# Holomorphic Functions of Several Variables: An Introduction by Ludger / Kaup, Burchard Kaup, Michael Bridgland, Gottfried By Ludger / Kaup, Burchard Kaup, Michael Bridgland, Gottfried Barthel

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Extra resources for Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory

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1); in the construction of F, we introduce a new parameter 2, with which we can reduce the verification of(12 B. 1) to the case n = 1. Consider the function ) '. (w,A)i—. , if w is real or 121 = 1. ii) imq,(w,2) and iii) Q(w,O) = w. iv) Now consider the holomorphic mapping it follows from ii) and iv) that (with an obvious abuse of notation) > / •(Px(T'rl{im = \ <1v is defined (and continuous) on Px V. For Then function we define a on P by setting For c>O. 1): for fixed XE > b(x, < v- {f: X —. C; f is holomorphic near is also in Now this it follows from ii) and iv) that, on P'.

11 Domains of convergence of power series and Reinhardt domains The domain of convergence of a power series of one variable is a disk. If P = a power series in n variables, then Abel's Lemma implies that the set A of points z' converges absolutely and the set B of points in C" at which in C" at which f converges conditionally have the same interior X; we call X the domain of convergence of P. Hence, P is convergent if X is nonempty; for nonconvergent P, both A and B may contain nonzero points.

B) X is polynomially convex. c) X is holomorphically convex. "That is, f—f(O) is P-linear. Function Theory on Domains in C" 38 d) X isa logarithmically convex complete Reinhardt domain. e) X is the domain of convergence of a power series. Proof The implications "a) b) c)" are trivial. (X) that is at no xc C" \ X holomorphically extendible. 4, the Taylor series P off at 0 converges on g; hence, A' = X, and X is thus the domain of convergence of P. 2. d) a) For compact K X, we have to is compact in X.