1.1. Preface.

This book is based on courses of lectures given to undergraduates in the Universities of Edinburgh and St Andrews, and is intended to provide an easy introduction to the methods of the theory of functions of a complex variable. The reader is assumed to have a knowledge of the elements of the theory of functions of a real variable, such as is contained, for example in Hardy's Course of Pure Mathematics. An acquaintance with the easier parts of Bromwich's Infinite Series would prove advantageous, but is not essential.

The first six chapters contain an exposition, based on Cauchy's Theorem, of the properties of the one-valued differentiable functions of a complex variable. In the rest of the book the problem of conformal representation, the elements of the theory of integral functions and the behaviour of some of the special functions of analysis are discussed by the methods developed in the earlier part. The book concludes with the classical proof of Picard's Theorem.

No attempt has been made to give the book an encyclopaedic character. My object has been to interest the reader and to encourage him to study further some of the more advanced parts of the subject; suggestions for further reading have been made at the end of each chapter.

I am especially indebted to Mr W L Ferrar, who read the manuscript of the whole book in its original and revised forms, and suggested many improvements. My grateful thanks are also due to Professor E T Whittaker, F.R.S., for his kindly criticism during the early stages of the preparation of this work and for his constant encouragement.

Finally, I have to thank Dr H S Ruse and Professor J M Whittaker for their careful reading of the proof sheets and many valuable suggestions.

E. T. C.
GREENWICH,
July 1935

1.2. Contents.

PREFACE

I. COMPLEX NUMBERS

II. THE CONVERGENCE OF INFINITE SERIES

III. FUNCTIONS OF A COMPLEX VARIABLE

IV. CAUCHY'S THEOREM

V. UNIFORM CONVERGENCE

VI. THE CALCULUS OF RESIDUES

VII. INTEGRAL FUNCTIONS

VIII. CONFORMAL REPRESENTATION

IX. THE GAMMA FUNCTION

X. THE HYPERGEOMETRIC FUNCTIONS

XI. LEGENDRE FUNCTIONS

XII. BESSEL FUNCTIONS

XIII. THE ELLIPTIC FUNCTIONS OF WEIERSTRASS

XIV. JACOBI'S ELLIPTIC FUNCTIONS

XV. ELLIPTIC MODULAR FUNCTIONS AND PICARD'S THEOREM

INDEX

1.3. The first Section.

The introduction of complex numbers into algebra

In arithmetic, we understand by a real number a magnitude which can be expressed as a decimal fraction. If the decimal terminates or recurs, the real number is said to be rational, since it is then the ratio of two whole numbers. But if the decimal does not terminate or recur, the number is not the ratio of two whole numbers and is said to be irrational. We shall assume that the reader is acquainted with Dedekind's method of founding the theory of rational and irrational numbers on a sound logical basis.

Elementary algebra is concerned with the application of the operations of arithmetic to symbols representing real numbers. The result of any sequence of such operations is always a real number.

A difficulty, however, soon arises in the theory of equations. If $a, b$, and $c$ are real numbers, the quadratic equation $ax^{2} + 2bx + c = 0$ has two distinct roots if $b^{2} > ca$ and two equal roots when $b^{2} = ca$. But if $b^{2} < ca$, there is no real number $x$ which satisfies the equation, since the square of every real number is positive. It is customary to introduce a new symbol $√(-1)$, whose square is defined to be -1, and then to show that the equation is formally satisfied by taking

          $ax = -b ± √(ac-b^{2})√(-1)$,

when $b^{2} < ca$.

When this new symbol has been added to the algebra of real numbers, every quadratic equation is formally satisfied by two expressions of the form $\alpha + \beta√(-1)$, where $\alpha$ and $\beta$ are real numbers. Such expressions are called complex numbers.

