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Analytic functions:

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Main Authors:	Saks, Stanislaw (Author), Zygmund, Antoni 1900-1992 (Author)
Format:	Book
Language:	English Polish
Published:	Amsterdam [u.a.] Elsevier Publishing Company [u.a.] 1971
Edition:	Third edition
Subjects:	Fonctions analytiques Analytic functions Funktionentheorie Analytische Funktion
Online Access:	Inhaltsverzeichnis
Physical Description:	XIV, 504 Seiten graph. Darst.
ISBN:	0444408738

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MARC


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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
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isbn	0444408738
language	English Polish
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publisher	Elsevier Publishing Company [u.a.]
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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
url	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
work_keys_str_mv	AT saksstanislaw funkcjeanalityczne AT zygmundantoni funkcjeanalityczne AT saksstanislaw analyticfunctions AT zygmundantoni analyticfunctions

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