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 THE present book is intended to form a supplement to Vol. I of “Functions of a Complex Variable and Some of Their Applications” by B. A. Fuchs and B.V. Shabat. It is intended for engineers, and also for students and postgraduate students of the higher technical colleges, who Wish to make themselves more familiar with some special questions in the theory of the functions of a complex variable and its applications to those chapters of mathematical analysis which play an important part in the solution of technical and physical problems (such as differential equations, the operational calculus, special functions and questions of stability).    The present book, together with the above-mentioned book by B.A. Fuchs and B.V. Shabat, will introduce the reader to the principles of the theory of the functions of a complex variable. It will acquaint him, although, of course, not exhaustively, with the special sections of this theory, which from the point of view of applications are the most important, and will give him an idea how to use its methods for the solution of applied problems. The reader is therefore assumed to have a knowledge of the foundations of complex analysis, which can be acquired from the book by B.A. Fuchs and B.V. Shabat mentioned above. Since the present book contains a large number of references to it, it Will be referred to simply as F.C.V.   For Chapters I, II and V of the present book it is sufficient to know the bases of the theory of the functions of a complex variable as set out in the version of the abbreviated study of F.C.V., which, as is stated in the Foreword to that book, gives an introduction to the elementary theory of functions. For Chaps. III and IV the second version of the abbreviated study of F.C.V., which gives the necessary introduction to operational calculus, is sufficient. The knowledge necessary can also be gained from other books containing the foundations of the general theory of the functions of a complex variable.   The book deals with the following questions: the analytic theory of differential equations (Chaps. I and II); the Laplace transformation and its applications (Chaps. III and IV) and Hurwitz’s problem for polynomials (Chap. V).   Chapter I is of a preliminary character and is devoted to algebraic functions. Principal attention is given to the study of these functions in the neighbourhoods of their regular and singular points and to their expansions into series of powers of the variable (integral and fractional, positive and negative).   Chapter II begins with a short section dealing with the analytic functions f (w, z) of two complex variables. Then differential equations of the form dw/dz = f(w, z) are considered for cases where the function f(w, z) is regular in the neighbourhood of the initial values wo, z0 of the variables w, z, where it has at w = wo, z = z0 a pole or, lastly, a point of indeterminateness. Linear differential equations of the second order are then considered, and the results obtained are applied to the Euler—Bessel equation and its integrals (cylinder functions).   Chapter III sets out the fundamental properties of the Laplace transformation and the principles of its application to the study of special functions and to the integration of differential equations. It must be emphasized that this material is treated here as a chapter on the theory of the functions of a complex variable. The authors have sought to deal simply with the most important facts on the subject, but at the same time more precisely and more strictly than is usually done in handbooks on the operational calculus for engineers. It has not been the authors’ aim to set out the actual apparatus of the operational calculus, since this can be found in many books.   Chapter IV is devoted to contour integration and asymptotic expansions. As is well known, these are of great practical value and have in general been inadequately dealt with in other books on the subject. In addition, an account is given here of the fundamental facts of the theory of asymptotic expansions, and examples of these expansions are analysed in detail. It should be noted that for the study of these questions, some knowledge is necessary of the theory of many-valued and, in particular, algebraic functions. This is to be found in the first chapter.   Chapter V treats Hurwitz’s problem for polynomials. Although it is not connected with the preceding material, the authors are agreed that its importance for a large number of applications makes its consideration essential in a book dealing with the special problems of the theory of functions of a complex variable.   In each chapter the reader will find a large number of worked examples. These are intended to serve not only as illustrations of the theoretical conclusions, but also as models for solving independently any similar problems encountered. The engineer does not usually meet the questions dealt with in this book until his problems have been already formulated mathematically. The authors have not therefore found it necessary to consider problems with marked physical content. Nevertheless, in a number of places they have emphasized the links that exist between the mathematical theories treated and the corresponding technical disciplines.   Chapters I and II were written by B. A. Fuchs, Chapters III and IV by V.I. Levin, and Chapter V by both authors together. However, the close contact constantly maintained between them during their work on the book makes them equally responsible for the volume as a whole. 