If we suppose that the symbol $√(-1)$ obeys all the laws of algebra, save that its square is -1, we can develop a consistent algebra of complex numbers which includes the algebra of real numbers as a particular case and which also possesses a character of completeness which is lacking in the simpler theory. This completeness is well illustrated by the theorem which states that every equation of degree $n$ has precisely $n$ roots - a result which is true in the algebra of complex numbers but is false in the algebra of real numbers.

The present book is concerned essentially with the application of the methods of the differential and integral calculus to complex numbers.

1.4. Review by: William Thomas Reid.
Science, New Series 84 (2166) (1936), 21-22.

This volume is based on lectures given to undergraduates in the Universities of Edinburgh and St Andrews. Assuming a knowledge of mathematical analysis such as is contained, for example, in Hardy's "Course of Pure Mathematics" (Cambridge, fifth edition, 1928), the book affords an introduction to a number of branches of the theory of functions of a complex variable. ...

The miscellaneous examples occurring at the end of each chapter include many significant theorems and contribute materially to the attractiveness of the book.

The matrix definition of complex numbers introduced in Chapter I is worthy of note. The treatment of analytic functions is based on the Cauchy integral theorem. A function is said to be analytic in a domain if it is single-valued and differentiable at every point of this domain, save possibly for a finite number of exceptional points. In the opinion of the reviewer, this is one of the less preferable uses of the greatly overworked term "analytic." Cauchy's theorem is proved for a polygonal contour by a method due to E H Moore. Reference is then made to a paper of S Pollard for the details of the passage from a polygonal contour to a general rectifiable simply closed curve. Pollard's work is dependent upon the theory of chains of regions as developed by de la Vallee Poussin. In addition to the statement and indication of proof of the more general form of Cauchy's theorem, there is given an elementary proof of the theorem for the case of a simply closed curve which consists of a finite number of arcs having continuously turning tangents. The proof is the usual one involving Green's theorem; however, Green's theorem is not proved in either of the two references cited by the author for a region bounded by such a general contour. The inclusion of more extensive bibliographies on the proofs of Cauchy's theorem and Green's theorem would have enhanced the value of Chapter IV.

Chapters VII and VIII are good introductions to their respective subjects. In considering these chapters, however, the student is to be reminded of the admonition given in the preface that the book is not intended to be encyclopaedic in character. One of the most interesting sections of the volume is that dealing with the application of saddle-point integration to the problem of determining asymptotic expansions of the Bessel functions. The work on modular functions culminates with the proof of Picard's theorem, together with the extension of this theorem due to Carathéodory. By way of conclusion, references are given to the elementary proofs of Picard's theorem due to Borel, Bloch and Nevanlinna.

For the most part, and in all the basic work, the proofs given are modern in character and sufficiently complete in detail for the thoughtful reader. The reviewer noted a few places, however, which are likely to trouble the student. For example, on page 16 it is stated without proof that a non-decreasing bounded sequence tends to a limit. In a later section, on page 21, it is proved that an arbitrary sequence of real numbers has maximum and minimum limits. One wonders why the order of these two sections was not interchanged. In the preliminary discussion of Cauchy's theorem on page 59 the author has omitted the assumption that the domain be simply connected. It is to be remarked that in many places the term "domain" is used without specifying whether an open domain or a closed domain is meant. In several statements made in Chapter IV it is necessary to interpret "domain" as meaning "open domain."

Due to the comprehensive treatment of the subject-matter of Chapters X-XII, this book will appeal to those interested in differential equations. In evaluating the work as a whole, however, the reviewer feels that an unduly large proportion of the book is devoted to the study of particular functions, rather than to the general methods of function theory. This opinion is strengthened when one considers the omissions. The subject of analytic extension and the definition of the complete analytic function, using the terminology of the author, receive a very meagre treatment. The entire geometrical aspect of analytic function theory is missing, and the terms "Riemann surface" and "algebraic function" do not appear in the book. There is no mention of analytic functions of two or more complex variables. These comments are not to be interpreted as adverse criticism. They are intended to emphasise, however, that this work shares with many other books on the subject the property of not affording in itself a complete and balanced introduction to this important field of mathematical analysis.

