# Analytic functions

• 1952

top

## Abstract

top
CONTENTS PREFACE................................... III PREFACE TO THE ENGLISH EDITION................................... VII INTRODUCTION. THEORY OF SETS § 1. Fundamental definitions................................... 1 § 2. Denumerable sets................................... 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 sequences................................... 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 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’s theorem for a rectangle................................... 112 § 5. 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(1 - 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................................... 153 § 10. Rouché’s theorem................................... 155 § 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 § 5. Schwarz’s lemma................................... 222 § 6. Riemann’s theorem................................... 225 § 7. Radó’s theorem................................... 231 § 8. The Schwarz-Christoffel formulae................................... 233 CHAPTER VI. ANALYTIC FUNCTION § 1. Introductory remarks................................... 238 § 2. Analytic element................................... 239 § 3. Analytic continuation along a curve................................... 246 § 4. Analytic functions................................... 247 § 5. Inverse of an analytic function................................... 254 § 6. Analytic functions arbitrarily continuable in a region................................... 255 § 7. Theorem of Poincaré-Volterra................................... 258 § 8. An analytic function as an abstract space................................... 259 § 9. Analytic functions in an annular neighbourhood of a point................................... 261 § 10. Analytic functions in an annular neighbourhood as an abstract space................................... 264 § 11. Critical points................................... 265 § 12. Algebraic critical points................................... 267 § 13. Auxiliary theorems of algebra................................... 268 § 14. Functions with algebraic critical points................................... 271 § 15. Algebraic functions................................... 275 § 16. Riemann surfaces................................... 277 CHAPTER VII. ENTIRE FUNCTIONS AND FUNCTIONS MEROMORPHIC IN THE ENTIRE OPEN PLANE § 1. Infinite products................................... 286 § 2. Weierstrass’s theorem on the decomposition of entire functions into products................................... 295 § 3. Mittag-Leffler’s theorem on the decomposition of meromorphic functions into simple fractions................................... 301 § 4. Cauchy’s method of decomposing meromorphic functions into simple fractions................................... 305 § 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’s 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................................... 354 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 § 4. The function p(z)................................... 368 § 5. Differential equation of the function p(z)................................... 371 § 6. The function ζ(z) and σ(z)................................... 375 § 7. Construction of elliptic functions by means of the function σ(z)................................... 378 § 8. Expression of elliptic functions in terms of the functions ζ(z) and σ(z)................................... 380 § 9. Algebraic addition theorem for the function p(z)................................... 384 § 10. Algebraic relations between elliptic functions................................... 386 § 11. The modular function J(τ)................................... 387 § 12. Further properties of the function J(τ)................................... 392 § 13.Solution of the system of equations ${g}_{2}\left(\omega ,{\omega }^{\text{'}}\right)=a$, ${g}_{3}\left(\omega ,{\omega }^{\text{'}}\right)=b$................................... 403 § 14. Elliptic integrals................................... 404 CHAPTER IX. THE FUNCTIONS Γ(s) AND ζ(s) DIRICHLET SERIES § 1. The function Γ(s)................................... 411 § 2. The function B(p,q)................................... 416 § 3. Hankel’s formulae for the function Γ(s)................................... 418 § 4. Stirling’s formula................................... 420 § 5. The function ζ(s) of Riemann................................... 424 § 6. Functional equation of the function ζ(s)................................... 428 § 7. Roots of the function ζ(s)................................... 429 § 8. Dirichlet series................................... 432 INDEX................................... 441 ERRATA................................... 446

## How to cite

top

Stanisław Saks, and Antoni Zygmund. Analytic functions. 1952. <http://eudml.org/doc/219298>.

