Analytic functions

Stanisław Saks ; Antoni Zygmund

GDML_Books, (1952), p.

Résumé

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

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