Classical topics in complex function theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
New York, NY [u.a.]
Springer
1998
|
| Series: | Graduate Texts in Mathematics
172 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 349 Seiten Ill., graph. Darst. |
| ISBN: | 0387982213 |
Staff View
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| 245 | 1 | 0 | |a Classical topics in complex function theory |c Reinhold Remmert |
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Record in the Search Index
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| adam_text | Contents
Preface to the Second German Edition vii
Preface to the First German Edition vii
Acknowledgments x
Advice to the reader x
A Infinite Products and Partial Fraction Series 1
1 Infinite Products of Holomorphic Functions 3
§1. Infinite Products 4
1. Infinite products of numbers 4
2. Infinite products of functions 6
§2. Normal Convergence 7
1. Normal convergence 7
2. Normally convergent products of holomorphic
functions 9
3. Logarithmic differentiation 10
§3. The Sine Product sinivz = irzH^il z2/v2) 12
1. Standard proof 12
2. Characterization of the sine
by the duplication formula 14
3. Proof of Euler s formula using Lemma 2 15
4*. Proof of the duplication formula for Euler s product,
following Eisenstein 16
5. On the history of the sine product 17
xii Contents
§4*. Euler Partition Products 18
1. Partitions of natural numbers and Euler products . 19
2. Pentagonal number theorem. Recursion formulas for
p(n) and r(n) 20
3. Series expansion of lXli(l + ? z) m Powers of 2 . . 22
4. On the history of partitions and the pentagonal
number theorem 24
§5*. Jacobi s Product Representation of the Series
¦/(*,«) ^E^ oXV 25
1. Jacobi s theorem 25
2. Discussion of Jacobi s theorem 26
3. On the history of Jacobi s identity 28
Bibliography 30
2 The Gamma Function 33
§1. The Weierstrass Function A(z) = zelzX[v^1{l +z/v)e zl / 36
1. The auxiliary function
#(*):= *n~=i(l + */ )e */1 36
2. The entire function A(z) := e zH(z) 37
§2. The Gamma Function 39
1. Properties of the F function 39
2. Historical notes 41
3. The logarithmic derivative 42
4. The uniqueness problem 43
5. Multiplication formulas 45
6*. Holder s theorem 46
7*. The logarithm of the F function 47
§3. Euler s and Hankel s Integral Representations of T(z) .... 49
1. Convergence of Euler s integral 49
2. Euler s theorem 51
3*. The equation 52
4*. Hankel s loop integral 53
§4. Stirling s Formula and Gudermann s Series 55
1. Stieltjes s definition of the function /j,(z) 56
2. Stirling s formula 58
3. Growth of T(x + iy) for y +oo 59
4*. Gudermann s series 60
5*. Stirling s series 61
6*. Delicate estimates for the remainder term 63
7*. Binet s integral 64
8*. Lindelof s estimate 66
§5. The Beta Function 67
1. Proof of Euler s identity 68
2. Classical proofs of Euler s identity 69
Bibliography 70
Contents xiii
3 Entire Functions with Prescribed Zeros 73
§1. The Weierstrass Product Theorem for C 74
1. Divisors and principal divisors 74
2. Weierstrass products 75
3. Weierstrass factors 76
4. The Weierstrass product theorem 77
5. Consequences 78
6. On the history of the product theorem 79
§2. Discussion of the Product Theorem 80
1. Canonical products 81
2. Three classical canonical products 82
3. The (7 function 83
4. The p function 85
5*. An observation of Hurwitz 85
Bibliography 86
4* Holomorphic Functions with Prescribed Zeros 89
§1. The Product Theorem for Arbitrary Regions 89
1. Convergence lemma 90
2. The product theorem for special divisors 91
3. The general product theorem 92
4. Second proof of the general product theorem .... 92
5. Consequences 93
§2. Applications and Examples 94
1. Divisibility in the ring O(G). Greatest common
divisors 94
2. Examples of Weierstrass products 96
3. On the history of the general product theorem ... 97
4. Glimpses of several variables 97
