Convergence on the Levi-Civita Field and Study of Power Series
AbstractConvergence under various topologies and analytical properties of power
series on Levi-Civita fields are studied. A radius of convergence is
established that asserts convergence under a weak topology and reduces to
the conventional radius of convergence for real power series. It also
asserts strong (order) convergence for points the distance of which from the
center is infinitely smaller than the radius of convergence.
In addition to allowing the introduction of common transcendental functions,
power series are shown to behave similar to real power series. Besides being
infinitely often differentiable and re-expandable around other points, it is
shown that power series satisfy a general intermediate value theorem as well
as a maximum theorem and a mean value theorem.
K. Shamseddine, M. Berz, Lecture Notes in Pure and Applied Mathematics 222 (2000) 283-299
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