Book:E.C. Titchmarsh/The Theory of Functions/Second Edition
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E.C. Titchmarsh: The Theory of Functions (2nd Edition)
Published $\text {1939}$, Oxford University Press
- ISBN 978-0-198-53349-8
Subject Matter
Contents
- Chapter 1: Infinite Series, Products, and Integrals
- 1.1 Uniform convergence of series
- 1.2 Series of complex terms. Power series
- 1.3 Series which are not uniformly convergent
- 1.4 Infinite products
- 1.5 Infinite integrals
- 1.6 Double series
- 1.7 Integration of series
- 1.8 Repeated integrals. The Gamma-function
- 1.88 Differentiation of integrals
- Chapter 2: Analytic Functions
- 2.1 Functions of a complex variable
- 2.2 The complex differential calculus
- 2.3 Complex integration. Cauchy's theorem
- 2.4 Cauchy's integral. Taylor's series
- 2.5 Cauchy's inequality. Liouville's thorem
- 2.6 The zeros of an analytic function
- 2.7 Laurent series. Singularities
- 2.8 Series and integrals of analytic functions
- 2.9 Remark on Laurent Series
- Chapter 3: Residues, Contour Integration, Zeros
- 3.1 Residues. Contour integration
- 3.2 Meromorphic functions. Integral functions
- 3.3 Summation of certain series
- 3.4 Poles and zeros of a meromorphic function
- 3.5 The modulus, and real and imaginary parts, of an analytic function
- 3.6 Poisson's integral. Jensen's theorem
- 3.7 Carleman's theorem
- 3.8 A theorem of Littlewood
- Chapter 4: Analytic Continuation
- 4.1 General theory
- 4.2 Singularities
- 4.3 Riemann surfaces
- 4.4 Functions denned by integrals. The Gamma-function. The Zeta-function
- 4.5 The principle of reflection
- 4.6 Hadamard's multiplication theorem
- 4.7 Functions with natural boundaries
- Chapter 5: The Maximum-Modulus Theorem
- 5.1 The maximum-modulus theorem
- 5.2 Schwarz's theorem. Vitali's theorem. Montel's theorem
- 5.3 Hadamard's three-circles theorem
- 5.4 Mean values of $\left\vert{f \left({z}\right)}\right\vert$
- 5.5 The Borel-Carathedory inequality
- 5.6 The Phragmen-Lindelof theorems
- 5.7 The Phragmen-Lindelof function $h \left({0}\right)$
- 5.8 Applications
- Chapter 6: Conformal Representation
- 6.1 General theory
- 6.2 Linear transmormations
- 6.3 Various transformations
- 6.4 Simple (schlicht) functions
- 6.5 Application of the principle of reflection
- 6.6 Representation of a polygon on a half-plane
- 6.7 General existence theorems
- 6.8 Further properties of simple functions
- Chapter 7: Power Series with a Finite Radius of Convergence
- 7.1 The circle of convergence
- 7.2 Position of the singularities
- 7.3 Convergence of the series and regularity of the function
- 7.4 Over-convergence. Gap theorems
- 7.5 Asymptotic behaviour near the circle of convergence
- 7.6 Abel's theorem and its converse
- 7.7 Partial sums of a power series
- 7.8 The zeros of partial sums
- Chapter 8: Integral Functions
- 8.1 Factorization of integral functions
- 8.2 Functions of finite order
- 8.3 The coefficients in the power series
- 8.4 Examples
- 8.5 The derived function
- 8.6 Functions with real zeros only
- 8.7 The minimum modulus
- 8.8 The a-points of an integral function. Picard's theorem
- 8.9 Meromorphic functions
- Chapter 9: Dirichlet Series
- 9.1 Introduction. Convergence. Absolute convergence
- 9.2 Convergence of the series and regularity of the function
- 9.3 Asymptotic behaviour
- 9.4 Functions of finite order
- 9.5 The mean-value formula and half-plane
- 9.6 The uniqueness theorem. Zeros
- 9.7 Representation of functions by Dirichlet series
- Chapter 10: The Theory of Measure and the Lebesgue Integral
- 10.1 Riemann integration
- 10.2 Sets of points. Measure
- 10.3 Measurable functions
- 10.4 The Lebesgue integral of a bounded function
- 10.5 Bounded convergence
- 10.6 Comparison between Riemann and Lebesgue integrals
- 10.7 The Lebesgue integral of an unbounded function
- 10.8 General convergence theorems
- 10.9 Integrals over an infinite range
- Chapter 11: Differentiation and Integration
- 11.1 Introduction
- 11.2 Differentiation throughout an interval. Non-differentiable functions
- 11.3 The four derivates of a function
- 11.4 Functions of bounded variation
- 11.5 Integrals
- 11.6 The Lebesgue set
- 11.7 Absolutely continuous functions
- 11.8 Integration of a differential coefficient
- Chapter 12: Further Theorems on Lebesgue Integration
- 12.1 Integration by parts
- 12.2 Approximation to an integrable function. Change of the independent variable
- 12.3 The second mean-value theorem
- 12.4 The Lebesgue class $L^p$
- 12.5 Mean convergence
- 12.6 Repeated integrals
- Chapter 13: Fourier Series
- 13.1 Trigonometrical series and Fourier series
- 13.2 Dirichlet's integral. Convergence tests
- 13.3 Summation by arithmetic means
- 13.4 Continuous functions with divergent Fourier series
- 13.5 Integration of Fourier series. Parseval's theorem
- 13.6 Functions of the class L*. Bessel's inequality. The Riesz-Fischer theorem
- 13.7 Properties of Fourier coefficients
- 13.8 Uniqueness of trigonometrical series
- 13.9 Fourier integrals
- Bibliography
- General Index