... This book will doubtless be widely used as an introductory text on the subject.

1.5. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 20 (237) (1936), 72.

The first six chapters of this book deal with the classical properties of regular functions; the next two discuss developments which, in their method of treatment at any rate, belong essentially to this century, integral functions and conformal representation. Then we have the gamma function, hypergeometric functions, Legendre and Bessel functions, the elliptic functions of Weierstrass and Jacobi, and finally the elliptic modular functions leading round again to the modern ideas which have grown out of Picard's theorem that an integral function takes every finite value with at most one exception. The exposition is clear and easy to follow, and the author has provided us with a much-needed account of methods which are of the utmost value in modern function-theory - for example, the principle of the maximum modulus, and saddle-point integration - though we feel that more stress might have been laid on methods and less on functions; but this is possibly a matter of taste.

Professor Copson has based his book on courses of lectures given in the Universities of Edinburgh and St Andrews; such courses must be as far as possible self-contained and this perhaps accounts for some fifty pages on convergence and uniform convergence, an amount of space which we could wish had been available at later stages in the book's progress. Even so, uniform convergence comes so late (Chapter V), that Chapter III has to include a rather dreary proof that the power series inside its circle of convergence has a sum which is a regular function. ...

This volume will probably take its place as the standard English introduction to the subject. It is printed in the admirable style to which the Oxford Press has recently accustomed us.

1.6. Review by: Theodor Estermann.
Science Progress (1933-) 31 (121) (1936), 158-159.

Compared with other text-books on functions of a complex variable, this one makes easy reading, a characteristic which will be appreciated particularly by undergraduates. Apart from the general theory, it deals with the Gamma function, Legendre and Bessel functions, hypergeometric and elliptic functions, and briefly with integral functions. Proofs of some of the more difficult theorems are omitted, but ample references are given concerning them. The book is rich in examples, and there is a good index at the end.

The book is, on the whole, well written, but unfortunately no more accurate in details than other books of this kind that I have seen. Thus, on page 59, it is stated that, if $f(z)$ is regular in a domain $D$, then the integral of $f(z)$, taken along a path contained in $D$, depends only on the end-points of the path. This is true for simply connected domains only.

1.7. Review by: Albert Ernest Maxwell.
The Mathematical Gazette 55 (391), 87.

First edition, 1935. Reprints 1944, 1946, 1948, 1050, 1955, 1957, 1960, 1961, 1962, 1970. Need a reviewer say more? The readers have spoken.

2.1. From the Publisher.

Certain functions, capable of expansion only as a divergent series, may nevertheless be calculated with great accuracy by taking the sum of a suitable number of terms. The theory of such asymptotic expansions is of great importance in many branches of pure and applied mathematics and in theoretical physics. Solutions of ordinary differential equations are frequently obtained in the form of a definite integral or contour integral, and this tract is concerned with the asymptotic representation of a function of a real or complex variable defined in this way. After a preliminary account of the properties of asymptotic series, the standard methods of deriving the asymptotic expansion of an integral are explained in detail and illustrated by the expansions of various special functions. These methods include integration by parts, Laplace's approximation, Watson's lemma on Laplace transforms, the method of steepest descents, and the saddle-point method. The last two chapters deal with Airy's integral and uniform asymptotic expansions.

2.2. Preface.

In 1943, at the request of the Admiralty Computing Service, I wrote a short monograph on The Asymptotic Expansion of a Function Defined by a Definite Integral or Contour Integral. This was one of a series of monographs intended for use in Admiralty Research Establishments, on topics which appeared to be inadequately covered in easily accessible literature. It evidently met a need of the time, since a revised edition was issued in 1946 and had a wide circulation.

