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

Annales mathématiques Blaise Pascal (2005)

  • Volume: 12, Issue: 2, page 309-329
  • ISSN: 1259-1734

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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  1. N. L. Alling, Foundations of analysis over surreal number fields, (1987), North Holland Zbl0621.12001MR886475
  2. M. Berz, Analysis on a nonarchimedean extension of the real numbers, (1994) 
  3. M. Berz, Calculus and numerics on Levi-Civita fields, Computational Differentiation: Techniques, Applications, and Tools (1996), 19-35, SIAM, Philadelphia Zbl0878.65013MR1431039
  4. M. Berz, Cauchy Theory on Levi-Civita fields, Contemporary Mathematics, American Mathematical Society 319 (2003), 39-52 Zbl1045.12006MR1977437
  5. M. Berz, Analytical and Computational Methods for the Levi-Civita fields, Lecture Notes in Pure and Applied Mathematics (Proceedings of the Sixth International Conference on P-adic Analysis, July 2-9, 2000, ISBN 0-8247-0611-0), 21-34, Marcel Dekker Zbl1001.12009MR1838279
  6. W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196 Zbl0004.09802
  7. D. Laugwitz, Tullio Levi-Civita’s Work on Nonarchimedean Structures (with an Appendix: Properties of Levi-Civita Fields), Atti Dei Convegni Lincei 8: Convegno Internazionale Celebrativo Del Centenario Della Nascita De Tullio Levi-Civita (1975), Academia Nazionale dei Lincei, Roma 
  8. T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti Ist. Veneto di Sc., Lett. ed Art. 7a, 4 (1892) 
  9. T. Levi-Civita, Sui numeri transfiniti, Rend. Acc. Lincei 5a, 7 (1898), 91-113 
  10. L. Neder, Modell einer Leibnizschen Differentialrechnung mit aktual unendlich kleinen Größen, Mathematische Annalen 118 (1941-1943), 718-732 Zbl0027.38904MR10180
  11. W. F. Osgood, Functions of real variables, (1938), G. E. Stechert & CO., New York Zbl0087.26801
  12. S. Priess-Crampe, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, (1983), Springer, Berlin Zbl0558.51012MR704186
  13. P. Ribenboim, Fields: algebraically closed and others, Manuscripta Mathematica 75 (1992), 115-150 Zbl0767.12001MR1160093
  14. W. H. Schikhof, Ultrametric calculus: an introduction to p-adic analysis, (1985), Cambridge University Press Zbl0553.26006MR791759
  15. K. Shamseddine, M. Berz, Exception handling in derivative computation with non-archimedean calculus, Computational Differentiation: Techniques, Applications, and Tools (1996), 37-51, SIAM, Philadelphia Zbl0878.65014MR1431040
  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
  17. K. Shamseddine, M. Berz, Measure theory and integration on the Levi-Civita field, Contemporary Mathematics 319 (2003), 369-387 Zbl1130.12301MR1977457
  18. K. Shamseddine, M. Berz, Convergence on the Levi-Civita field and study of power series, Lecture Notes in Pure and Applied Mathematics (Proceedings of the Sixth International Conference on P-adic Analysis, July 2-9, 2000, ISBN 0-8247-0611-0), 283-299, Marcel Dekker Zbl0985.26014MR1838300
  19. K. Shamseddine, New elements of analysis on the Levi-Civita field, (1999), East Lansing, Michigan, USA 

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