 THE present book is intended to form a supplement to Vol. I of “Functions of a Complex Variable and Some of Their Applications” by B. A. Fuchs and B.V. Shabat. It is intended for engineers, and also for students and postgraduate students of the higher technical colleges, who Wish to make themselves more familiar with some special questions in the theory of the functions of a complex variable and its applications to those chapters of mathematical analysis which play an important part in the solution of technical and physical problems (such as differential equations, the operational calculus, special functions and questions of stability).    The present book, together with the above-mentioned book by B.A. Fuchs and B.V. Shabat, will introduce the reader to the principles of the theory of the functions of a complex variable. It will acquaint him, although, of course, not exhaustively, with the special sections of this theory, which from the point of view of applications are the most important, and will give him an idea how to use its methods for the solution of applied problems. The reader is therefore assumed to have a knowledge of the foundations of complex analysis, which can be acquired from the book by B.A. Fuchs and B.V. Shabat mentioned above. Since the present book contains a large number of references to it, it Will be referred to simply as F.C.V.   For Chapters I, II and V of the present book it is sufficient to know the bases of the theory of the functions of a complex variable as set out in the version of the abbreviated study of F.C.V., which, as is stated in the Foreword to that book, gives an introduction to the elementary theory of functions. For Chaps. III and IV the second version of the abbreviated study of F.C.V., which gives the necessary introduction to operational calculus, is sufficient. The knowledge necessary can also be gained from other books containing the foundations of the general theory of the functions of a complex variable.   The book deals with the following questions: the analytic theory of differential equations (Chaps. I and II); the Laplace transformation and its applications (Chaps. III and IV) and Hurwitz’s problem for polynomials (Chap. V).   Chapter I is of a preliminary character and is devoted to algebraic functions. Principal attention is given to the study of these functions in the neighbourhoods of their regular and singular points and to their expansions into series of powers of the variable (integral and fractional, positive and negative).   Chapter II begins with a short section dealing with the analytic functions f (w, z) of two complex variables. Then differential equations of the form dw/dz = f(w, z) are considered for cases where the function f(w, z) is regular in the neighbourhood of the initial values wo, z0 of the variables w, z, where it has at w = wo, z = z0 a pole or, lastly, a point of indeterminateness. Linear differential equations of the second order are then considered, and the results obtained are applied to the Euler—Bessel equation and its integrals (cylinder functions).   Chapter III sets out the fundamental properties of the Laplace transformation and the principles of its application to the study of special functions and to the integration of differential equations. It must be emphasized that this material is treated here as a chapter on the theory of the functions of a complex variable. The authors have sought to deal simply with the most important facts on the subject, but at the same time more precisely and more strictly than is usually done in handbooks on the operational calculus for engineers. It has not been the authors’ aim to set out the actual apparatus of the operational calculus, since this can be found in many books.   Chapter IV is devoted to contour integration and asymptotic expansions. As is well known, these are of great practical value and have in general been inadequately dealt with in other books on the subject. In addition, an account is given here of the fundamental facts of the theory of asymptotic expansions, and examples of these expansions are analysed in detail. It should be noted that for the study of these questions, some knowledge is necessary of the theory of many-valued and, in particular, algebraic functions. This is to be found in the first chapter.   Chapter V treats Hurwitz’s problem for polynomials. Although it is not connected with the preceding material, the authors are agreed that its importance for a large number of applications makes its consideration essential in a book dealing with the special problems of the theory of functions of a complex variable.   In each chapter the reader will find a large number of worked examples. These are intended to serve not only as illustrations of the theoretical conclusions, but also as models for solving independently any similar problems encountered. The engineer does not usually meet the questions dealt with in this book until his problems have been already formulated mathematically. The authors have not therefore found it necessary to consider problems with marked physical content. Nevertheless, in a number of places they have emphasized the links that exist between the mathematical theories treated and the corresponding technical disciplines.   Chapters I and II were written by B. A. Fuchs, Chapters III and IV by V.I. Levin, and Chapter V by both authors together. However, the close contact constantly maintained between them during their work on the book makes them equally responsible for the volume as a whole. 

