By Marek Jarnicki

The tale of individually holomorphic features all started approximately a hundred years in the past. through the moment half the nineteenth century, it grew to become recognized individually non-stop functionality isn't really inevitably non-stop as a functionality of all variables. before everything of the 20 th century, the examine of individually holomorphic capabilities began because of the basic paintings of Osgood and Hartogs. This ebook offers the 1st self-contained and entire presentation of the examine of individually holomorphic features, from its beginnings to present examine. many of the effects provided have by no means been released earlier than in publication shape. The textual content is split into components. the 1st half offers with individually holomorphic services, "without singularities". the second one half addresses the placement of current singularities. A dialogue of the classical effects regarding individually holomorphic features ends up in the main basic consequence, the classical move theorem in addition to quite a few extensions and generalizations, to extra advanced "crosses". also, a number of purposes for different sessions of "separately typical" features are given. a superior heritage in simple advanced research is a prerequisite. To make the booklet self contained, all of the effects are accumulated in specified introductory chapters and stated at the start of every part. This e-book is addressed to scholars and researchers in different advanced variables in addition to mathematicians and theoretical physicists attracted to this zone of arithmetic. A book of the eu Mathematical Society (EMS). dispensed in the Americas by way of the yankee Mathematical Society.

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The function Œ0; 2 /n 3 Â 7! a; r/ ! a;r/ . r12 / : : : . 7. ˝/, a 2 ˝. 8. X /. If u1 Ä u2 almost everywhere in X , then u1 Ä u2 everywhere. 9. uI a; r/: ˝, r 2 Rn>0 . 10. X /, and u 6Á integrable; in particular, the set u 1 . 1/ is of zero measure. 11. supi2I ui / is psh in X . y/, x 2 X . 12. 1 u / is psh on X . 13 (Hartogs’ lemma for plurisubharmonic functions). 14 (Regularization). z/ ´ 1 z ˚. w/; Dn The function u" is called the "-regularization of u. 15. X /, u 6Á u" & u pointwise in X when " & 0.

Cn /). e. Cn /). Cn / such that X is a relatively compact open set in X 0 and p D p 0 jX . 2. Cn /. This is the standard identification of open sets in Cn with Riemann regions. Cn /, then p is an open mapping. X / is open in Cn . a/ is a discrete subset of X . U; pjU /U , where U runs over all univalent open subsets of X , introduces on X an atlas of an n-dimensional complex manifold. y//. X /. Then Y D X . Cn /. Consequently, a Riemann region is countable at infinity iff it has an at most countable number of connected components.

1. Cn /. X; p/ is a Riemann–Stein domain; 54 2 Prerequisites D Dp D x ! X ´ X (cf. Œ0; 1/ D/ D D (iii) for any continuous mapping f W Œ0; 1 D ! X for which the mapping p B f extends to a holomorphic mapping g W D D ! Dn 1 A. ; 1//, 0 < r; < 1, any biholomorphic mapping f W T ! T / X extends to a holomorphic mapping fO W Dn ! X; p/ ! Y; q/ such that for any domain T D Tr; and a biholomorphic mapping f W T ! T / X such that ' B f extends to a biholomorphic mapping gy W Dn ! Dn / Y , there exists a holomorphic mapping fO W Dn !