By Bruce P. Palka

This e-book presents a rigorous but user-friendly advent to the idea of analytic features of a unmarried complicated variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal must haves past a legitimate wisdom of calculus. ranging from uncomplicated definitions, the textual content slowly and punctiliously develops the information of advanced research to the purpose the place such landmarks of the topic as Cauchy's theorem, the Riemann mapping theorem, and the theory of Mittag-Leffler should be handled with no sidestepping any problems with rigor. The emphasis all through is a geometrical one, so much suggested within the large bankruptcy facing conformal mapping, which quantities primarily to a "short direction" in that very important quarter of advanced functionality idea. every one bankruptcy concludes with a big variety of workouts, starting from effortless computations to difficulties of a extra conceptual and thought-provoking nature

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IV) THE AXIOM OF COMPLETENESS ( CONTINUITY ) If X and Y are nonempty subsets of lR having the property that x � y for every x E X and every y E Y, then there exists c E lR such that x � c � y for all x E X and y E Y. We now have a complete list of axioms such that any set on which these axioms hold can be considered a concrete realization or model of the real numbers. This definition does not formally require any preliminary knowledge about numbers, and from it "by turning on mathematical thought" we should, again formally, obtain as theorems all the other properties of real numbers.

The term "function" has a variety of useful synonyms in different areas of mathematics, depending on the nature of the sets X and Y: mapping, transformation, morphism, operator, functional. The commonest is mapping, and we shall also use it frequently. For a function (mapping) the following notations are standard: f : X -+ Y, When it is clear from the context what the domain and range of a function are, one also uses the notation x t-+ f(x) or y = f(x) , but more frequently a function in general is simply denoted by the single symbol f.

We shall make no use of this axiom in our construction of analysis. Axioms 1 ° -7° constitute the axiom system known as the Zermelo-Fraenkel ax ioms. 2 1 To this system another axiom is usually added, one that is independent of Axioms 1 ° -7° and used very frequently in analysis. 2 0 J . von Neumann (1903-1957) - American mathematician who worked in func tional analysis, the mathematical foundations of quantum mechanics, topological groups, game theory, and mathematical logic. He was one of the leaders in the creation of the first computers.