By Takeo Ohsawa

ISBN-10: 0821820982

ISBN-13: 9780821820988

One of many ways to the learn of features of numerous complicated variables is to take advantage of equipment originating in genuine research. during this concise ebook, the writer offers a lucid presentation of the way those equipment produce quite a few international lifestyles theorems within the conception of capabilities (based at the characterization of holomorphic features as susceptible recommendations of the Cauchy-Riemann equations). Emphasis is on fresh effects, together with an $L^2$ extension theorem for holomorphic capabilities, that experience introduced a deeper knowing of pseudoconvexity and plurisubharmonic capabilities. according to Oka's theorems and his schema for the grouping of difficulties, the e-book covers subject matters on the intersection of the speculation of analytic features of a number of variables and mathematical research. it's assumed that the reader has a easy wisdom of advanced research on the undergraduate point. The e-book could make a very good supplementary textual content for a graduate-level direction on advanced research.

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**Additional info for Analysis of Several Complex Variables**

**Sample text**

V(,, z) - m Sup K-zI

As preparation for this, we describe below the regularization of plurisubharmonic functions. 2. Let be a plurisubharmonic function that is locally integrable on fl, and set E(z) := f 'Y(z + ()µe(()dV( for z E f? ^ Clearly, & E C°°(QE). 9. lE), and pE \ 0 as e \ 0. PROOF. The monotonicity of the family of functions ,0E follows Y'E is shown as from that of Ill (i , t). The plurisubharmonicity of 42 3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS follows: 1 2. 27r 1 J0 = te(z + re'Bw)dO r ;ii (()dV0(z + re0w - e()dO J f27T e()dV( where r is chosen to be sufficiently small.

Take a sequence a j of points in D that converges to a and satisfies aj E Dj+1 \Dj. Let mad (z) be such a monomial as above that is determined for each domain Dj and point aj. 15) sup Ickmak (Z) "k I < zEDk Then the series 1 . 2k 00 E cjma, (z)-' j=1 converges uniformly on compact sets in D. 6 implies F E A(D). 15), we clearly have IF(ak)I > k - 1 for each k. This results in lim IF(z)j = oo. 3. " General open sets that possess this property will be discussed later. It seems interesting, at this point, to study the convergence m range of a series > fk whose terms are homogeneous polynomials fk k=0 of degree k as a slight generalization of a power series, but we do not do so in this book.

### Analysis of Several Complex Variables by Takeo Ohsawa

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