Cauchy's integral theorem on a finitely generated, real, commutative, and associative algebra (Q1376312)

Let \(R[\alpha] =[\alpha_1, \dots, \alpha_n]\) be a real, unitary, finitely generated, commuative, and associative algebra. The author develops machinery for doing classical analysis on \(R[\alpha]\). In particular, the notions of ``continuous function'', ``derivative'', and ``Riemann integral'' are defined. As a consequence of these constructs, the author is able to prove a Cauchy integral theorem: \(\oint_\gamma f(z) dz=0\) for suitable functions \(f\) and for closed curves \(\gamma\).