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Analytical and Computational Methods for the Levi-Civita Field
Abstract
A field extension R of the real numbers is presented. It has similar algebraic properties as R; for example, all roots of positive numbers exist, and the structure C obtained by adjoining the imaginary unit is algebraically complete. The set can be totally ordered and contains infinitely small and infinitely large quantities. Under the topology induced by the ordering, the set is Cauchy complete, and it is shown that R is the smallest totally ordered algebraically and Cauchy complete extension of R. Furthermore, There is a natural way to extend any other real function under preservation of its smoothness properties, and as shown in an accompanying paper, power series have identical convergence properties as in R. In addition to these common functions, delta functions can be introduced directly. A calculus involving continuity, differentiability and integrability is developed. Central concepts like the intermediate value theorem and Rolle's theorem hold under slightly stronger conditions. It is shown that, up to infinitely small errors, derivatives are differential quotients, i.e. slopes of infinitely small secants. While justifying the intuitive concept of derivatives of the fathers of analysis, it also offers a practical way of calculating exact derivatives numerically.
M. Berz, Lecture Notes in Pure and Applied Mathematics 222 (2000) 21-34
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