Logarithmic-exponential series

Lou Van Den Dries, Angus MacIntyre, David Marker

Research output: Contribution to journalArticlepeer-review


We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of "logarithmic-exponential series" (LE-series), which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered.

Original languageEnglish (US)
Pages (from-to)61-113
Number of pages53
JournalAnnals of Pure and Applied Logic
Issue number1-2
StatePublished - Jul 20 2001

ASJC Scopus subject areas

  • Logic


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