Book:E.C. Titchmarsh/The Theory of Functions/Second Edition

Subject Matter

 * Analysis

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