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We study in detail the one-variable local theory of functions holomorphicover a finite-dimensional commutative associative unital $\mathbb{C}$-algebra$\mathcal{A}$, showing that it shares a multitude of features with theclassical one-variable Complex Analysis, including the validity of the Jacobianconjecture for $\mathcal{A}$-holomorphic regular maps and a generalizedHomological Cauchy's Integral Formula. In fact, in doing so we replace$\mathcal{A}$ by a morphism $\varphi: \mathcal{A} \to \mathcal{B}$ in thecategory of finite-dimensional commutative associative unital$\mathbb{C}$-algebras in a natural manner, paving a way to establishing anappropriate category of Funktionentheorien (ger. function theories). We alsotreat the very instructive case of non-unital finite-dimensional commutativeassociative $\mathbb{R}$-algebras as far as it serves above agenda.

Functions Holomorphic over Finite-Dimensional Commutative Associative Algebras 1: One-Variable Local Theory I