The theory of functions / by E. C. Titchmarsh.
Editor: Oxford : Oxford University Press, 1939Edición: 2nd edDescripción: x, 454 p. ; 23 cmOtra clasificación: 30-01 (26-01 28-01)CHAPTER I INFINITE SERIES, PRODUCTS, AND INTEGRALS 1.1. Uniform convergence of series [2] 1.2. Series of complex terms. Power series [8] 1.3. Series which are not uniformly convergent [11] 1.4. Infinite products [13] 1.5. Infinite integrals [19] 1.6. Double series [27] 1.7. Integration of series [36] 1.8. Repeated integrals. The Gamma-function [48] 1.88. Differentiation of integrals [59] CHAPTER II ANALYTIC FUNCTIONS 2.1. Functions of a complex variable [64] 2.2. The complex differential calculus [70] 2.3. Complex integration. Cauchy’s theorem [71] 2.4. Cauchy’s integral. Taylor’s series [80] 2.5. Cauchy’s inequality. Liouville’s thorem [84] 2.6. The zeros of an analytic function [87] 2.7. Laurent series. Singularities [89] 2.8. Series and integrals of analytic functions [95] 2.9. Remark on Laurent Series [101] CHAPTER III RESIDUES, CONTOUR INTEGRATION, ZEROS 3.1. Residues. Contour integration [102] 3.2. Meromorphic functions. Integral functions [110] 3.3. Summation of certain series [114] 3.4. Poles and zeros of a meromorphic function [115] 3.5. The modulus, and real and imaginary parts, of an analytic function [119] 3.6. Poisson’s integral. Jensen’s theorem [124] 3.7. Carleman’s theorem [130] 3.8. A theorem of Littlewood [132] CHAPTER IV ANALYTIC CONTINUATION 4.1. General theory [138] 4.2. Singularities [143] 4.3. Riemann surfaces [146] 4.4. Eunctions defined by integrals. The Gamma-function. The Zeta-function [147] 4.5. The principle of reflection [155] 4.6. Hadamard’s multiplication theorem [157] 4.7. Functions with natural boundaries [159] CHAPTER V THE MAXIMUM-MODULUS THEOREM 5.1. The maximum-modulus theorem [165] 5.2. Schwarz’s theorem. Vitali’s theorem. Montel’s theorem [168] 5.3. Hadamard’s three-circles theorem [172] 5.4. Mean values of |f(z)| [173] 5.5. The Borel-Carathedory inequality [174] 5.6. The Phragmen-Lindelof theorems [176] 5.7. The Phragmen-Lindelof function h(ϴ) [181] 5.8. Applications [185] CHAPTER VI CONFORMAL REPRESENTATION 6.1. General theory [188] 6.2. Linear transformations [190] 6.3. Various transformations [195] 6.4. Simple (schlicht) functions [198] 6.5. Application of the principle of reflection [203] 6.6. Representation of a polygon on a half-plane [205] 6.7. General existence theorems [207] 6.8. Further properties of simple functions [209] CHAPTER VII POWER SERIES WITH A FINITE RADIUS OF CONVERGENCE 7.1. The circle of convergence [213] 7.2. Position of the singularities [214] 7.3. Convergence of the series and regularity of the function [217] 7.4. Over-convergence. Gap theorems [220] 7.5. Asymptotic behaviour near the circle of convergence [224] 7.6. Abel’s theorem and its converse [229] 7.7. Partial sums of a power series [235] 7.8. The zeros of partial sums [238] CHAPTER VIII INTEGRAL FUNCTIONS 8.1. Factorization of integral functions [246] 8.2. Functions of finite order [248] 8.3. The coefficients in the power series [253] 8.4. Examples [253] 8.5. The derived function [265] 8.6. Functions with real zeros only [268] 8.7. The minimum modulus [273] 8.8. The a-points of an integral function. Picard’s theorem [277] 8.9. Meromorphic functions 284 b CHAPTER IX DIRICHLET SERIES 9.1. Introduction. Convergence. Absolute convergence [289] 9.2. Convergence of the series and regularity of the function [294] 9.3. Asymptotic behaviour [295] 9.4. Functions of finite order [298] 9.5. The mean-value formula and half-plane [303] 9.6. The uniqueness theorem. Zeros [309] 9.7. Representation of functions by Dirichlet series [313] CHAPTER X THE THEORY OF