# 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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