Analytic functions:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English Polish |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier Publishing Company [u.a.]
1971
|
| Ausgabe: | Third edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 504 Seiten graph. Darst. |
| ISBN: | 0444408738 |
Internformat
MARC
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| 240 | 1 | 0 | |a Funkcje analityczne |
| 245 | 1 | 0 | |a Analytic functions |c by S. Saks and A. Zygmund |
| 250 | |a Third edition | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier Publishing Company [u.a.] |c 1971 | |
| 300 | |a XIV, 504 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Fonctions analytiques | |
| 650 | 4 | |a Analytic functions | |
| 650 | 0 | 7 | |a Funktionentheorie |0 (DE-588)4018935-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Analytische Funktion |0 (DE-588)4142348-3 |2 gnd |9 rswk-swf |
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| 689 | 1 | 0 | |a Funktionentheorie |0 (DE-588)4018935-1 |D s |
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Datensatz im Suchindex
| _version_ | 1804117103756705792 |
|---|---|
| adam_text | CONTENTS
Page
PREFACE V
PREFACES TO ENGLISH EDITIONS VIII
INTRODUCTION. THEORY OF SETS.
§ 1. Fundamental definitions ....... 1
§ 2. Denumerable gets ........ 3
§ 3. Abstract topological space ...... 4
{ 4. Closed and open sets ....... 6
§ 5. Connected sets . . . . . . . .11
§ 6. Compact sets ¦ . . . . . . . .13
§ 7. Continuous transformations . . . . . .14
§ 8. The plane ......... 17
§ 9. Connected sets in the plane . . . . . .25
§ 10. Square nets in the plane ....... 32
§ 11. Real and complex functions . . . . . .36
§ 12. Curves ......... 38
§ 13. Cartesian product of sets . . . . . .40
CHAPTER I. FUNCTIONS OF A COMPLEX VARIABLE.
§ 1. Continuous functions ....... 44
§ 2. Uniformly and almost uniformly convergent nequeiices . . 46
§ 3. Normal families of functions . . . . . .49
§ 4. Equi continuous functions . . . . . .53
§ 5. The total differential . . . . . . .55
§ 6. The derivative in the complex domain. Cauchy Riemann equations 57
§ 7. The exponential function . . . . . . .60
§ 8. Trigonometric functions . . . . . . .62
§ 9. Argument ......... 68
§ 10. Logarithm ......... 72
§ 11. Branches of the logarithm, argument and power . . .74
§ 12. Angle between half lines . . . • • • .77
§ 13. Tangent to a curve . . . ¦ • • .79
§ 14. Homographic transformations . . • ¦ ¦ .80
§ 15. Similarity transformations . • • • ¦ .87
§ 16. Regular curves ........ 91
§ 17. Curvilinear integrals . . ¦ • • • .92
§ 18. Examples ......... 95
Xir Contents.
CHAPTER II. HOLOMORPHIC FUNCTIONS.
§ 1. The derivative in the complex domain . . . . .98
§ 2. Primitive function ........ 100
§ 3. Differentiation of an integral with respect to a complex variable 107
§ 4. Cauchy 8 theorem for a rectangle ..... 109
§ 6. Cauchy s formula for a system of rectangles . . . .112
§ 6. Almost uniformly convergent sequences of holomorphic functions 116
§ 7. Theorem of Stieltjes Osgood . . . . . .119
§ 8. Morera s theorem . . . . . . . .120
CHAPTER III. MEROMORPHIC FUNCTIONS.
§ 1. Power series in the circle of convergence . . . .125
§ 2. Abel s theorem . . . . . . . .128
§ 3. Expansion of Log(l — z) ....... 134
§ 4. Laurent s series. Annulus of convergence . . . .137
§ 5. Laurent expansion in an annular neighbourhood . . .140
§ 6. Isolated singular points . . . . . .143
§ 7. Regular, meromorphic, and rational functions . . .145
§ 8. Roots of a meromorphic function . . . . .150
§ 9. The logarithmic derivative . . . . . .163
§ 10. Rouche s theorem . . . . . . . .166
§ 11. Hurwitz s theorem ........ 158
§ 12. Mappings defined by meromorphic functions . . . .161
§ 13. Holomorphic functions of two variables . . . .165
§ 14. Weierstrass s preparation theorem ..... 167
CHAPTER IV. ELEMENTARY GEOMETRICAL METHODS OF THE
THEORY OF FUNCTIONS.