The Admiralty monograph has long been unobtainable, and several of my friends have urged me to write this more extensive book on the same general lines. There are few theorems; the aim is the modest one of explaining the methods which are available, and illustrating them by means of a few of the more important special functions.

I must express my thanks to Professor Arthur Erdélyi for the generous advice and encouragement he has given me during the writing of this book.

2.3. Review by: Yudell Leo Luke.
Mathematics of Computation 20 (93) (1966), 182.

Certain important functions may often be represented by asymptotic series which are usually divergent. Nevertheless, the functions may be calculated to some level of accuracy by taking the sum of a suitable number of terms. In some situations, the sequence obtained by a certain weighting of the sequence of partial sums of an asymptotic series converges. Solutions of ordinary differential equations can often be expressed in the form of a definite integral or a contour integral. Thus, the subject of asymptotics is very important to both pure and applied mathematicians.

This volume gives an excellent treatment of asymptotic expansions of transcendents defined by integrals. After an introductory account of the properties of asymptotic expansions (Chapters 1 and 2), the standard methods of deriving asymptotic expansions are explained in detail and illustrated with special functions. These techniques include integration by parts (Chapter 3), the method of stationary phase (Chapter 4), Laplace's approximation (Chapter 5), Laplace's integral and Watson's lemma (Chapter 6), the method of steepest descent (Chapter 7) and the saddle-point method (Chapter 8). Chapter 9 treats Airy's integral by various methods. For the most part, the expansions discussed are not uniform. Uniform asymptotic expansions is the subject of Chapter 10.

Professor Copson's volume presupposes only a knowledge of the more elementary notions of real and complex variable theory. The subject matter is within the capabilities of undergraduate students.

The volume is very readable and suitable for self-study or as an academic text- book. In this connection, the utility of the text would have been considerably enhanced by the inclusion of exercises.

2.4. Review by: Swarupchand Mohanlal Shah.
The American Mathematical Monthly 73 (9) (1966), 1031-1032.

This monograph gives an introduction to the theory of asymptotic representation of a function defined by a definite integral or a contour integral. A knowledge of the elementary parts of the theory of functions is assumed, in particular complex integration and special functions such as the Gamma function and Bessel functions. After a preliminary account of the properties of asymptotic series, the standard methods of deriving the asymptotic expansion of an integral are explained in detail.

The chapter headings are: (1) Introduction (Historical Remarks), (2) Preliminaries, (3) Integration by parts, (4) The method of stationary phase, (5) The method of Laplace, (6) Watson's lemma, (7) The method of steepest descents, (8) The saddle-point method, (9) Airy's integral, (10) Uniform asymptotic expansions.

Chapters 5-7 contain applications to the Gamma function, the logarithmic integral, Bessel functions and the error function. There are no exercises and the applications of asymptotic methods to differential equations are not considered. As mentioned by the author in the preface, there are few theorems and the aim is to explain the available methods and to illustrate them by means of a few of the more important special functions. The typography is splendid. The reviewer feels that this monograph is a welcome addition to the existing literature on the subject.

2.5. Review by: William Thomas Reid.
Quarterly of Applied Mathematics 24 (4) (1967), 385-386.

This book is a valuable addition to the small but slowly growing list of works dealing with various aspects of asymptotic analysis. It is a revised and enlarged version of the author's well-known monograph on The Asymptotic Expansion of a Function Defined by a Definite Integral or Contour Integral.

That monograph, which first appeared in 1943 and in a revised edition in 1946, has been out of print for many years and the appearance of the present book is therefore particularly welcome.

The first part of the book deals with general properties of asymptotic sequences and series, integration by parts, the method of stationary phase, and the method of Laplace. Watson's lemma is then introduced and used to discuss Debye's method of steepest descents and Riemann's saddle-point method.

