Cauchy Theory on Levi-civita Fields

We develop the basic elements of a Cauchy theory on the complex Levi-Civita field, which constitutes the smallest algebraically closed nonArchimedean extension of the complex numbers. We introduce a concept of analyticity based on differentiation, and show that it leads to local expandability in power series. We show that analytic functions can be integrated over suitable piecewise smooths paths in the sense of integrals developed in an accompanying paper. It is then shown that the resulting path integrals allow the formulation of a workable Cauchy theory in a rather similar way as in the conventional case. In particular, we obtain a Cauchy theorem and the Cauchy formula for analytic functions.