Functions Of A Complex Variable

JUNCTIONS OF A COMPLEX VARIABLE BY E. J. TOWNSEND, PH. D. PROFESSOR OF MATHEMATICS, UNIVERSITY OF ILLINOIS NEW YOEK HENRY HOLT AND COMPANY 1915 COPYRIGHT, 1915, BY HENRY HOLT AND COMPANY September, 192 Stanbope iftress F. H. GILSON COMPANY BOSTON, U. S. A. PEEFACB THE present volume is based on a course of lectures given by the author for a number of years at the University of Illinois. It is in tended 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 concern ing 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 Greens theorem, 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 factthat the continuity of the derivative follows from its existence, and consequently the Cauchy-Goursat and allied theorems can be dem onstrated without any assumption as to such continuity. Likewise, it can thus be shown that Laplaces differential 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 property of single-valued functions, reserving the use of the term analytic for use in connection with functions derived from a given element by means of analytic continuation. While the Cauchy-Riemann viewpoint is that first introduced, attention is called to the Weierstrass development in the iii iv PREFACE chapter on series, and in subsequent discussions either definition of an analytic function is used as best suits the purpose in hand. 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. The author wishes to express his appreciation of the helpful sug gestions which have been given to him by Professor J. L. Markley of the University of Michigan, Professor A. Dresden of theUniversity of Wisconsin, Professor W. A. Hurwitz of Cornell University, and to Dr. Otto Dunkel of the University of Missouri, who have read the proof sheets. He is also under obligations to his colleagues Dr. Denton and Dr. Kempner, who have read the manuscript. Finally, he wishes to express especially his obligations to Dr. George Rut ledge, who has rendered him valuable assistance in the preparation of the manuscript. E. J. TOWNSEND. UNIVERSITY OF ILLINOIS July, 1915 CONTENTS CHAPTER I REAL AND COMPLEX NUMBERS ARTICLJE PAGE 1. Rational Numbers 1 2...