The Cauchy Integral Formula in Hermitian, Quaternionic and $$\mathfrak {osp}(4|2)$$ Clifford Analysis

Abstract

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space $${\mathbb {R}}^{2n}$$ of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on $${\mathbb {R}}^{2n}$$ , quaternionic Clifford analysis arose as a further refinement by introducing a so-called hypercomplex structure $${\mathbb {Q}}$$ , i.e. three complex structures ( $${\mathbb {I}}$$ , $${\mathbb {J}}$$ , $${\mathbb {K}}$$ ) which follow the quaternionic multiplication rules, on $${\mathbb {R}}^{4p}$$ , the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Cauchy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for $$\mathfrak {osp}(4|2)$$ -monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.