 THE present book is intended to form a supplement to Vol. I of “Functions of a Complex Variable and Some of Their Applications” by B. A. Fuchs and B.V. Shabat. It is intended for engineers, and also for students and postgraduate students of the higher technical colleges, who Wish to make themselves more familiar with some special questions in the theory of the functions of a complex variable and its applications to those chapters of mathematical analysis which play an important part in the solution of technical and physical problems (such as differential equations, the operational calculus, special functions and questions of stability).    The present book, together with the above-mentioned book by B.A. Fuchs and B.V. Shabat, will introduce the reader to the principles of the theory of the functions of a complex variable. It will acquaint him, although, of course, not exhaustively, with the special sections of this theory, which from the point of view of applications are the most important, and will give him an idea how to use its methods for the solution of applied problems. The reader is therefore assumed to have a knowledge of the foundations of complex analysis, which can be acquired from the book by B.A. Fuchs and B.V. Shabat mentioned above. Since the present book contains a large number of references to it, it Will be referred to simply as F.C.V.   For Chapters I, II and V of the present book it is sufficient to know the bases of the theory of the functions of a complex variable as set out in the version of the abbreviated study of F.C.V., which, as is stated in the Foreword to that book, gives an introduction to the elementary theory of functions. For Chaps. III and IV the second version of the abbreviated study of F.C.V., which gives the necessary introduction to operational calculus, is sufficient. The knowledge necessary can also be gained from other books containing the foundations of the general theory of the functions of a complex variable.   The book deals with the following questions: the analytic theory of differential equations (Chaps. I and II); the Laplace transformation and its applications (Chaps. III and IV) and Hurwitz’s problem for polynomials (Chap. V).   Chapter I is of a preliminary character and is devoted to algebraic functions. Principal attention is given to the study of these functions in the neighbourhoods of their regular and singular points and to their expansions into series of powers of the variable (integral and fractional, positive and negative).   Chapter II begins with a short section dealing with the analytic functions f (w, z) of two complex variables. Then differential equations of the form dw/dz = f(w, z) are considered for cases where the function f(w, z) is regular in the neighbourhood of the initial values wo, z0 of the variables w, z, where it has at w = wo, z = z0 a pole or, lastly, a point of indeterminateness. Linear differential equations of the second order are then considered, and the results obtained are applied to the Euler—Bessel equation and its integrals (cylinder functions).   Chapter III sets out the fundamental properties of the Laplace transformation and the principles of its application to the study of special functions and to the integration of differential equations. It must be emphasized that this material is treated here as a chapter on the theory of the functions of a complex variable. The authors have sought to deal simply with the most important facts on the subject, but at the same time more precisely and more strictly than is usually done in handbooks on the operational calculus for engineers. It has not been the authors’ aim to set out the actual apparatus of the operational calculus, since this can be found in many books.   Chapter IV is devoted to contour integration and asymptotic expansions. As is well known, these are of great practical value and have in general been inadequately dealt with in other books on the subject. In addition, an account is given here of the fundamental facts of the theory of asymptotic expansions, and examples of these expansions are analysed in detail. It should be noted that for the study of these questions, some knowledge is necessary of the theory of many-valued and, in particular, algebraic functions. This is to be found in the first chapter.   Chapter V treats Hurwitz’s problem for polynomials. Although it is not connected with the preceding material, the authors are agreed that its importance for a large number of applications makes its consideration essential in a book dealing with the special problems of the theory of functions of a complex variable.   In each chapter the reader will find a large number of worked examples. These are intended to serve not only as illustrations of the theoretical conclusions, but also as models for solving independently any similar problems encountered. The engineer does not usually meet the questions dealt with in this book until his problems have been already formulated mathematically. The authors have not therefore found it necessary to consider problems with marked physical content. Nevertheless, in a number of places they have emphasized the links that exist between the mathematical theories treated and the corresponding technical disciplines.   Chapters I and II were written by B. A. Fuchs, Chapters III and IV by V.I. Levin, and Chapter V by both authors together. However, the close contact constantly maintained between them during their work on the book makes them equally responsible for the volume as a whole. 

