Polish Virtual Library of Science

Mathematical Collection

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Monografie Matematyczne

Volume 28

Analytic functions

Stanisław Saks, Antoni Zygmund

Warszawa-Wrocław 1952

Contents

INTRODUCTION	THEORY OF SETS	§ 1. Fundamental definitions § 2. Denumerable sets § 3. Abstract topological space § 4. Closed and open sets § 5. Connected sets § 6. Compact sets § 7. Continuous transformations § 8. The plane § 9. Connected sets in the plane § 10. Square nets in the plane § 11. Real and complex functions § 12. Curves § 13. Cartesian product of sets
CHAPTER I	FUNCTIONS OF A COMPLEX VARIABLE	§ 1. Continuous functions § 2. Uniformly and almost uniformly convergent sequences § 3. Normal families of functions § 4. Equi-continuous functions § 5. The total differential § 6. The derivative in the complex domain. Cauchy-Riemann equations § 7. The exponential function § 8. Trigonometric functionsTrigonometric functions § 9. Argument § 10. Logarithm § 11. Branches of the logarithm, argument and power § 12. Angle between half-lines § 13. Tangent to a curve § 14. Homographic transformations § 15. Similarity transformations § 16. Regular curves § 17. Curvilinear integrals § 18. Examples
CHAPTER II	HOLOMORPHIC FUNCTIONS	§ 1. The derivative in the complex domain § 2. Primitive function § 3. Differentiation of an integral with respect to a complex variable § 4. Cauchy's theorem for a rectangle § 5. Cauchy's formula for a system of rectangles § 6. Almost uniformly convergent sequences of holomorphic functions § 7. Theorem of Stieltjes-Osgood § 8. Morera's theorem
CHAPTER III	MEROMORPHIC FUNCTIONS	§ 1. Power series in the circle of convergence § 2. Abel's theorem § 3. Expansion of Log (1 - z) § 4. Laurent's series. Annulus of convergence § 5. Laurent expansion in an annular neighbourhood § 6. Isolated singular points § 7. Regular, meromorphic, and rational functions § 8. Roots of a meromorphic function § 9. The logarithmic derivative § 10. Rouché's theorem § 11. Hurwitz's theorem § 12. Mappings defined by meromorphic functions § 13. Holomorphic functions of two variables § 14. Weierstrass's preparation theorem
CHAPTER IV	ELEMENTARY GEOMETRICAL METHODS OF THE THEORY OF FUNCTIONS	§ 1. Translation of poles § 2. Runge's theorem. Cauchy's theorem for a simply connected region § 3. Branch of the logarithm § 4. Jensen's formula § 5. Increments of the logarithm and argument along a curve § 6. Index of a point with respect to a curve § 7.Theorem on residues § 8. The method of residues in the evaluation of definite integrals § 9. Cauchy's theorem and formula for an annulus § 10. Analytical definition of a simply connected region vv § 11. Jordan's theorem for a closed polygon § 12. Analytical definition of the degree of connectivity of a region
CHAPTER V	CONFORMAL TRANSFORMATIONS	§ 1. Definition § 2. Homographic transformationsv § 3. Symmetry with respect to a circumference § 4. Blaschke's factors § 5. Schwarz's lemma § 6. Riemann's theorem § 7. Radó's theorem § 8. The Schwarz-Christoffel formulae
CHAPTER VI	ANALYTIC FUNCTIONS	§ 1. Introductory remarks § 2. Analytic element § 3. Analytic continuation along a curve § 4. Analytic functions § 5. Inverse of an analytic function § 6. Analytic functions arbitrarily continuable in a region § 7. Theorem of Poincaré-Volterra § 8. An analytic function as an abstract space § 9. Analytic functions in an annular neighbourhood of a point § 10. Analytic functions in an annular neighbourhood as an abstract space § 11. Critical points § 12. Algebraic critical points § 13. Auxiliary theorems of algebra § 14. Functions with algebraic critical points § 15. Algebraic functions § 16. Riemann surfaces
CHAPTER VII	ENTIRE FUNCTIONS AND FUNCTIONS MEROMOR-PHIC IN THE ENTIRE OPEN PLANE	§ 1. Infinite products § 2. Weierstrass's theorem on the decomposition of entire functions into products § 3. Mittag-Leffler's theorem on the decomposition of meromorphic functions into simple fractions § 4. Cauchy's method of decomposing meromorphic functions into simple fractions § 5. Examples of expansions of entire and meromorphic functions § 6. Order of an entire function § 7. Dependence of the order of an entire function on the coefficients of its Taylor series expansion § 8. The exponent of convergence of the roots of an entire function § 9. Canonical product § 10. Hadamard's theorem § 11. Borel's theorem on the roots of entire functions § 12. The small theorem of Picard § 13. Schottky's theorem. Montel's theorem. Picard's great theorem § 14. Landau's theorem
CHAPTER VIII	ELLIPTIC FUNCTIONS	§ 1. General remarks about periodic functions § 2. Expansion of a periodic function in a Fourier series § 3. General theorems on elliptic functions § 4. The function p(z) § 5. Differential equation of the function p(z) § 6. The function ... § 7. Construction of elliptic functions by means of the function ... § 8. Expression of elliptic functions in terms of the functions ... § 9. Algebraic addition theorem for the function p(z) § 10. Algebraic relations between elliptic functions § 11. The modular function ... § 12. Further properties of the function ... § 13.Solution of the system of equations ... § 14. Elliptic integrals
CHAPTER IX	THE FUNCTIONS .... DIRICHLET SERIE	§ 1. The function ... § 2. The function B(p, q) § 3. Hankel's formulae for the function ... § 4. Stirling's formula § 5. The function ... of Riemann § 6. Functional equation of the function ... § 7. Roots of the function ... § 8. Dirichlet series
Materiały redakcyjne		Preface, preface to the english edition, index, errata, contents