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From the reviews of the first printing, published as Volume 7 of the Encyclopaedia of Mathematical Sciences: "...... In this volume, we find an introductory essay entitled "Remarkable Facts of Complex Analysis" by Vitushkin... This is followed by articles by G.M.Khenkin on integral formulas in complex analysis, by E.M.Chirka on complex analytic sets, by Vitushkin on the geometry of hypersurfaces and by P.Dolbeault, on the theory of residues in several variables. ... 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; all contain substantial lists of references." Bulletin of the American Mathematical 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 "... comprehensive and authoritative survey of results in contemporary mathematics, indications of the directions of its future development, the material presented around pivotal facts whose understanding enables one to have a general view of the area, no proofs or only the outline of proofs, and extensive bibliographies. ... Browsing through this collection of surveys gives one a feeling of awe and admiration. Truly, complex analysis is vigorously alive. ... They are highly recommended to everyone with an interest in complex analysis." Medelingen van het wiskundig genootshap, 1992.

Complex analysis --- 517.53 --- Functions of a complex variable --- Mathematical analysis. --- Functions of complex variables. --- Engineering & Applied Sciences --- Applied Mathematics --- 517.53 Functions of a complex variable --- Functions of several complex variables --- Fonctions de plusieurs variables complexes --- Analysis (Mathematics). --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables

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In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten­ sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re­ spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials.

Potentiaaltheorie --- Potential theory (Mathematics) --- Potentiel [Théorie du ] --- 517.53 --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functions of a complex variable --- 517.53 Functions of a complex variable --- Mathematical analysis. --- Analysis (Mathematics). --- Applied mathematics. --- Engineering mathematics. --- Potential theory (Mathematics). --- Mathematical physics. --- Functions of complex variables. --- Analysis. --- Mathematical and Computational Engineering. --- Applications of Mathematics. --- Potential Theory. --- Theoretical, Mathematical and Computational Physics. --- Functions of a Complex Variable. --- Complex variables --- Elliptic functions --- Functions of real variables --- Physical mathematics --- Physics --- Engineering --- Engineering analysis --- 517.1 Mathematical analysis --- Mathematics