§3. Bounded Functions on E and Their Divisors 99
1. Generalization of Schwarz s lemma 99
2. Necessity of the Blaschke condition 100
3. Blaschke products 100
4. Bounded functions on the right half plane 102
Appendix to Section 3: Jensen s Formula 102
Bibliography 104
5 Iss sa s Theorem. Domains of Holomorphy 107
§1. Iss sa s Theorem 107
1. Bers s theorem 108
2. Iss sa s theorem 109
3. Proof of the lemma 109
4. Historical remarks on the theorems of
Bers and Iss sa 110
5*. Determination of all the valuations on M{G) .... Ill
xiv Contents
§2. Domains of Holomorphy 112
1. A construction of Goursat 113
2. Well distributed boundary sets. First proof of the
existence theorem 115
3. Discussion of the concept of domains of holomorphy 116
4. Peripheral sets. Second proof of the existence
theorem 118
5. On the history of the concept of domains of
holomorphy 119
6. Glimpse of several variables 120
§3. Simple Examples of Domains of Holomorphy 120
1. Examples for E 120
2. Lifting theorem 122
3. Cassini regions and domains of holomorphy 122
Bibliography 123
6 Functions with Prescribed Principal Parts 125
§1. Mittag Leffler s Theorem for C 126
1. Principal part distributions 126
2. Mittag Leffier series 127
3. Mittag Leffler s theorem 128
4. Consequences 128
5. Canonical Mittag Leffier series. Examples 129
6. On the history of Mittag Leffler s theorem for C . . 130
§2. Mittag Leffler s Theorem for Arbitrary Regions 131
1. Special principal part distributions 131
2. Mittag Leffler s general theorem 132
3. Consequences 133
4. On the history of Mittag Leffler s general theorem . 134
5. Glimpses of several variables 135
§3*. Ideal Theory in Rings of Holomorphic Functions 135
1. Ideals in O(G) that are not finitely generated .... 136
2. Wedderburn s lemma (representation of 1) 136
3. Linear representation of the gcd. Principal ideal
theorem 138
4. Nonvanishing ideals 138
5. Main theorem of the ideal theory of O(G) 139
6. On the history of the ideal theory of holomorphic
functions 140
7. Glimpses of several variables 141
Bibliography 142
Contents xv
B Mapping Theory 145
7 The Theorems of Montel and Vitali 147
§1. Montel s Theorem 148
1. Montel s theorem for sequences 148
2. Proof of Montel s theorem 150
3. Montel s convergence criterion 150
4. Vitali s theorem 150
5*. Pointwise convergent sequences of holomorphic
functions 151
§2. Normal Families 152
1. Montel s theorem for normal families 152
2. Discussion of Montel s theorem 153
3. On the history of Montel s theorem 154
4*. Square integrable functions and normal families . . . 154
§3*. Vitali s Theorem 156
1. Convergence lemma 156
2. Vitali s theorem (final version) 157
3. On the history of Vitali s theorem 158
§4*. Applications of Vitali s theorem 159
1. Interchanging integration and differentiation .... 159
2. Compact convergence of the F integral 160
3. Miintz s theorem 161
§5. Consequences of a Theorem of Hurwitz 162
Bibliography 164
8 The Riemann Mapping Theorem 167
§1. Integral Theorems for Homotopic Paths 168
1. Fixed endpoint homotopic paths 168
2. Freely homotopic closed paths 169
3. Null homotopy and null homology 170
4. Simply connected domains 171
5*. Reduction of the integral theorem 1 to a lemma ... 172
6*. Proof of Lemma 5* 174
§2. The Riemann Mapping Theorem 175
1. Reduction to Q domains 175
2. Existence of holomorphic injections 177
3. Existence of expansions 177
4. Existence proof by means of an extremal principle . 178
5. On the uniqueness of the mapping function 179
6. Equivalence theorem 180
§3. On the History of the Riemann Mapping Theorem 181
1. Riemann s dissertation 181
2. Early history 183
3. From Caratheodory Koebe to Fejer Riesz 184
xvi Contents
4. Caratheodory s final proof 184
5. Historical remarks on uniqueness and boundary
behavior 186
6. Glimpses of several variables 187
§4. Isotropy Groups of Simply Connected Domains 188
