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The present volume is based on a course of lectures given by the author for a number of years at the University of I llinois. It is intended as an introductory course suitable for first year graduate students and assumes a knowledge of only such fundamental principles of analysis as the student will have had upon completing the usual first course in calculus. Such additional information concerning functions of real variables as is needed in the development of the subject has been introduced as a regular part of the text. Thus a discussion of the general properties of line-integrals, a proof of Green stheorem, etc., have been included. The material chosen deals for the most part with the general properties of functions of a complex variable, and but little is said concerning the properties of some of the more special classes of functions, as for example elliptic functions, etc., since in a first course these subjects can hardly be treated in a satisfactory manner. The course presupposes no previous knowledge of complex numbers and the order of development is much as that commonly followed in the calculus of real variables. Integration is introduced early, in connection with differentiation. In fact the first statement of the necessary and sufficient condition that a function is holomorphic in a given region is made in terms of an integral. By this order of arrangement, it is possible to establish early in the course the fact that the continuity of the derivative follows from its existence, and consequently the Cauchy-G oursat and allied theorems can be demonstrated without any assumption as to such continuity. Likewise, it can thus be shown that Laplace sdifferential equation is satisfied without making the usual assumptions as to the existence of the derivatives of second order. The term holomorphic, often omitted, has been used as expressing an important pr
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Excerpt from Functions of a Complex Variable
In Chapter IV much use is made of mapping, thus enabling us to consider in connection with the definition of certain elementary functions some of their more important uses in physics. For the same reason in Chapter V the consideration of linear fractional transformation is especially emphasized and discussed as a kinematic problem. The discussion of series in Chapter VI lays the foundation for the consideration of the fundamental properties of single-valued functions discussed in the following chapter. In the final chapter, it is pointed out how these properties may be extended to the con sideration Of multiple-valued functions.
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Reseña del editor
The present volume is based on a course of lectures given by the author for a number of years at the University of I llinois. It is intended as an introductory course suitable for first year graduate students and assumes a knowledge of only such fundamental principles of analysis as the student will have had upon completing the usual first course in calculus. Such additional information concerning functions of real variables as is needed in the development of the subject has been introduced as a regular part of the text. Thus a discussion of the general properties of line-integrals, a proof of Green stheorem, etc., have been included. The material chosen deals for the most part with the general properties of functions of a complex variable, and but little is said concerning the properties of some of the more special classes of functions, as for example elliptic functions, etc., since in a first course these subjects can hardly be treated in a satisfactory manner. The course presupposes no previous knowledge of complex numbers and the order of development is much as that commonly followed in the calculus of real variables. Integration is introduced early, in connection with differentiation. In fact the first statement of the necessary and sufficient condition that a function is holomorphic in a given region is made in terms of an integral. By this order of arrangement, it is possible to establish early in the course the fact that the continuity of the derivative follows from its existence, and consequently the Cauchy-G oursat and allied theorems can be demonstrated without any assumption as to such continuity. Likewise, it can thus be shown that Laplace sdifferential equation is satisfied without making the usual assumptions as to the existence of the derivatives of second order. The term holomorphic, often omitted, has been used as expressing an important pr
(Typographical errors above are due to OCR software and don't occur in the book.)
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