A Glimpse of Several Complex Variables

Functions of Several Complex Variables At a naive level, the analysis of several real variables is much like the real analysis of one variable; the main difference is that one deals with n-tuples of reals instead of scalars and one needs matrices to keep track of information. Of course deeper study...

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Veröffentlicht in:	Complex Analysis S. 161 - 188
1. Verfasser:	Steven G. Krantz
Format:	Buchkapitel
Sprache:	Englisch
Veröffentlicht:	Washington DC The Mathematical Association of America 2004 Mathematical Association of America
Ausgabe:	2
Schlagworte:	Analytic functions Complex variables Mathematical expressions Mathematical functions Mathematical values Mathematical variables Mathematics Pure mathematics Real and complex analysis Real variables Topological theorems Topology distance-decreasing property of holomorphic mappings normal family Hartogs domain Riemann mapping theorem holomorphic function Riemann's philosophy unitary domain of convergence of a power series biholomorphisms of complete circular domains differential ball Weierstrass's theorem Jacobian matrix set of uniqueness for a holomorphic function automorphism group biholomorphic mapping several complex variables complex Osgood's theorem Carathéodory, Constantin Euclidean invariance Liouville's theorem Cartan's theorem Jacobian matrix complex Moser, Jurgen complete circular domain Hartogs function of two complex variables Kobayashi, S neighborhood in two complex variables Poincaré, Henri function of two complex variables, continuity of bidisc invariance of the Carathéodory indicatrix Schwarz lemma Bell, Steven R Levi, E.E unitary biholomorphisms metric Fefferman, Charles distance-decreasing property domain Chern, S. S Cauchy
ISBN:	9780883850350, 0883850354
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	Functions of Several Complex Variables At a naive level, the analysis of several real variables is much like the real analysis of one variable; the main difference is that one deals with n-tuples of reals instead of scalars and one needs matrices to keep track of information. Of course deeper study reveals much complexity and ri chness in analysis of several real variables. It is noteworthy that, for the most part, this richness was discovered rather late in the history of the subject-mostly in the last fifty years. The history of analysis of several complex variables is quite different. Early in the subject, in the first decade of the twentieth century, two remarkable discoveries indicated that this area has an incredible depth and variety which one complex variable does not even hint at. Even today we have only scratched the surface of several complex variables. Let us briefly discuss the two developments which established several complex variables as a subject in its own right. The first is related to the Riemann mapping theorem. As we have discussed throughout the present volume, Riemann's theorem asserts that, with the single exception of the plane, any domain topologically equivalent to the disc is conformally equivalent to it. One might expect an analogue to this result in two complex variables.
ISBN:	9780883850350 0883850354
DOI:	10.5948/UPO9780883859681.008