE-599)DNB987816055 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035085953 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:08:55Z |
| indexdate | 2024-07-09T21:21:52Z |
| institution | BVB |
| isbn | 0387789324 9780387789323 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016754134 |
| oclc_num | 233933476 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-355 DE-BY-UBR DE-706 DE-11 DE-188 DE-824 |
| owner_facet | DE-703 DE-20 DE-355 DE-BY-UBR DE-706 DE-11 DE-188 DE-824 |
| physical | XVIII, 433 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series2 | CMS books in mathematics |
| spelling | Mureşan, Marian ca. 20./21. Jh. Verfasser (DE-588)1127871331 aut A concrete approach to classical analysis Marian Mureşan New York, NY Springer 2009 XVIII, 433 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier CMS books in mathematics Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Integralrechnung (DE-588)4027232-1 gnd rswk-swf Differentialrechnung (DE-588)4012252-9 s Integralrechnung (DE-588)4027232-1 s DE-604 Analysis (DE-588)4001865-9 s Erscheint auch als Online-Ausgabe 978-0-387-78933-0 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754134&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mureşan, Marian ca. 20./21. Jh A concrete approach to classical analysis Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd Analysis (DE-588)4001865-9 gnd Integralrechnung (DE-588)4027232-1 gnd |
| subject_GND | (DE-588)4012252-9 (DE-588)4001865-9 (DE-588)4027232-1 |
| title | A concrete approach to classical analysis |
| title_auth | A concrete approach to classical analysis |
| title_exact_search | A concrete approach to classical analysis |
| title_exact_search_txtP | A concrete approach to classical analysis |
| title_full | A concrete approach to classical analysis Marian Mureşan |
| title_fullStr | A concrete approach to classical analysis Marian Mureşan |
| title_full_unstemmed | A concrete approach to classical analysis Marian Mureşan |
| title_short | A concrete approach to classical analysis |
| title_sort | a concrete approach to classical analysis |
| topic | Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung (DE-588)4012252-9 gnd Analysis (DE-588)4001865-9 gnd Integralrechnung (DE-588)4027232-1 gnd |
| topic_facet | Calculus, Integral Differential calculus Mathematical analysis Differentialrechnung Analysis Integralrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754134&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT muresanmarian aconcreteapproachtoclassicalanalysis |