Real Analysis Exchange

Transseries for Beginners

G. A. Edgar

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From the simplest point of view, transseries concern manipulations on formal series, or 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---also known as the transline---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: dynamical systems, 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.

Article information

Real Anal. Exchange, Volume 35, Number 2 (2009), 253-310.

First available in Project Euclid: 22 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] 30A84

Hardy field ordered field differential field log-exp series Ecalle


Edgar, G. A. Transseries for Beginners. Real Anal. Exchange 35 (2009), no. 2, 253--310.

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