By Toshio Nishino

``Kiyoshi Oka, first and foremost of his examine, looked the gathering of difficulties which he encountered within the examine of domain names of holomorphy as huge mountains which separate this day and the next day. hence, he believed that there may be no crucial development in research with out mountain climbing over those mountains ... this e-book is a priceless preliminary step for the reader that allows you to comprehend the mathematical international which was once created through Kiyoshi Oka.'' --from the Preface This e-book explains leads to the idea of features of a number of complicated variables that have been ordinarily demonstrated from the overdue 19th century throughout the heart of the 20th 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 elements. the 1st is the learn of analytic features in univalent domain names in ${\mathbf C}^n$. the following Oka proved that 3 ideas are similar: 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, whilst acknowledged thoroughly, could be solved in domain names of holomorphy enjoyable the perfect stipulations. the second one a part of Oka's paintings confirmed a mode for the learn of analytic capabilities outlined in a ramified area over ${\mathbf C}^n$ within which the department issues are regarded as inside issues of the area. the following analytic services in an analytic area are taken care of, that's a moderate generalization of a ramified area over ${\mathbf C}^n$. In writing the e-book, the author's aim used to be to carry to readers a true realizing of Oka's unique papers. This quantity is an English translation of the unique eastern version, released through the collage of Tokyo Press (Japan). it should make an appropriate path textual content for complex graduate point introductions to a number of advanced variables.

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**Example text**

OLONIORPHIC FUNCTIONS from the maximum principle for harmonic functions that for any y > 0. log If(z)I 5 u(z) - u(--)< -i log If(z)I, z E A. Hence u(z) = u(z) on w. Similarly, we have i(z) = v(z) in w for u(z) = 3'f(z). We set f (z) := 11(z) + i%(z) in A. Since f (z) = f (z) in w. it follows that f (z) is holomorphic in A. On the other hand, both f(z) and f(z) are continuous on A and the zero set of f (z) is isolated in ,: hence. e is isolated and f (z) f (z) in J. 5. Holomorphic Mappings. We let z = (z,.....

Therefore w(z,w) is a holomorphic function in A and w(z, w) 0 0 in 0° x r. 1 that w(z, w) 34 0 on A. Summarizing these results, we have the following theorem.

Contrary to the case of one complex variable, there exist domains D in C". n > 1, with C" \D having non-empty interior but such that every bounded holomorphic function in D is constant. 2) that the complement of a ball has this property. 1. HOLO\MORPHIC FUNCTIONS AND DOMAINS OF HOLOMORPHY 16 2. Identity theorem. Let f (z) and gg(z) be holomorphic functions in a domain D in C". If f (z) = g(z) for all z in a non-empty open set 6 in D. then f (z) __ g(z) in D. Hence, analytic continuation of holomorphic functions in several complex variables can be performed as in the case of one complex variable.