Complex Analysis (eBook)

The authors' aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.

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This bookpresents fundamental material that should be part of the education of every practicing mathematician. This material will also be of interest to computer scientists, physicists, and engineers. Complex analysis is also known as function theory. In this text we address the theory of complex-valued functions of a single complex variable. This is a prerequisite for the study of many current and rapidlydevelopingareasofmathematics, includingthetheoryofseveral andin?nitely many complex variables, thetheoryofgroups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other many other subjects in mathematics, and also to other sciences as an area where the classic and the modern techniques meet and bene?t from each other. We will try to illustrate this in the applications we give. Because function theory has been used by generations of practicing mathematicians working in a number of di?erent ?elds, the basic - sults have been developed and redeveloped from a number of di?erent perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the subject and explain and interpret standard de?nitions and results using the most convenient tools from analysis, geometry, and algebra.