 THIS book is intended for undergraduate and postgraduate students of higher technical institutes and for engineers wishing to increase their knowledge of theory. The aim influenced the manner of presentation: the authors decided against a “theorem-proof” sequence, but at the same time considered it inadvisable to transform the book into a work of reference. Considerable attention is paid to the fundamental ideas of the theory of functions of a complex variable; on those occasions when the proof of a theorem is omitted the authors have tried to illustrate, by means of examples, the significance of the assumptions and the conclusion. The authors also attach great importance to the development of skill in the practical application of the methods described; for this reason many examples are included in the text and a set of exercises appears at the end of each chapter. These exercises are provided with answers, and hints for solution are given for some. The authors stress that the independent solution of these problems is essential if a real mastery of the methods of function theory is to be attained (exercises marked * may be regarded as optional).   As a whole the book comprises a thorough coverage of this important complex analysis subject and can justly be described as an unusual and skilful synthesis of topics from pure and applied mathematics. It is intended for undergraduate and post-graduate students of higher technical institutes, and for engineers, wishing to increase their knowledge of theory.   Considerable attention is paid to the fundamental ideas of the theory of functions of a complex variable; in certain instances the proof of a theorem is omitted and the authors have illustrated, by means of examples, the significance of the assumptions and the conclusion. Importance is attached to the development of skill in the practical application of the methods described; for this reason many examples are included in the text and a set of exercises appears at the end of each chapter. These exercises are provided with answers and hints for solution are given for some. The topics are discussed under eight main headings: the fundamental ideas of complex analysis; conformal mappings; elementary functions; applications to the theory of plane fields; the integral representation of a regular function; representation of regular functions by series; applications of the theory of residues; and mapping of polygonal domains.   For readers who cannot study the book in its entirety the authors indicate the following three ways of making a partial study. The first method (giving an introduction to the theory of functions) includes the Introduction, Chapter I (omitting Art. 9), Chapter II (omitting Art. 17), Arts. 24—25 and 29—31 of Chapter III, Arts. 46—53 of Chapter V, and Arts. 58—61, 64—67, and 69 of Chapter VI; attention may also be directed to selected illustrative examples from Chapters IV and VII. A second course (for students interested in the analytical theory only—for example, those wishing to proceed to a study of the operational calculus) would consist of Chapter II (omitting Arts. 18—23), the definitions of the elementary functions in Chapter III (Arts. 29—31), Arts. 34—37 of Chapter IV, Chapter V (omitting Arts. 55—57), and Chapter VI (omitting Art. 70); Chapter VII should be studied in its entirety, Chapter VIII may be omitted. The third variant would suit those interested only in conformal transformations and their applications; here it would be possible to omit Arts. 54-57 of Chapter V, Arts. 66-68 of Chapter VI, and the whole of Chapter VII.   We shall dwell on certain peculiarities in our treatment (here we are addressing ourselves primarily to teachers who might recommend this book to their pupils). The Introduction surveys arithmetical operations with complex numbers; the authors have abandoned the method of introducing complex numbers usually adopted in the middle school since that method leads the pupil to regard complex numbers as being literally 'imaginary"; in this book, complex numbers are defined as vectors, or points of a plane, on which certain defined operations can be performed. The authors recognize the imperfections of this approach; however, they consider it the most appropriate for the purposes of this book since it would make considerable demands on the reader if these numbers were introduced as the elements of an abstract algebraic field having certain properties.   Chapter I expounds the fundamentals of complex analysis. As the authors desire to impart concrete ideas to their readers the function concept is introduced simultaneously with that of the corresponding transformation. Other basic concepts are also treated geometrically. The treatment emphasizes that the point at infinity on the sphere of complex numbers plays a part which is of equal importance to that played by the finite points.   In view of its special importance, the concept of conformal transformations is made the subject of a complete chapter (Chapter II). An account of fundamental definitions and theorems is followed by a detailed study of the bilinear transformation. Acquaintance with the properties of these transformations should prepare the reader for the account, given in the final article of this chapter, of the general principles of the theory of conformal transformations.   