1. Examples 188
2. The group AutaG for simply connected domains
G ^ C 189
3*. Mapping radius. Monotonicity theorem 189
Appendix to Chapter 8: Caratheodory Koebe Theory 191
§1. Simple Properties of Expansions 191
1. Expansion lemma 191
2. Admissible expansions. The square root method . . 192
3*. The crescent expansion 193
§2. The Caratheodory Koebe Algorithm 194
1. Properties of expansion sequences 194
2. Convergence theorem 195
3. Koebe families and Koebe sequences 196
4. Summary. Quality of convergence 197
5. Historical remarks. The competition between
Caratheodory and Koebe 197
§3. The Koebe Families fCm and Kx 198
1. A lemma 198
2. The families /Cm and K.^ 199
Bibliography for Chapter 8 and the Appendix 201
9 Automorphisms and Finite Inner Maps 203
§1. Inner Maps and Automorphisms 203
1. Convergent sequences in Hoi G and Aut G 204
2. Convergence theorem for sequences of
automorphisms 204
3. Bounded homogeneous domains 205
4*. Inner maps of M and homotheties 206
§2. Iteration of Inner Maps 206
1. Elementary properties 207
2. H. Cartan s theorem 207
3. The group AutaG for bounded domains 208
4. The closed subgroups of the circle group 209
5*. Automorphisms of domains with holes. Annulus
theorem 210
§3. Finite Holomorphic Maps 211
1. Three general properties 212
2. Finite inner maps of E 212
3. Boundary lemma for annuli 213
Contents xvii
4. Finite inner maps of annuli 215
5. Determination of all the finite maps
between annuli 216
§4*. Rado s Theorem. Mapping Degree 217
1. Closed maps. Equivalence theorem 217
2. Winding maps 218
3. Rado s theorem 219
4. Mapping degree 220
5. Glimpses 221
Bibliography 221
C Selecta 223
10 The Theorems of Bloch, Picard, and Schottky 225
§1. Bloch s Theorem 226
1. Preparation for the proof 226
2. Proof of Bloch s theorem 227
3*. Improvement of the bound by the solution of an
extremal problem 228
4*. Ahlfors s theorem 230
5*. Landau s universal constant 232
§2. Picard s Little Theorem 233
1. Representation of functions that omit two values . . 233
2. Proof of Picard s little theorem 234
3. Two amusing applications 235
§3. Schottky s Theorem and Consequences 236
1. Proof of Schottky s theorem 237
2. Landau s sharpened form of
Picard s little theorem 238
3. Sharpened forms of Montel s
and Vitali s theorems 239
§4. Picard s Great Theorem 240
1. Proof of Picard s great theorem 240
2. On the history of the theorems of this chapter . . . 240
Bibliography 241
11 Boundary Behavior of Power Series 243
§1. Convergence on the Boundary 244
1. Theorems of Fatou, M. Riesz, and Ostrowski .... 244
2. A lemma of M. Riesz 245
3. Proof of the theorems in 1 247
4. A criterion for nonextendibility 248
Bibliography for Section 1 249
§2. Theory of Overconvergence. Gap Theorem 249
xviii Contents
1. Overconvergent power series 249
2. Ostrowski s overconvergence theorem 250
3. Hadamard s gap theorem 252
4. Porter s construction of overconvergent series .... 253
5. On the history of the gap theorem 254
6. On the history of overconvergence 255
7. Glimpses 255
Bibliography for Section 2 256
§3. A Theorem of Fatou Hurwitz Polya 257
1. Hurwitz s proof 258
2. Glimpses 259
Bibliography for Section 3 259
§4. An Extension Theorem of Szego 260
1. Preliminaries for the proof of (Sz) 260
2. A lemma 262
3. Proof of (Sz) 263
4. An application 263
5. Glimpses 265
Bibliography for Section 4 266
12 Runge Theory for Compact Sets 267
§1. Techniques 268
1. Cauchy integral formula for compact sets 269
2. Approximation by rational functions 271
3. Pole shifting theorem 272
§2. Runge Theory for Compact Sets 273
1. Runge s approximation theorem 273
2. Consequences of Runge s little theorem 275
3. Main theorem of Runge theory for compact sets . . 276
§3. Applications of Runge s Little Theorem 278
1. Pointwise convergent sequences of polynomials that
do not converge compactly everywhere 278
2. Holomorphic imbedding of the unit disc in C3 .... 281
