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Über dieses Buch

From the reviews: "... In sum, the volume under review is the first quarter of an important work that surveys an active branch of modern mathematics. Some of the individual articles are reminiscent in style of the early volumes of the first Ergebnisse series and will probably prove to be equally useful as a reference; ...for the appropriate reader, they will be valuable sources of information about modern complex analysis." Bulletin of the Am.Math.Society, 1991
"... This remarkable book has a helpfully informal style, abundant motivation, outlined proofs followed by precise references, and an extensive bibliography; it will be an invaluable reference and a companion to modern courses on several complex variables." ZAMP, Zeitschrift für Angewandte Mathematik und Physik, 1990



I. Remarkable Facts of Complex Analysis

The present article gives a short survey of results in contemporary complex analysis and its applications. The material presented is concentrated around several pivotal facts whose understanding enables one to have a general view of this area of analysis.
A. G. Vitushkin

II. The Method of Integral Representations in Complex Analysis

Let D be a domain in the complex plane ℂ1 with rectifiable boundary ∂D and f a complex valued function, continuous on \( \overline D \) together with its Cauchy–Riemann derivative:
$$ \frac{{\partial f}} {{\partial \overline z }} = \frac{1} {2}\left( {\frac{{\partial f}} {{\partial x}} + i\frac{{\partial f}} {{\partial y}}} \right),{\text{ }}z = x + iy. $$
G. M. Khenkin

III. Complex Analytic Sets

The theory of complex analytic sets is part of the modern geometric theory of functions of several complex variables. Traditionally, the presentation of the foundations of the theory of analytic sets is introduced in the algebraic language of ideals in Noetherian rings as, for example, in the books of Hervé [23] or Gunning-Rossi [19]. However, the modern methods of this theory, the principal directions and applications, are basically related to geometry and analysis (without regard to the traditional direction which is essentially related to algebraic geometry). Thus, at the beginning of this survey, the geometric construction of the local theory of analytic sets is presented. Its foundations are worked out in detail in the book of Gunning and Rossi [19] via the notion of analytic cover which together with analytic theorems on the removal of singularities leads to the minimum of algebraic apparatus necessary in order to get the theory started.
E. M. Chirka

IV. Holomorphic Mappings and the Geometry of Hypersurfaces

The principal topic of this paper is non-degenerate (in the sense of Levi) hypersurfaces of complex manifolds and the automorphisms of such hypersur-faces. The material on strictly pseudoconvex hypersurfaces is presented most completely. We discuss in detail a form of writing the equations of the hyper-surface which allows one to carry out a classification of hypersurfaces. Certain biholomorphic invariants of hypersurfaces are considered. Especially, we consider in detail a biholomorphically invariant family of curves called chains. A lot of attention is given to constructing a continuation of a holomorphic mapping.
A. G. Vitushkin

V. General Theory of Multidimensional Residues

Let X be a Riemann surface and let ω be a meromorphic differential form of degree 1 on X; in the neighborhood of a point where z is a local coordinate, we have ω = f(z)dz, where f is a meromorphic function. Let Y = {a l } lI be the set of poles of ω, wnd res a j (ω) the Cauchy residue of ω at a j . Then, for J a finite subset of I, if for any jJ, γ j is a positively oriented circle whose center is a j and if which γ j is the boundary does not meet Y in any point different from a j and if (n j ) jJ is a family of elements of ℤ, ℝ or ℂ, we have the residue formula.
P. Dolbeault


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