# Analytical properties of power series on Levi-Civita fields

Khodr Shamseddine[1]; Martin Berz[2]

• [1] Western Illinois University Department of Mathematics Macomb, IL 61455 USA
• [2] Michigan State University Department of Physics and Astronomy East Lansing, MI 48824 USA
• Volume: 12, Issue: 2, page 309-329
• ISSN: 1259-1734

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## Abstract

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A detailed study of power series on the Levi-Civita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and re-expandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable.

## How to cite

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Shamseddine, Khodr, and Berz, Martin. "Analytical properties of power series on Levi-Civita fields." Annales mathématiques Blaise Pascal 12.2 (2005): 309-329. <http://eudml.org/doc/10522>.

@article{Shamseddine2005,
abstract = {A detailed study of power series on the Levi-Civita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and re-expandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable.},
affiliation = {Western Illinois University Department of Mathematics Macomb, IL 61455 USA; Michigan State University Department of Physics and Astronomy East Lansing, MI 48824 USA},
author = {Shamseddine, Khodr, Berz, Martin},
journal = {Annales mathématiques Blaise Pascal},
keywords = {convergence criteria},
language = {eng},
month = {7},
number = {2},
pages = {309-329},
publisher = {Annales mathématiques Blaise Pascal},
title = {Analytical properties of power series on Levi-Civita fields},
url = {http://eudml.org/doc/10522},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Shamseddine, Khodr
AU - Berz, Martin
TI - Analytical properties of power series on Levi-Civita fields
JO - Annales mathématiques Blaise Pascal
DA - 2005/7//
PB - Annales mathématiques Blaise Pascal
VL - 12
IS - 2
SP - 309
EP - 329
AB - A detailed study of power series on the Levi-Civita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and re-expandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable.
LA - eng
KW - convergence criteria
UR - http://eudml.org/doc/10522
ER -

## References

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16. K. Shamseddine, M. Berz, Intermediate values and inverse functions on non-archimedean fields, International Journal of Mathematics and Mathematical Sciences 30 (2002), 165-176 Zbl0996.26020MR1905419
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19. K. Shamseddine, New elements of analysis on the Levi-Civita field, (1999), East Lansing, Michigan, USA

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