The generalized homological Cauchy integral formula for \(\mathcal{A}\)-holomorphic functions

Abstract

Given a finite-dimensional commutative associative unital \(\mathbb{C}\)-algebra \(\mathcal{A}\) and \(U \subseteq \mathcal{A}\) open (in the underlying analytic topology), a (totally) differentiable function \(f \colon U \to \mathcal{A}\) is called \(\mathcal{A}\)-holomorphic if \(Df\) is also \(\mathcal{A}\)-linear. For the local case \(\mathcal{A} = (\mathcal{A},\mathfrak{m},\mathbb{C})\) we prove for this class of functions an analogue of the homological Cauchy Integral Formula, substantially strengthening a previous result of G. B. Rizza ([10], [11]). On the way to do so, we give a complete characterization of the corresponding analogue of the index (winding number) of (singular) 1-cycles in \(U\), thus improving on previous partial results and calculations of J. Edenhofer ([4]) and V. Shpakivskyi ([13]). We also prove (maximal) \(\mathcal{A}\)-analytic continuation of \(\mathcal{A}\)-holomorphic functions, thus in particular characterizing the domains of \(\mathcal{A}\)-holomorphy.