Synopsis
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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Paperback. Condition: New. Print on Demand. This book delves into the fascinating world of complex numbers and functions, a field of mathematics that has revolutionized our understanding of numbers and their relationships. The author takes the reader on a journey through the evolution of the concept of numbers, starting with the familiar integers and fractions, then expanding to explore irrational numbers, which cannot be expressed as simple ratios. The book's core focus is on complex numbers, which involve the imaginary unit "i," the square root of -1. By introducing this new unit, the book demonstrates how mathematics extends beyond the realm of real numbers, opening up a whole new dimension of possibilities. The author skillfully navigates the complex arithmetic of adding, subtracting, multiplying, and dividing complex numbers, providing clear explanations and geometric interpretations. The book then delves into the key concept of functions of a complex variable, exploring their properties and classifications, such as rational, irrational, algebraic, and transcendental functions. The book explores essential concepts like limits and convergence, laying the groundwork for a deeper understanding of the behavior of complex functions. It also highlights the importance of mapping and linear fractional transformations, demonstrating their practical applications in various fields. Ultimately, this book offers a comprehensive and accessible introduction to the captivating world of complex variables and functions, revealing the elegance and power of this fundamental branch of mathematics. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9781440055980_0
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