By Toshio Nishino

ISBN-10: 0821808168

ISBN-13: 9780821808160

``Kiyoshi Oka, before everything of his study, looked the gathering of difficulties which he encountered within the research of domain names of holomorphy as huge mountains which separate this present day and the next day to come. therefore, he believed that there can be no crucial growth in research with no mountaineering over those mountains ... this ebook is a precious preliminary step for the reader on the way to comprehend the mathematical international which used to be created via Kiyoshi Oka.'' --from the Preface This e-book explains ends up in the speculation of features of a number of complicated variables which have been generally confirmed from the overdue 19th century in the course of the heart of the 20 th century. within the paintings, the writer introduces the mathematical international created by means of his consultant, Kiyoshi Oka. during this quantity, Oka's paintings is split into components. the 1st is the examine of analytic capabilities in univalent domain names in ${\mathbf C}^n$. the following Oka proved that 3 thoughts are identical: domain names of holomorphy, holomorphically convex domain names, and pseudoconvex domain names; and furthermore that the Poincare challenge, the Cousin difficulties, and the Runge challenge, while acknowledged competently, could be solved in domain names of holomorphy pleasurable the precise stipulations. the second one a part of Oka's paintings proven a mode for the research of analytic capabilities outlined in a ramified area over ${\mathbf C}^n$ during which the department issues are regarded as inside issues of the area. the following analytic features in an analytic house are taken care of, that's a mild generalization of a ramified area over ${\mathbf C}^n$. In writing the publication, the author's objective used to be to deliver to readers a true realizing of Oka's unique papers. This quantity is an English translation of the unique jap variation, released by means of the college of Tokyo Press (Japan). it is going to make an appropriate path textual content for complex graduate point introductions to numerous complicated variables.

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**Extra info for Function Theory in Several Complex Variables (Translations of Mathematical Monographs)**

**Example text**

5) is harmonic in each complex variable zJ but is not necessarily pluriharmonic in z = (zl,... ,z"). 4. Elementary Properties of Holomorphic Functions. We list some elementary properties of holomorphic functions of several complex variables which are proved by the same methods as in the case of one complex variable. 1. Liouville's theorem. e.. a holomorphic function in all of C". If If (z)j is bounded in C", then f (z) is a constant in C". More generally, let A,. : Izj I < r (j = 1,... n) and let M(r) = Max{If(z)I I z E &r}.

Vi) 96 0 at any point w in the punctured disk r \ {b}. We then fix r > 0 (depending on p) such that the closed polydisk A = a x r, where iz,-ail<_r lies in D and f (z, w) # 0 on 0 x or. 1. For any fixed z' in A. w) of the variable w has v zeros in r (counted with multiplicity). PROOF. For each z' E 0 we let v(z') denote the number of zeros (counted f (z', w) in F. By the argument with multiplicity) of the holomorphic function w principle, we have 1 vW) = 27ri Jar of(z'. w)/aw 37 du. 2. IMPLIC'IT rum- IONS AND ANALXTIC SETS 38 Since f (z.

Let f (z) be a non-constant holomorphic function on a domain D and let E be the zero set of f (--) in D: E= {zEDIf(z)=0}. Such sets will be discussed in Chapter 2. 6 (Riemann). 9(z) be a holomorphic function on D \ E. If g(z) is bounded in D \ E. then g(z) may be holomorphically extended to all of D. PROOF. Fix a E E. It suffices to prove that g(z) has a holomorphic extension to a neighborhood of a in D. By using a suitable linear transformation in C", we may assume that the section E' C Cz of E over zj = aj (j = 1....

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