# n-times iterated Cauchy-Riemann operator

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!

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## 1 Answer

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?

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I suppose any further properties can be deduced from the one you gave without greater problems so yes, that's just about what I was looking for. Thanks! –  HeWhoHungers Jul 16 '11 at 19:19
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