On primitives of monogenic functions†

Abstract

Let Ω be a suitable open subset of Euclidean space , let Dx be the Cauchy–Riemann operator in and let M(Ω) be the space of left monogenic functions in Ω. Then it is shown that for each f ∈ M(Ω) there exists f ∈ M(Ω) such that , i.e. f admits a left monogenic primitive F in Ω. Furthermore, if for , M + (k) denotes the space of left monogenic homogeneous polynomials of degree k in , then for each Pk ∈ M + (k) a particular primitive P k+1 ∈ M + (k+1) of Pk is explicitly constructed. †Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.