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

By Ludger / Kaup, Burchard Kaup, Michael Bridgland, Gottfried Barthel

A De Gruyter name

Show description

Read or Download Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory PDF

Best mathematics books

Periodic solutions of nonlinear wave equations with general nonlinearities

Authored by means of top students, this accomplished, self-contained textual content provides a view of the cutting-edge in multi-dimensional hyperbolic partial differential equations, with a specific emphasis on difficulties during which glossy instruments of research have proved worthwhile. Ordered in sections of steadily expanding levels of hassle, the textual content first covers linear Cauchy difficulties and linear preliminary boundary worth difficulties, prior to relocating directly to nonlinear difficulties, together with surprise waves.

Chinese mathematics competitions and olympiads: 1981-1993

This booklet comprises the issues and recommendations of 2 contests: the chinese language nationwide highschool festival from 198182 to 199293, and the chinese language Mathematical Olympiad from 198586 to 199293. China has a great checklist within the overseas Mathematical Olympiad, and the e-book comprises the issues that have been used to spot the crew applicants and choose the chinese language groups.

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

Sample text

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.

Download PDF sample

Rated 4.71 of 5 – based on 27 votes