An Introduction to Complex Analysis

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Scharf, John
Alfred Murray
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An Introduction to Complex Analysis
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The purpose of this paper is to present an introductory survey of the basic ideas and concepts which form a foundation for the analytic study of the single-valued functions of a complex variable. In no way does it purport to be an in depth, rigorous, or complete analytic study. In it, however, many of the basic tools which are prerequisite to any meaningful study of complex functions are presented. In chapter one, the idea of a function of a complex variable is developed. The linear fractional transformation is presented in some detail for the purpose of demonstrating some basic geometric concepts, and to introduce a few fundamental ideas used in the study of conformal or isogonal mappings. In the remainder of the chapter, some of the common functions, such as the power and root functions and some transcendental functions are discussed, but in considerably less detail than is the linear fractional transformation. The discussion of these latter functions is mainly from an algebraic view point. Chapter two introduces the concept of an analytic function of a complex variable. The limit, continuity, and derivative of a single-valued complex function are defined. From these basic definitions, the fact that the Cauchy-Riemann conditions are necessary for the existence of a unique derivative is derived. It is then shown that under certain conditions, the Cauchy-Riemann conditions are not only necessary but also sufficient for the existence of a unique derivative. In chapter three, analytic functions are again the main topic of discussion. In this chapter, however, the discussion centers mainly around the concept of the complex integral. The integral of a complex function is defined and its direct relationship to the line integral in the two dimensional real plane is established. From this relationship Cauchy’s integral theorem for simply connected regions is developed. The theorem is then extended to multiply connected regions. With the aid of the extended theorem Cauchy’s integral formula is arrived at. .To conclude chapter three, it is then shown, by using the integral formula, that if a function is analytic it has derivitives of all orders. The pre^uisites for the study of this paper are very few. An understanding of the complex numbers, their field axioms, the complex arithmetic operations of addition, subtraction,, multiplication, and division along with the plane geometric interpretations of these is essential. Some familiarity with the complex number plane is also necessary. Along with these the reader should understand the calculus of real variables up through some basic ideas on partial derivatives. Some concepts which may not be immediate to the reader but which are essential to the understanding of the context of the paper are developed in four supplementary appendixes. The titles of these are: Regions; The Stereographic Projection; DeMoivre’s Theorem; and Line Integrals,
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Mathematics, Engineering & Computer Science