In the application of the method of steepest descents to Airy functions it is shown that a single application of the method together with the use of connection formulas suffices to determine all of the required expansions for all values of $arg z$. The expansions so obtained have overlapping sectors of validity in which the two expansions differ by a subdominant series. This situation is not inconsistent, according to the usual interpretation, but is merely an example of Stokes phenomenon. In some situations, however, it would appear desirable to have a criterion by which the expansions can be restricted to non-overlapping sectors. Such a criterion has recently been suggested by Olver based on his theory of error bounds for asymptotic expansions of functions defined by ordinary linear differential equations but a corresponding theory of error bounds for functions defined by contour integrals does not appear to exist at the present time.

Finally, the author discusses the important theory of Chester, Friedman, and Ursell for obtaining uniform expansions according to the method of steepest descents when the integrand has two nearly coincident saddle-points.

2.6. Review by: Edward M Wright.
The Mathematical Gazette 50 (372) (1966), 206.

In 1943 Professor Copson, at the request of the Admiralty Computing Service, wrote a short monograph on the asymptotic expansion of a function defined by a definite integral or contour integral. All those who have used this monograph, which was circulated principally in Government research establishments, will welcome the appearance of this more extensive book by the same author. It is far from easy to write a good book about asymptotic expansions. An elaborate, systematic, logical account of results in the form of theorems, especially if given with full generality, would be indescribably tedious. Most readers of a book on the subject have a particular problem to solve and want to find the most hopeful method as quickly as possible. It is perhaps because the perils of dullness in the subject are so obvious that one or two authors have produced such strikingly successful expositions. De Bruijn's and Jeffrey's well-known books are examples of such success and the one under review is another.

The author is primarily concerned with the asymptotic expansion of functions expressed in the form of integrals. There are very clear and useful expositions of the methods of stationary phase, and of steepest descent and the saddle-point. The book is a worthy member of the series of Cambridge Tracts and will be an essential and heavily used part of the library of anyone working in any field requiring the results of analysis.

3.1. From the Publisher.

Metric space topology, as the generalisation to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. Professor Copson's book, which is based on lectures given to third-year undergraduates at the University of St Andrews, provides a more leisurely treatment of metric spaces than is found in books on functional analysis, which are usually written at graduate student level. His presentation is aimed at the applications of the theory to classical algebra and analysis; in particular, the chapter on contraction mappings shows how it provides proof of many of the existence theorems in classical analysis.

3.2. Preface.

This book, based on lectures given at the University of St Andrews is intended to give honours students the background and training necessary before they start to study functional analysis.

There are many books on functional analysis; and some of them seem to go over the preliminaries to the subject far too quickly. The aim here is to provide a more leisurely approach to the theory of the topology of metric spaces, a subject which is not only the basis of functional analysis but also unifies many branches of classical analysis. The applications of the theory in Chapter 8 to problems in classical algebra and analysis show how much can be done without ever defining a normed vector space, a Banach space or a Hilbert space.

The reader is not expected to know much more classical analysis than is contained in Hardy's Pure Mathematics or Burkill's First Course in Mathematical Analysis. A knowledge of the elements of the theory of uniform convergence is assumed. Analytic functions and Lebesgue integrals are mentioned occasionally; their introduction provides more advanced applications of the theory of metric spaces, but adds nothing to the theory.

I am most grateful to Professor Arthur Erdélyi and to the Editors of the series of Cambridge Mathematical tracts for their kindly criticisms and suggestions.

3.3. Review by: Robert A Rankin.
Proceeding of the Edinburgh Mathematical Society 16 (3) (1969), 267.

The author's aim is to provide a more leisurely approach to the theory of the topology of metric spaces than is normally given in textbooks on functional analysis. In this he has been eminently successful and has produced a very readable book, which could be used by undergraduates either as a text for a course of lectures or for private study. A minimum of classical analysis is assumed and the subjects studied include complete metric spaces, connected and compact sets. Applications to spaces of functions are given, such as Arzelà's theorem and Tietze's extension theorem.

