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Theory of Functions

Edward C. Titchmarsh
Oxford University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a very clearly-written text covering primarily complex analysis, and including a number of hard-to-find results. This second edition was published in 1939 and has been in print continuously since then.

The mathematical prerequisites are low: a rigorous course in single-variable calculus is enough. The book refers throughout to particular sections of Hardy’s A Course of Pure Mathematics for background (Hardy’s book, despite its very general title, is merely a rigorous calculus book). The most valuable sections for modern readers are the numerous far-from-trivial examples (throughout the book), an in-depth exploration of the maximum modulus theorem (including Hadamard’s three-circle theorem and the Phragmén-Lindelöf theorem), a thorough study of entire functions (here called integral functions), and a detailed look at asymptotics and Tauberian theorems for power series.

This is very much a pure-mathematics book, with no applications given to other sciences and almost no applications to other areas of mathematics. It’s a peculiar feeling to read a detailed chapter on Dirichlet series and to realize that it never mentions their use in number theory! It is also a very analytical book, with no pictures or geometric arguments, not even in the chapter on conformal mappings: everything is done with formulas.

The three chapters on Lebesgue measure and integration are probably the least valuable part of the book today. This theory has been streamlined and now has better terminology, and the material here is found in most books on real analysis in a more concise and easier-to-understand form. The one chapter of Fourier analysis is less dated, but the material is found in many real analysis books, and in most Fourier analysis books that use the Lebesgue integral.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Chapter I. 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 II. 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 theorem
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 III. 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 IV. Analytic Continuation
4.1. General theory
4.2. Singularities
4.3. Riemann surfaces
4.4. Functions defined 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 V. 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 |f(z)|
5.5. The Borel-CarathŽodory inequality
5.6. The Phragmén-Lindelöf theorems
5.7. The Phragmén-Lindelöf function h(θ)
5.8. Applications

Chapter VI. Conformal Representation
6.1. General theory
6.2. Linear transformations
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 VII. 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 VIII. 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 IX. 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 X. 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 XI. 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 XII. Further Theorems On Lebesgue Integration
12.2. Approximation to an integrable function. Change of the independent variable
12.3. The second mean-value theorem
12.4. The Lebesgue class Lp
12.5. Mean convergence
12.6. Repeated integrals

Chapter XIII. 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 L2. Bessel’s inequality. The Riesz-Fischer theorem
13.7. Properties of Fourier coefficients
13.8. Uniqueness of trigonometrical series
13.9. Fourier integrals


General Index