Transseries for beginners

By Gerald A. Edgar. 2008

79 citations recorded by the source corpus.

From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. This paper is an exposition for the non-specialist mathematician. All a mathematician needs to know in order to apply transseries.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/df4714c38ced1fb0bb4de65fb933c4259713f64d

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Citations in the corpus

• The Resurgence of Instantons in String Theory (2011, background): “More general transseries may be constructed, with S(z) a transseries itself possibly further involving compositions with exponentials or with logarithms, but we refer to [45] for a complete discussion.”
• A primer on resurgent transseries and their asymptotics (2018, background): “The fact that the definitions above only require “formal sums” naturally leads to rather simple constructions, as we are now allowed to do essentially any formal (nonlinear) operations on transseries we may wish, from algebraic to differential to integral, without worrying about any convergence issu”