MEASURE AND THE LEBESGUE INTEGRAL 10.1. Riemann integration [318] 10.2. Sets of points. Measure [319] 10.3. Measurable functions [330] 10.4. The Lebesgue integral of a bounded function [332] 10.5. Bounded convergence [337] 10.6. Comparison between Riemann and Lebesgue integrals [339] 10.7. The Lebesgue integral of an unbounded function [341] 10.8. General convergence theorems [345] 10.9. Integrals over an infinite range [347] CHAPTER XI DIFFERENTIATION AND INTEGRATION 11.1. Introduction [349] 11.2. Differentiation throughout an interval. Non-differentiable functions [350] 11.3. The four derivates of a function [354] 11.4. Functions of bounded variation [355] 11.5. Integrals [359] 11.6. The Lebesgue set [362] 11.7. Absolutely continuous functions [364] 11.8. Integration of a differential coefficient [367] CHAPTER XII FURTHER THEOREMS ON LEBESGUE INTEGRATION 12.1. Integration by parts [375] 12.2. Approximation to an integrable function. Change of the independent variable [376] 12.3. The second mean-value theorem [379] 12.4. The Lebesgue class Lp [381] 12.5. Mean convergence [386] 12.6. Repeated integrals [390] CHAPTER XIII FOURIER SERIES 13.1. Trigonometrical series and Fourier series [399] 13.2. Dirichlet’s integral. Convergence tests [402] 13.3. Summation by arithmetic means [411] 13.4. Continuous functions with divergent Fourier series [416] 13.5. Integration of Fourier series. Parseval’s theorem [419] 13.6. Functions of the class L2. Bessel’s inequality. The Riesz-Fischer theorem [422] 13.7. Properties of Fourier coefficients [425] 13.8. Uniqueness of trigonometrical series [427] 13.9. Fourier integrals [432] BIBLIOGRAPHY [445] GENERAL INDEX [453]
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Reimpresión con correcciones, 1952.
CHAPTER I --
INFINITE SERIES, PRODUCTS, AND INTEGRALS --
1.1. Uniform convergence of series [2] --
1.2. Series of complex terms. Power series [8] --
1.3. Series which are not uniformly convergent [11] --
1.4. Infinite products [13] --
1.5. Infinite integrals [19] --
1.6. Double series [27] --
1.7. Integration of series [36] --
1.8. Repeated integrals. The Gamma-function [48] --
1.88. Differentiation of integrals [59] --
CHAPTER II --
ANALYTIC FUNCTIONS --
2.1. Functions of a complex variable [64] --
2.2. The complex differential calculus [70] --
2.3. Complex integration. Cauchy’s theorem [71] --
2.4. Cauchy’s integral. Taylor’s series [80] --
2.5. Cauchy’s inequality. Liouville’s thorem [84] --
2.6. The zeros of an analytic function [87] --
2.7. Laurent series. Singularities [89] --
2.8. Series and integrals of analytic functions [95] --
2.9. Remark on Laurent Series [101] --
CHAPTER III --
RESIDUES, CONTOUR INTEGRATION, ZEROS --
3.1. Residues. Contour integration [102] --
3.2. Meromorphic functions. Integral functions [110] --
3.3. Summation of certain series [114] --
3.4. Poles and zeros of a meromorphic function [115] --
3.5. The modulus, and real and imaginary parts, of an analytic function [119] --
3.6. Poisson’s integral. Jensen’s theorem [124] --
3.7. Carleman’s theorem [130] --
3.8. A theorem of Littlewood [132] --
CHAPTER IV --
ANALYTIC CONTINUATION --
4.1. General theory [138] --
4.2. Singularities [143] --
4.3. Riemann surfaces [146] --
4.4. Eunctions defined by integrals. The Gamma-function. The Zeta-function [147] --
4.5. The principle of reflection [155] --
4.6. Hadamard’s multiplication theorem [157] --
4.7. Functions with natural boundaries [159] --
CHAPTER V --
THE MAXIMUM-MODULUS THEOREM --
5.1. The maximum-modulus theorem [165] --
5.2. Schwarz’s theorem. Vitali’s theorem. Montel’s theorem [168] --
5.3. Hadamard’s three-circles theorem [172] --
5.4. Mean values of |f(z)| [173] --
