n-times iterated Cauchy-Riemann operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:23:49Z http://mathoverflow.net/feeds/question/70515 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70515/n-times-iterated-cauchy-riemann-operator n-times iterated Cauchy-Riemann operator HeWhoHungers 2011-07-16T18:48:06Z 2011-07-16T19:05:02Z <p>Are there an results on functions annihilated by the n-times iterated Cauchy-Riemann operator ${\partial\over\partial\bar z}$, aka functions $f$ that for some $n\in\mathbb{N}$ satisfy the following equation? $${\partial^n f\over\partial\bar z^n}=0$$ EDIT: I have posted the same questions only a very short period of time ago, when my web browser suddenly froze and I didn't believe the question got posted successively, therefor posting it again here. If such repetition goes against any site rules or regulations, as I would suspect it does, may somebody in power to do so please remove one of the posts. Thank you!</p> http://mathoverflow.net/questions/70515/n-times-iterated-cauchy-riemann-operator/70516#70516 Answer by Robert Bryant for n-times iterated Cauchy-Riemann operator Robert Bryant 2011-07-16T19:05:02Z 2011-07-16T19:05:02Z <p>You can prove by induction that, if $f$ satisfies your equation, then there exist holomorphic functions $f_0,\ldots,f_{n-1}$ on the domain of $f$ such that $$f = f_0(z) + f_1(z)\ \bar z + \cdots + f_{n-1}(z)\ {\bar z}^{n-1}.$$ Is this the kind of answer you had in mind?</p>