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.
@article{AMBP_2005__12_2_309_0,
author = {Shamseddine, Khodr and Berz, Martin},
title = {Analytical properties of power series on {Levi-Civita} fields},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {309--329},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {12},
number = {2},
year = {2005},
doi = {10.5802/ambp.209},
zbl = {1087.26020},
mrnumber = {1760545},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ambp.209/}
}
TY - JOUR AU - Shamseddine, Khodr AU - Berz, Martin TI - Analytical properties of power series on Levi-Civita fields JO - Annales mathématiques Blaise Pascal PY - 2005 SP - 309 EP - 329 VL - 12 IS - 2 PB - Annales mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.209/ DO - 10.5802/ambp.209 LA - en ID - AMBP_2005__12_2_309_0 ER -
%0 Journal Article %A Shamseddine, Khodr %A Berz, Martin %T Analytical properties of power series on Levi-Civita fields %J Annales mathématiques Blaise Pascal %D 2005 %P 309-329 %V 12 %N 2 %I Annales mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.209/ %R 10.5802/ambp.209 %G en %F AMBP_2005__12_2_309_0
Shamseddine, Khodr; Berz, Martin. 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. http://www.numdam.org/articles/10.5802/ambp.209/
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