Chapter III introduces the most important of the elementary functions. Here the authors have tried to explain geometrically the procedures for separating the regular (single-valued) branches of many-valued functions. The discussion is, naturally, restricted to particular functions; the idea of the general many-valued analytic function and its regular branches is introduced in Chapter VI. An important aim of this chapter (and the Exercises following it) is the development of the reader's skill in selecting elementary functions which accomplish the required conformal mappings of given domains.   Chapter IV is concerned with the complex potential for a plane vector field and with the application of the simplest methods of function theory to the analysis of such a field. As this is the stage at which problems of an applied character make their first appearance the authors have thought it advisable to give an introductory account of the theory of plane fields before proceeding to the solution of these problems; also, the authors feel that a unified treatment of the complex potentials of the most important plane fields will assist the reader in applying function-theoretic methods to technical problems. In later chapters applied problems usually appear as illustrative examples following the exposition of mathematical methods.   An account of the fundamental apparatus of the theory of regular functions is given in Chapters V and VI: Chapter V deals with basic integral theorems, Chapter VI with expansions in series. Chapter VI also introduces the general concept of an analytic function as the set of all possible analytic continuations of a given regular function-element.   Chapters VII and VIII are devoted to applications of the theory— analytic applications in Chapter VII and geometric in Chapter VIII.   The first of these chapters depends mainly on the theory of residues and contains numerous worked examples illustrating general methods for evaluating definite integrals; the authors considered it inadvisable to set up special lemmas for the evaluation of particular types of integral (a procedure which might well be followed in certain other courses) and recommend that the student base each evaluation on general principles. The same chapter contains examples illustrating the representation of functions by contour integrals; it is hoped that these will assist the reader when he proceeds to the study of the operational calculus.   THIS book results from a complete rewriting and revision of a translation of the second (1957) Russian edition. The original was often rather condensed in presentation and contained a large number of errors and misprints. Accordingly, I have made numerous changes and additions, both in the text and in the solutions of the Exercises. It would be difficult to indicate all these additions since nearly every article has been extended in some measure; typical of the changes made are the expanded account of the complex curvilinear integral in Art. 46, the alternative proof of the maximum-modulus principle sketched in Art. 53, the counter-example at the end of Art. 56 and the tightening of the argument in Arts. 77 to 80. My main contribution, perhaps, has been the provision of a rigorous proof, in Art. 85, of the extension of the Schwarz—Christoffel formula covering the important cases of zero and non-zero angles at a vertex at infinity (the heuristic discussion given in the Russian edition was incomplete and could not easily be made into a strict proof); it is hoped that this will fill a gap in the literature: most of the standard texts give examples concerning these cases, but none, it appears, gives general proofs of the required extensions. References to standard texts in English have been added at appropriate points. This said, it must be remarked that the book remains essentially that written by Fuchs and Shabat. The authors have produced an unusual and skilful synthesis of topics from pure and applied mathematics. A sound, useful account of analytic function theory is given in Chapters V, VI and VII, the field covered corresponding roughly with the content of the usual honours degree course; features of interest are the introductory account of harmonic functions and Dirichlet’s problem in Arts. 54—57 and the excellent collection of worked examples in Arts. 72—75.   As an introduction to the theory and application of conformal mappings the book is outstanding: I know of no other text which can match the account of elementary mappings given in Chapters II and III, and the discussion of the four main types of plane potential problem in Chapter IV. To anyone who has lectured on the topics discussed in Chapter VIII, it will be clear that in, Arts. 82 to 88, the authors have made a valuable contribution to the teaching literature; here, again, it is fair to claim that no other text to date can match the combination of theory and application appearing in the accounts of the Schwarz reflection principle and the Schwarz—Christoif el formula. The wealth of worked examples makes the book specially useful to the student who must work on his own.   Certain standard notational conventions have been adopted: square brackets are used to denote closed intervals, round brackets to denote open intervals; and ez denotes exp(z). Brackets have also been used to represent line segments in the complex plane: for example, [ — i, l+i] denotes the closed line segment joining the points — i and 1 + i. It should be noted that symbols such as Arg z, Log z are used to denote, not only the corresponding multi-valued functions, but also, on occasion, particular branches of these functions differing (usually) from the corresponding principal branches.