Perhaps the most interesting chapter in the book is the one dealing with fixed point theorems and their applications to systems of linear equations, differential equations, integral equations, the implicit function theorem and other topics. This is a very valuable collection of results and illustrates admirably the power and use of abstract theorems on metric spaces. There is a short final chapter on Banach and Hilbert spaces. Numerous example for the student are included at the ends of the first eight chapters.

3.4. Review by: Claude A Rogers.
The Mathematical Gazette 53 (385) (1969), 342.

This book gives a very clear and detailed introduction into some aspects of the theory of metric spaces with detailed discussions of examples drawn almost exclusively from elementary classical analysis. The treatment culminates in the contraction mapping theorem and its application to prove a variety of existence theorems. The material could all be taught to second year undergraduates and I believe most of it should be.

The book will certainly make this much easier to do. Unfortunately the author seems to have written the book with an under- graduate audience continually in mind and has failed to bring to the subject the illumination that can sometimes be obtained from a more advanced point of view. While he proves a number of important results (the Baire Category Theorem, Arzelà's Theorem, the Extension Theorem of Tietze and Urysohn) their importance and significance are not made clear. No adequate recommendations are made for further reading.

Despite these limitations the book is strongly recommended to undergraduates who wish to appreciate the power of abstract analysis as a tool for the proof of results in classical analysis.

4.1. From the Publisher.

In this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first order and of linear partial differential equations of the second order, using the methods of classical analysis. In spite of the advent of computers and the applications of the methods of functional analysis to the theory of partial differential equations, the classical theory retains its relevance in several important respects. Many branches of classical analysing have their origins in the rigorous discussion of problems in applies mathematics and theoretical physics, and the classical treatment of the theory of partial differential equations still provides the best method of treating many physical problems. A knowledge of the classical theory is essential for pure mathematics who intend to undertake research in this field, whatever approach they ultimately adopt. The numerical analyst needs a knowledge of classical theory in order to decide whether a problem has a unique solution or not.

4.2. Preface.

This book has been written in memory of my father-in-law the late Professor Sir Edmund Whittaker, in gratitude for all the help and encouragement he gave me for over thirty years. Today is the hundredth anniversary of his birth.

When I went to Edinburgh as a young lecturer in 1922, I was surprised to find how different the curriculum was from that in Oxford. It included such topics as Lebesgue integration, matrix theory, numerical analysis, Riemannian geometry, of which I knew nothing. I was particularly impressed by Whittaker's lectures on partial differential equations to undergraduate and postgraduate students, far different from the standard English textbooks of the time. This book is not based on Whittaker's lectures; yet without his inspiration it would never have been written.

I have frequently given courses of lectures on partial differential equations and have always regretted that there was no book to which I could refer my students. Friends told me that the remedy was to write one myself; and here it is, a presentation of some of the theory by the methods of classical analysis.

There are few references to original sources. After lecturing on the subject for so many years, I could not now say whence the material came. On page 277 will be found a list of the books which I have read with profit, many of them more advanced than this.

E.T.C.
St Andrews
24 October 1973

4.3. Review by: Constantine Dafermos.
Quarterly of Applied Mathematics 34 (1) (1976), 124.

The study of partial differential equations has made considerable progress in the last 30 years and, as a result, the methodology itself of the subject has radically changed. Even so, the classical aspects of the qualitative theory should be familiar to the serious student of the field. The content, style and notation of Copson's book have a strong classical flavour. The following topics are covered: equations of first order; the Cauchy-Kowalewsky theorem and the theory of characteristics; Riemann's method and hyperbolic equations of second order; classical potential theory in two variables, subharmonic functions and barriers; elliptic second order equations in two and three variables; the equation of heat conduction. The book also contains a fairly detailed exposition of M Riesz's theory of integrals of fractional order and its application to hyperbolic equations, a subject not usually discussed in elementary texts of Differential Equations.

The book is dedicated to the memory of the author's father-in-law, Professor Sir Edmund Whittaker, F.R.S.