Analytical properties of power series on Levi-Civita fields
Annales mathématiques Blaise Pascal, Tome 12 (2005) no. 2, pp. 309-329.

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.

DOI : 10.5802/ambp.209
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
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Khodr Shamseddine; Martin Berz. Analytical properties of power series on Levi-Civita fields. Annales mathématiques Blaise Pascal, Tome 12 (2005) no. 2, pp. 309-329. doi : 10.5802/ambp.209. https://ambp.centre-mersenne.org/articles/10.5802/ambp.209/

[1] N. L. Alling Foundations of analysis over surreal number fields, North Holland, 1987 | MR | Zbl

[2] M. Berz Analysis on a nonarchimedean extension of the real numbers (1994) no. MSUCL-933 (Lecture Notes, 1992 and 1995 Mathematics Summer Graduate Schools of the German National Merit Foundation.)

[3] M. Berz; M. Berz; C. Bischof; G. Corliss; A. Griewank Calculus and numerics on Levi-Civita fields, Computational Differentiation: Techniques, Applications, and Tools (1996), pp. 19-35 | MR | Zbl

[4] M. Berz Cauchy Theory on Levi-Civita fields, Contemporary Mathematics, American Mathematical Society, Volume 319 (2003), pp. 39-52 | MR | Zbl

[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), pp. 21-34 | MR | Zbl

[6] W. Krull Allgemeine Bewertungstheorie, J. Reine Angew. Math., Volume 167 (1932), pp. 160-196 | DOI | Zbl

[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)

[8] T. Levi-Civita Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti Ist. Veneto di Sc., Lett. ed Art., Volume 7a, 4 (1892), pp. 1765

[9] T. Levi-Civita Sui numeri transfiniti, Rend. Acc. Lincei, Volume 5a, 7 (1898), pp. 91-113

[10] L. Neder Modell einer Leibnizschen Differentialrechnung mit aktual unendlich kleinen Größen, Mathematische Annalen, Volume 118 (1941-1943), pp. 718-732 | DOI | MR | Zbl

[11] W. F. Osgood Functions of real variables, G. E. Stechert & CO., New York, 1938 | Zbl

[12] S. Priess-Crampe Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, Springer, Berlin, 1983 | MR | Zbl

[13] P. Ribenboim Fields: algebraically closed and others, Manuscripta Mathematica, Volume 75 (1992), pp. 115-150 | DOI | MR | Zbl

[14] W. H. Schikhof Ultrametric calculus: an introduction to p-adic analysis, Cambridge University Press, 1985 | MR | Zbl

[15] K. Shamseddine; M. Berz; M. Berz; C. Bischof; G. Corliss; A. Griewank Exception handling in derivative computation with non-archimedean calculus, Computational Differentiation: Techniques, Applications, and Tools (1996), pp. 37-51 | MR | Zbl

[16] K. Shamseddine; M. Berz Intermediate values and inverse functions on non-archimedean fields, International Journal of Mathematics and Mathematical Sciences, Volume 30 (2002), pp. 165-176 | DOI | MR | Zbl

[17] K. Shamseddine; M. Berz Measure theory and integration on the Levi-Civita field, Contemporary Mathematics, Volume 319 (2003), pp. 369-387 | MR | Zbl

[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), pp. 283-299 | MR | Zbl

[19] K. Shamseddine New elements of analysis on the Levi-Civita field, Michigan State University (1999) (Ph. D. Thesis)

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