From: "David C. Ullrich" Subject: Re: Simple several complex variables question Date: Thu, 24 Feb 2000 12:52:32 -0600 Newsgroups: sci.math Summary: 'Analytic in each variable' implies analytic Mike McCarty wrote: > In article <88vktu$n8l$1@nntp8.atl.mindspring.net>, > Daniel Giaimo wrote: > ) Is it possible for a map f:C^2->C to be holomorphic in each variable > )separately, but fail to be holomorphic? Can it fail to be continuous? Of > > Please define "holomorphic" as applied to functions of two complex > variables. You can find a definition in any book on several complex variables. There are lots of versions of the definition, all equivalent - possibly the apparently strongest one is "has a convergent power series representation about each point", the weakest one that looks like it has a chance of being equivalent is "continuous and holomorphic in each variable separately". And (incredibly) "holomorphic in each variable separately" is also equivalent. When you think about how extremely simple it is to exhibit a discontinuous function in R^2 that's separately differentiable it's clear that the last can't be right, but it is. > > )course, if C is replaced by R then the answer to both is yes, but I don't > )think(?) that the standard examples will extend to C^2 in a holomorphic way. > ) > )--Daniel Giaimo > ) > ) > > -- > ---- > char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);} > This message made from 100% recycled bits. > I don't speak for Alcatel <- They make me say that.