@book{StanisławSaks1952,
abstract = {CONTENTS PREFACE................................... III PREFACE TO THE ENGLISH EDITION................................... VII INTRODUCTION. THEORY OF SETS § 1. Fundamental definitions................................... 1 § 2. Denumerable sets................................... 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 sequences................................... 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 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’s theorem for a rectangle................................... 112 § 5. 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(1 - 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................................... 153 § 10. Rouché’s theorem................................... 155 § 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 § 5. Schwarz’s lemma................................... 222 § 6. Riemann’s theorem................................... 225 § 7. Radó’s theorem................................... 231 § 8. The Schwarz-Christoffel formulae................................... 233 CHAPTER VI. ANALYTIC FUNCTION § 1. Introductory remarks................................... 238 § 2. Analytic element................................... 239 § 3. Analytic continuation along a curve................................... 246 § 4. Analytic functions................................... 247 § 5. Inverse of an analytic function................................... 254 § 6. Analytic functions arbitrarily continuable in a region................................... 255 § 7. Theorem of Poincaré-Volterra................................... 258 § 8. An analytic function as an abstract space................................... 259 § 9. Analytic functions in an annular neighbourhood of a point................................... 261 § 10. Analytic functions in an annular neighbourhood as an abstract space................................... 264 § 11. Critical points................................... 265 § 12. Algebraic critical points................................... 267 § 13. Auxiliary theorems of algebra................................... 268 § 14. Functions with algebraic critical points................................... 271 § 15. Algebraic functions................................... 275 § 16. Riemann surfaces................................... 277 CHAPTER VII. ENTIRE FUNCTIONS AND FUNCTIONS MEROMORPHIC IN THE ENTIRE OPEN PLANE § 1. Infinite products................................... 286 § 2. Weierstrass’s theorem on the decomposition of entire functions into products................................... 295 § 3. Mittag-Leffler’s theorem on the decomposition of meromorphic functions into simple fractions................................... 301 § 4. Cauchy’s method of decomposing meromorphic functions into simple fractions................................... 305 § 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’s 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................................... 354 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 § 4. The function p(z)................................... 368 § 5. Differential equation of the function p(z)................................... 371 § 6. The function ζ(z) and σ(z)................................... 375 § 7. Construction of elliptic functions by means of the function σ(z)................................... 378 § 8. Expression of elliptic functions in terms of the functions ζ(z) and σ(z)................................... 380 § 9. Algebraic addition theorem for the function p(z)................................... 384 § 10. Algebraic relations between elliptic functions................................... 386 § 11. The modular function J(τ)................................... 387 § 12. Further properties of the function J(τ)................................... 392 § 13.Solution of the system of equations $g_2(ω,ω^\{\prime \})=a$, $g_3(ω,ω^\{\prime \})=b$................................... 403 § 14. Elliptic integrals................................... 404 CHAPTER IX. THE FUNCTIONS Γ(s) AND ζ(s) DIRICHLET SERIES § 1. The function Γ(s)................................... 411 § 2. The function B(p,q)................................... 416 § 3. Hankel’s formulae for the function Γ(s)................................... 418 § 4. Stirling’s formula................................... 420 § 5. The function ζ(s) of Riemann................................... 424 § 6. Functional equation of the function ζ(s)................................... 428 § 7. Roots of the function ζ(s)................................... 429 § 8. Dirichlet series................................... 432 INDEX................................... 441 ERRATA................................... 446},
author = {Stanisław Saks, Antoni Zygmund},
keywords = {complex functions},
language = {eng},
title = {Analytic functions},
url = {http://eudml.org/doc/219298},
year = {1952},
}

TY - BOOK
AU - Stanisław Saks
AU - Antoni Zygmund
TI - Analytic functions
PY - 1952
AB - CONTENTS PREFACE................................... III PREFACE TO THE ENGLISH EDITION................................... VII INTRODUCTION. THEORY OF SETS § 1. Fundamental definitions................................... 1 § 2. Denumerable sets................................... 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 sequences................................... 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 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’s theorem for a rectangle................................... 112 § 5. 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(1 - 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................................... 153 § 10. Rouché’s theorem................................... 155 § 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 § 5. Schwarz’s lemma................................... 222 § 6. Riemann’s theorem................................... 225 § 7. Radó’s theorem................................... 231 § 8. The Schwarz-Christoffel formulae................................... 233 CHAPTER VI. ANALYTIC FUNCTION § 1. Introductory remarks................................... 238 § 2. Analytic element................................... 239 § 3. Analytic continuation along a curve................................... 246 § 4. Analytic functions................................... 247 § 5. Inverse of an analytic function................................... 254 § 6. Analytic functions arbitrarily continuable in a region................................... 255 § 7. Theorem of Poincaré-Volterra................................... 258 § 8. An analytic function as an abstract space................................... 259 § 9. Analytic functions in an annular neighbourhood of a point................................... 261 § 10. Analytic functions in an annular neighbourhood as an abstract space................................... 264 § 11. Critical points................................... 265 § 12. Algebraic critical points................................... 267 § 13. Auxiliary theorems of algebra................................... 268 § 14. Functions with algebraic critical points................................... 271 § 15. Algebraic functions................................... 275 § 16. Riemann surfaces................................... 277 CHAPTER VII. ENTIRE FUNCTIONS AND FUNCTIONS MEROMORPHIC IN THE ENTIRE OPEN PLANE § 1. Infinite products................................... 286 § 2. Weierstrass’s theorem on the decomposition of entire functions into products................................... 295 § 3. Mittag-Leffler’s theorem on the decomposition of meromorphic functions into simple fractions................................... 301 § 4. Cauchy’s method of decomposing meromorphic functions into simple fractions................................... 305 § 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’s 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................................... 354 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 § 4. The function p(z)................................... 368 § 5. Differential equation of the function p(z)................................... 371 § 6. The function ζ(z) and σ(z)................................... 375 § 7. Construction of elliptic functions by means of the function σ(z)................................... 378 § 8. Expression of elliptic functions in terms of the functions ζ(z) and σ(z)................................... 380 § 9. Algebraic addition theorem for the function p(z)................................... 384 § 10. Algebraic relations between elliptic functions................................... 386 § 11. The modular function J(τ)................................... 387 § 12. Further properties of the function J(τ)................................... 392 § 13.Solution of the system of equations $g_2(ω,ω^{\prime })=a$, $g_3(ω,ω^{\prime })=b$................................... 403 § 14. Elliptic integrals................................... 404 CHAPTER IX. THE FUNCTIONS Γ(s) AND ζ(s) DIRICHLET SERIES § 1. The function Γ(s)................................... 411 § 2. The function B(p,q)................................... 416 § 3. Hankel’s formulae for the function Γ(s)................................... 418 § 4. Stirling’s formula................................... 420 § 5. The function ζ(s) of Riemann................................... 424 § 6. Functional equation of the function ζ(s)................................... 428 § 7. Roots of the function ζ(s)................................... 429 § 8. Dirichlet series................................... 432 INDEX................................... 441 ERRATA................................... 446
LA - eng
KW - complex functions
UR - http://eudml.org/doc/219298
ER -

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.