§ 1. Translation of poles ....... 171
§ 2. Runge s theorem. Cauchy s theorem for a simply connected region 176
§ 3. Branch of the logarithm ....... 179
§ 4. Jensen s formula . . . . . . . .181
§ 5. Increments of the logarithm and argument along a curve . .183
§ 6. Index of a point with respect to a curve . . . .186
§ 7. Theorem on residues ....... 189
§ 8. The method of residues in the evaluation of definite integrals . 194
§ 9. Cauchy s theorem and formula for an annulus . . . 196
§ 10. Analytical definition of a simply connected region . . . 204
§ 11. Jordan s theorem for a closed polygon . . • . . 206
§ 12. Analytical definition of the degree of connectivity of a region . 209
CHAPTER V. CONFORMAL TRANSFORMATIONS.
§ 1. Definition 214
§ 2. Homographic transformations ...... 216
§ 3. Symmetry with respect to a circumference .... 217
| 4. Blaschke s factors ........ 220
Contents. XIII
§ 5. Schwarz s lemma ........ 222
§ 6. Riemann s theorem ....... 226
§ 7. Rado s theorem 231
§ 8. The Schwarz Christoffel formulae . . . . .233
CHAPTER VI. ANALYTIC FUNCTIONS.
§ 1. Introductory remarks . . . . . . .238
§ 2. Analytic element ........ 339
| 3. Analytic continuation along a curve ..... 246
§ 4. Analytic functions ........ 247
§ 5. Inverse of an analytic function ...... 264
§ 6. Analytic functions arbitrarily continuable in a region . . 266
§ 7. Theorem of Poincare Volterra ...... 268
{ 8. An analytic function as an abstract space .... 269
§ 9. Analytic functions in an annular neighbourhood of a point . .261
§ 10. An analytic function in an annular neighbourhood as an abstract space 264
§ 11. Critical points ........ 266
§ 12. Algebraic critical points ....... 267
§ 13. Auxiliary theorems of algebra . ... . . . 268
§ 14. Functions with algebraic critical points .... 271
§ 16. Algebraic functions ........ 276
§ 16. Riemann surfaces . . ..... 277
CHAPTER VII. ENTIRE FUNCTIONS AND FUNCTIONS MEROMOR
PHIC IN THE ENTIRE OPEN PLANE.
{ 1. Infinite products ........ 286
§ 2. Weierstrass s theorem on the decomposition of entire functions into
products ......... 296
§ 3. Mittag Leffler s theorem on the decomposition of meromorphic
functions into simple fractions . . . . . .301
§ 4. Cauchy s method .of decomposing meromorphic functions into sim¬
ple fractions . . . . . . . . . 306
§ 5. Examples of expansions of entire and meromorphic functions . 309
§ 6. Order of an entire function . . . . . .319
§ 7. Dependence of the order of an entire function on the coefficients
of its Taylor series expansion ...... 324
§ 8. The exponent of convergence of the roots of an entire function . 327
§ 9. Canonical product ........ 329
§ 10. Hadamard s theorem ....... 332
§ 11. Borel 8 theorem on the roots of entire functions . . . 338
§ 12. The small theorem of Picard . . . . . .341
§ 13. Schottky s theorem. Montel s theorem. Picard s great theorem . 346
{ 14. Landau s theorem ........ 364
CHAPTER VIII. ELLIPTIC FUNCTIONS.
§ 1. General remarks about periodic functions .... 356
§ 2. Expansion of a periodic function in a Fourier series . . 360
§ 3. General theorems on elliptic functions . . • ¦ .363
Xrv Contents.
