By Bruce P. Palka

This booklet presents a rigorous but easy advent to the idea of analytic features of a unmarried advanced variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal necessities past a legitimate wisdom of calculus. ranging from uncomplicated definitions, the textual content slowly and punctiliously develops the tips of complicated 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 might be taken care of with no sidestepping any problems with rigor. The emphasis all through is a geometrical one, such a lot suggested within the wide bankruptcy facing conformal mapping, which quantities primarily to a "short path" in that very important region of complicated functionality idea. every one bankruptcy concludes with a big variety of routines, starting from undemanding computations to difficulties of a extra conceptual and thought-provoking nature

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Then as x → p, (i) (ii) (iii) (iv) ( f + g)(x) → q + r and c f (x) → cq (for any real constant c); if Y = Rk , ( f · g)(x) → q · r ; when Y = R, if g = 0 on E and r = 0, then ( f /g)(x) → q/r ; when Y = Rk , f (x) → q iff f i (x) := f (x)i → qi for all i = 1, . . , k. The meaning of (iv) is that convergence of a vector valued function f to the vector q is equivalent to the convergence of its components to the corresponding components of q. The above properties of limits follow easily from the definition.

Exercise 5(d)). This is a way to construct a complete metric space, namely Z , in which X is injected isometrically as a dense subset, namely π(X ). Such a space Z is unique up to an isometric equivalence, and is called the completion of X . A more classical way of doing this is by taking the set of all equivalence classes of Cauchy sequences under some natural equivalence relation, with a metric induced by the metric of X (cf. 4). 40 1 Continuity 9. Let (X, d) be a metric space, and consider the metric space X k of Exercise 7, Sect.

1 Definition Let X be a metric space, and E ⊂ X . An open cover of E is a family of open sets Vα (α ∈ I ) such that Vα . 6) α∈I A subcover of the above cover of E is a subfamily of the above family, whose union contains E. The subcover is said to be finite if it has finitely many sets in it. The set E is said to be compact if every open cover of E has a finite subcover. A finite subset of X is compact. Indeed, if E = {x1 , . . 6), then for each j = 1, . . , n, there exists α j ∈ I such that x j ∈ Vα j .