§4. Discussion of the Cauchy Integral Formula for
Compact Sets 283
1. Final form of Theorem 1.1 284
2. Circuit theorem 285
Bibliography 287
13 Runge Theory for Regions 289
§1. Runge s Theorem for Regions 290
1. Filling in compact sets. Runge s proof of
Mittag Leffler s theorem 291
2. Runge s approximation theorem 292
3. Main theorem of Cauchy function theory 292
Contents xix
4. On the theory of holes 293
5. On the history of Runge theory 294
§2. Runge Pairs 295
1. Topological characterization of Runge pairs 295
2. Runge hulls 296
3. Homological characterization of Runge hulls.
The Behnke Stein theorem 297
4. Runge regions 298
5*. Approximation and holomorphic extendibility .... 299
§3. Holomorphically Convex Hulls and Runge Pairs 300
1. Properties of the hull operator 300
2. Characterization of Runge pairs by means of
holomorphically convex hulls 302
Appendix: On the Components of Locally Compact Spaces. Sura
Bura s Theorem 303
1. Components 303
2. Existence of open compact sets 304
3. Filling in holes 305
4. Proof of Sura Bura s theorem 305
Bibliography 306
14 Invariance of the Number of Holes 309
§1. Homology Theory. Separation Lemma 309
1. Homology groups. The Betti number 310
2. Induced homomorphisms. Natural properties .... 311
3. Separation of holes by closed paths 312
§2. Invariance of the Number of Holes. Product Theorem
for Units 313
1. On the structure of the homology groups 313
2. The number of holes and the Betti number 314
3. Normal forms of multiply connected
domains (report) 315
4. On the structure of the multiplicative
group O(G)X 316
5. Glimpses 318
Bibliography 318
Short Biographies 321
Symbol Index 329
Name Index 331
Subject Index 337
|
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| owner | DE-20 DE-384 DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-706 DE-521 DE-739 DE-11 DE-188 DE-92 |
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| physical | XIX, 349 Seiten Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Graduate Texts in Mathematics |
| series2 | Graduate Texts in Mathematics |
| spelling | Remmert, Reinhold 1930-2016 Verfasser (DE-588)131654764 aut Funktionentheorie Classical topics in complex function theory Reinhold Remmert New York, NY [u.a.] Springer 1998 XIX, 349 Seiten Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate Texts in Mathematics 172 Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 Funktion Mathematik (DE-588)4071510-3 s Mehrere komplexe Variable (DE-588)4169285-8 s 1\p DE-604 Graduate Texts in Mathematics 172 (DE-604)BV000000067 172 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007882805&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Remmert, Reinhold 1930-2016 Classical topics in complex function theory Graduate Texts in Mathematics Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4071510-3 (DE-588)4169285-8 |
| title | Classical topics in complex function theory |
| title_alt | Funktionentheorie |
| title_auth | Classical topics in complex function theory |
| title_exact_search | Classical topics in complex function theory |
| title_full | Classical topics in complex function theory Reinhold Remmert |
| title_fullStr | Classical topics in complex function theory Reinhold Remmert |
| title_full_unstemmed | Classical topics in complex function theory Reinhold Remmert |
| title_short | Classical topics in complex function theory |
| title_sort | classical topics in complex function theory |
| topic | Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd |
| topic_facet | Functions of complex variables Funktionentheorie Funktion Mathematik Mehrere komplexe Variable |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007882805&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT remmertreinhold funktionentheorie AT remmertreinhold classicaltopicsincomplexfunctiontheory |