5.5. The Borel-Carathedory inequality [174] --
5.6. The Phragmen-Lindelof theorems [176] --
5.7. The Phragmen-Lindelof function h(ϴ) [181] --
5.8. Applications [185] --
CHAPTER VI --
CONFORMAL REPRESENTATION --
6.1. General theory [188] --
6.2. Linear transformations [190] --
6.3. Various transformations [195] --
6.4. Simple (schlicht) functions [198] --
6.5. Application of the principle of reflection [203] --
6.6. Representation of a polygon on a half-plane [205] --
6.7. General existence theorems [207] --
6.8. Further properties of simple functions [209] --
CHAPTER VII --
POWER SERIES WITH A FINITE RADIUS OF CONVERGENCE --
7.1. The circle of convergence [213] --
7.2. Position of the singularities [214] --
7.3. Convergence of the series and regularity of the function [217] --
7.4. Over-convergence. Gap theorems [220] --
7.5. Asymptotic behaviour near the circle of convergence [224] --
7.6. Abel’s theorem and its converse [229] --
7.7. Partial sums of a power series [235] --
7.8. The zeros of partial sums [238] --
CHAPTER VIII --
INTEGRAL FUNCTIONS --
8.1. Factorization of integral functions [246] --
8.2. Functions of finite order [248] --
8.3. The coefficients in the power series [253] --
8.4. Examples [253] --
8.5. The derived function [265] --
8.6. Functions with real zeros only [268] --
8.7. The minimum modulus [273] --
8.8. The a-points of an integral function. Picard’s theorem [277] --
8.9. Meromorphic functions 284 b --
CHAPTER IX DIRICHLET SERIES --
9.1. Introduction. Convergence. Absolute convergence [289] --
9.2. Convergence of the series and regularity of the function [294] --
9.3. Asymptotic behaviour [295] --
9.4. Functions of finite order [298] --
9.5. The mean-value formula and half-plane [303] --
9.6. The uniqueness theorem. Zeros [309] --
9.7. Representation of functions by Dirichlet series [313] --
CHAPTER X --
THE THEORY OF MEASURE AND THE LEBESGUE INTEGRAL --
10.1. Riemann integration [318] --
10.2. Sets of points. Measure [319] --
10.3. Measurable functions [330] --
10.4. The Lebesgue integral of a bounded function [332] --
10.5. Bounded convergence [337] --
10.6. Comparison between Riemann and Lebesgue integrals [339] --
10.7. The Lebesgue integral of an unbounded function [341] --
10.8. General convergence theorems [345] --
10.9. Integrals over an infinite range [347] --
CHAPTER XI --
DIFFERENTIATION AND INTEGRATION --
11.1. Introduction [349] --
11.2. Differentiation throughout an interval. Non-differentiable functions [350] --
11.3. The four derivates of a function [354] --
11.4. Functions of bounded variation [355] --
11.5. Integrals [359] --
11.6. The Lebesgue set [362] --
11.7. Absolutely continuous functions [364] --
11.8. Integration of a differential coefficient [367] --
CHAPTER XII --
FURTHER THEOREMS ON LEBESGUE INTEGRATION --
12.1. Integration by parts [375] --
12.2. Approximation to an integrable function. Change of the independent variable [376] --
12.3. The second mean-value theorem [379] --
12.4. The Lebesgue class Lp [381] --
12.5. Mean convergence [386] --
12.6. Repeated integrals [390] --
CHAPTER XIII --
FOURIER SERIES --
13.1. Trigonometrical series and Fourier series [399] --
13.2. Dirichlet’s integral. Convergence tests [402] --
13.3. Summation by arithmetic means [411] --
13.4. Continuous functions with divergent Fourier series [416] --
13.5. Integration of Fourier series. Parseval’s theorem [419] --
13.6. Functions of the class L2. Bessel’s inequality. The Riesz-Fischer theorem [422] --
13.7. Properties of Fourier coefficients [425] --
13.8. Uniqueness of trigonometrical series [427] --
13.9. Fourier integrals [432] --
BIBLIOGRAPHY [445] --
GENERAL INDEX [453] --
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