On primitives of monogenic functions

Abstract

Let Ω be a suitable open subset of Euclidean space , let D_x 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 P_k  ∈ M ⁺ (k) a particular primitive P _k+1 ∈ M ⁺ (k+1) of P_k is explicitly constructed.

†Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Keywords:

• Monogenic functions
• Primitives
• Cauchy–Riemann operator

Keywords:

• AMS 2000 Mathematics Subject Classification: 30G35

Notes

†Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.