§ 4. The function p(«) . . . . . . . . 368
§ 5. Differential equation of the function p (z) . . . .371
§ 6. The function £(*) and a (z) . . . . . .375
$ 7. Construction of elliptic functions by means of the function a{z) . 378
§ 8. Expression of elliptic functions in terms of the functions f(«) and a(z) 380
§ 9. Algebraic addition theorem for the function p (z) . . 384
$ 10. Algebraic relations between elliptic functions .... 386
§ 11. The modular function J(r) . . . . . . 387
| 12. Further properties of the function J (t) . . . . 392
§ 13. Solution of the system of equations gt(a , to ) = a, g3(io, to ) = b . 403
§ 14. Elliptic integrals . . . . . . .404
CHAPTER IX. THE FUNCTIONS /» AND C(»). DIRICHLET SERIES.
§ 1. The function /*(«) ........ 411
§ 2. The function B(p,q) . . . . . . .416
§ 3. Hankel s formulae for the function T(«) .... 418
§ 4. Stirling s formula ........ 420
{ 5. The function C(«) of Riemann ...... 424
| 6. Functional equation of the function £(*) . . . 428
§ 7. Roots of the function £(«) . . . . . . 429
| 8. Dirichlet series ........ 432
CHAPTER X. HARMONIC AND SUBHARMONIC FUNCTIONS.
$ 1. Harmonic functions as real parts of holomorphic functions . 441
§ 2. Functions harmonic in a circle ...... 449
| 3. Functions harmonic in an annulus ..... 455
$ 4. Poisson s integral ........ 459
§ 5. The maximum condition for harmonic functions . . . 468
{ 6. An alternative definition of a harmonic function . . . 472
{ 7. Convex functions ......... 474
§ 8. Subharmonic functions . . • ¦ ¦ • ¦ ¦ 480
$ 9. Examples and applications of subharmonic functions . . 488
§ 10. Laplace s equation ........ 493
INDEX 497
|
| any_adam_object | 1 |
| author | Saks, Stanislaw Zygmund, Antoni 1900-1992 |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
| edition | Third edition |
| format | Book |
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| id | DE-604.BV002750632 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:49:05Z |
| institution | BVB |
| isbn | 0444408738 |
| language | English Polish |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001758557 |
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| physical | XIV, 504 Seiten graph. Darst. |
| publishDate | 1971 |
| publishDateSearch | 1971 |
| publishDateSort | 1971 |
| publisher | Elsevier Publishing Company [u.a.] |
| record_format | marc |
| spelling | Saks, Stanislaw Verfasser aut Funkcje analityczne Analytic functions by S. Saks and A. Zygmund Third edition Amsterdam [u.a.] Elsevier Publishing Company [u.a.] 1971 XIV, 504 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Fonctions analytiques Analytic functions Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Analytische Funktion (DE-588)4142348-3 gnd rswk-swf Analytische Funktion (DE-588)4142348-3 s DE-604 Funktionentheorie (DE-588)4018935-1 s 1\p DE-604 Zygmund, Antoni 1900-1992 Verfasser (DE-588)124637531 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001758557&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 | Saks, Stanislaw Zygmund, Antoni 1900-1992 Analytic functions Fonctions analytiques Analytic functions Funktionentheorie (DE-588)4018935-1 gnd Analytische Funktion (DE-588)4142348-3 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4142348-3 |
| title | Analytic functions |
| title_alt | Funkcje analityczne |
| title_auth | Analytic functions |
| title_exact_search | Analytic functions |
| title_full | Analytic functions by S. Saks and A. Zygmund |
| title_fullStr | Analytic functions by S. Saks and A. Zygmund |
| title_full_unstemmed | Analytic functions by S. Saks and A. Zygmund |
| title_short | Analytic functions |
| title_sort | analytic functions |
| topic | Fonctions analytiques Analytic functions Funktionentheorie (DE-588)4018935-1 gnd Analytische Funktion (DE-588)4142348-3 gnd |
| topic_facet | Fonctions analytiques Analytic functions Funktionentheorie Analytische Funktion |
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