In this article, we show that the Cauchy integral formula for a monogenic function \(f: {\mathbb {H}}\longrightarrow {\mathbb {H}}\) for which \(\text{ Im } f\subset {\mathbb {C}}\subset {\mathbb {H}}\) turns out to be the Bochner–Martinelli integral formula for an associated holomorphic functions \(g: {\mathbb {C}}^2\longrightarrow {\mathbb {C}}\). To this end, we need to interpret the holomorphic self-mapping of \({\mathbb {C}}^2\) as a monogenic functions \({\mathbb {H}}\rightarrow {\mathbb {H}}\) annihilated by a pair of Cauchy–Fueter type operators. We also need a concise version of the chain rule for quaternions as well as the explicit formulas among the various inner products in quaternions.

在本文中，我们展示了单基因函数\(f: {\mathbb {H}}\longrightarrow {\mathbb {H}}\)的柯西积分公式，其中\(\text{ Im } f\subset { \mathbb {C}}\subset {\mathbb {H}}\)是关联全纯函数的 Bochner-Martinelli 积分公式\(g: {\mathbb {C}}^2\longrightarrow {\mathbb {C}}\)。为此，我们需要将\({\mathbb {C}}^2\)的全纯自映射解释为单基因函数\({\mathbb {H}}\rightarrow {\mathbb {H}}\ )被一对 Cauchy–Fueter 类型算子消灭。我们还需要四元数链式法则的简明版本，以及四元数中各种内积之间的显式公式。