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 formulæ 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.

与复杂平面上的全纯函数理论一样，柯西积分公式已证明是高维欧氏空间中的单基因函数理论Clifford分析的基石。近年来，出现了Clifford分析的几个新分支。类似于偶数维欧空间\（{\ mathbb {R}} ^ {2n} \）中的Hermitian Clifford分析如何通过在\（{\ mathbb {R} } ^ {2n} \），通过引入所谓的超复杂结构\（{\ mathbb {Q}} \），即三个复杂结构（\（{\ mathbb {I}} \），\（{\ mathbb {J}} \），\（{\ mathbb {K}} \））遵循四元数乘法规则，位于\（{\ mathbb {R}} ^ {4p} \）上，尺寸现在是四倍。在相应的对称群U（n）和Sp（p）的作用下，两个微分算子分别导致一阶系统不变。它们同时存在的零解分别称为埃尔米特单基因函数和四元离子单基因函数。在这一贡献中，我们进一步阐述了关于厄米和四元单基因功能的柯西积分公式。此外，我们为\（\ mathfrak {osp}（4 | 2）\）建立Caychy积分公式-单基因函数，Clifford分析的最新分支，通过充分考虑潜在的辛对称性来完善四元离子的单基因性。