PREFACE................................... III ERRATA.................................... IV CHAPTER I. Trigonometrical series and Fourier series...... 1 1.1. Definitions. 1.2. Abel's transformation. 1.3. Orthogonal systems of functions. Fourier series. 1.4. The trigonometrical system. 1.5. Completness of the trigonometrical system. 1.6. Bessel's inequality. Farseval's relation. 1.7. Remarks on series and integrals. 1.8. Miscellaneous theorems and examples. CHAPTER II. Fourier coefficients. Tests for the convergence of Fourier series........... 14 2.1. Operations on Fourier series. 2.2. Modulus of continuity. Fourier coefficients. 2.3. Formulae for partial sums. 2.4. Dini's test. 2.5. Theorems on localization. 2.6. Functions of bounded variation. 2.7. Tests of Lebesgue and Dini-Lipschitz. 2.8. Tests of de la Vallée-Poussin, Young, and Hardy and Littlewood. 2.9. Miscellaneous theorems and examples. CHAPTER III. Summability of Fourier series.......... 40 3.1. Toeplitz matrices. Abel and Cesaro means. 3.2. Fejér's theorem. 3.3 Summability (C, r) of Fourier series and conjugate series. 3.4. Abel's summability. 3.5. The Cesaro summation of differentiated series. 3.6. Fourier sine series. 3.7. Convergence factors. 3.8. Summability of Fourier-Stieltjes series. 3.9. Miscellaneous theorems and examples. CHAPTER IV. Classes of functions and Fourier series.... 64 4.1. Inequalities 4.2. Mean convergence. The Riesz-Fischer theorem. 4.3. Classes B, C, S, and $L_φ$ of functions. 4.4. Parseval's relations. 4.5. Linear operations. 4.6. Transformations of Fourier series. 4.7. Miscellaneous theorems and examples. CHAPTER V. Properties of some special series....... 108 5.1. Series with coefficients monotonically tending to 0. 5.2. Approximate expressions for such series. 5.3. A power series. 5.4. Lacunary series. 5.5. Rademacher's series. 5.6. Applications of Rademacher's functions. 5.7. Miscellaneous theorems and examples. CHAPTER VI. The absolute convergence of trigonometrical series.......... 131 6.1. The Lusin-Denjoy theorem. 6.2. Fatou's theorems. 6.3. The absolute convergence of Fourier series. 6.4. Szidon's theorem on lacunary series. 6.5. The theorems of Wiener and Levy. 6.6. Miscellaneous theorems and examples. CHAPTER VII. Conjugate series and complex methods in the theory of Fourier series........... 145 7.1. Summability of conjugate series. 7.2. Conjugate series and Fourier series. 7.3. Mean convergence of Fourier series. 7.4. Privaloff's theorem. 7.5. Power series of bounded variation. 7.6. Miscellaneous theorems and examples. CHAPTER VIII. Divergence of Fourier series. Gibbs's phenomenon...................... 167 8.1. Continuous functions with divergent Fourier series. 8.2. A theorem of Faber and Lebesgue. 8.3. Lebesgue's constants. 8.4. Kolmogoroffs example. 8.5. Gibbs's phenomenon. 8.6. Theorems of Rogosinski. 8.7. Cramer's theorem. 8.8. Miscellaneous theorems and examples. CHAPTER IX. Further theorems on Fourier coefficients. Integration of fractional order............. 189 9.1. Remarks on the theorems of Hausdorff-Young and F. Riesz. 9.2. M. Riesz'a convexity theorems. 9.3. Proof of F. Riesz's theorem. 9.4. Theorems of Paley. 9.5. Theorems of Hardy and Littlewood. 9.6. Banach's theorems on lacunary coefficients. 9.7. Wiener's theorem on functions of bounded variation. 9.8. Integrals of fractional order. 9.9. Miscellaneous theorems and examples. CHAPTER X. Further theorems on the summability and convergence of Fourier series............. 237 10.1. An extension of Fejér's theorem. 10.2. Maximal theorems of Hardy and Littlewood. 10.3. Partial sums. 10.4. Summability C of Fourier series. 10.5. Miscellaneous theorems and examples. CHAPTER XI. Riemann's theory of trigonometrical series....................... 267 11.1. The Cantor-Lebesgue theorem and its generalization. 11.2. Riemann's and Fatou's theorems. 11.3. Theorems of uniqueness. 11.4. The principle of localization. Rajchman's theory of formal multiplication. 11.5. Sets of uniqueness and sets of multiplicity. 11.6. Uniqueness in the case of summable series. 11.7. Miscellaneous theorems and examples. CHAPTER XII. Fourier's integral....................... 306 12.1. Fourier's single integral. 12.2. Fourier's repeated integral. 12 3. Summability of integrals. 12.4. Fourier transforms. TERMINOLOGICAL INDEX, NOTATIONS........................ 320 BIBLIOGRAPHY.......................... 321
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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 $g_2(ω,ω')=a$, $g_3(ω,ω')=b$................................... 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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Le but de cette note est de déduire du théorème de Zermelo l'existence d'une fonction d'une variable réelle f(x) qui est discontinue sur tout ensemble de puissance du continu.
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Le but de cette note est de démontrer: Théorème: Une courbe C étant donnée dans le plan, tout faisceau F des droites tangents (de l'un ou des deux cotes) à cette courbe est de mesure nulle, sauf peut-être, le cas où le sommet du faisceau se trouve sur la courbe envisagée.
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