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Everything about Transseries
Mathematical field
In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges, corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity and other similar asymptotic expansions. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: TransseriesA 5-minute orientation
- Skim a recent paper: Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseriesInfluential work since 2022
- Pick up a classic book: Asymptotic Differential Algebra and Model Theory of TransseriesThe most-reprinted book on the subject
- Read the landmark paper: A primer on resurgent transseries and their asymptoticsCited 228 times
- Go sideways: List of logarithmic identitiesA closely related topic
- Go sideways: Characterizations of the exponential functionA closely related topic
📖 Definitions
Noun A series (sequence of partial sums of a given sequence) in which members of the sequence from which the series is defined are defined by a transcendental function.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
A primer on resurgent transseries and their asymptotics
Nonperturbative Ambiguities and the Reality of Resurgent Transseries
Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local $${\mathbb{C}\mathbb{P}^2}$$ C P 2
Fast summation of divergent series and resurgent transseries from Meijer- G approximants
Resurgent Transseries and the Holomorphic Anomaly
Transseries for Beginners
The large proper-time expansion of Yang-Mills plasma as a resurgent transseries
Global flow structure and exact formal transseries of the Gubser flow in kinetic theory
🆕 Influential since 2022
Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
Exact quantization conditions and full transseries structures for P T symmetric anharmonic oscillators
Exact instanton transseries for quantum mechanics
Novel features of attractors and transseries in nonconformal Bjorken flows
The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries
📚 Books worth reading
Asymptotic Differential Algebra and Model Theory of TransseriesMatthias Aschenbrenner, Lou van den Dries · 2017
Transseries and Real Differential AlgebraJoris Van der Hoeven · 2006
Transseries in difference and differential equationsGeert Roelof Kuik · 2003- 📕Asymptotic Differential Algebra and Model Theory of Transseries : (ams-195)Matthias Aschenbrenner, Lou van den Dries · 2017
Transseries and Real Differential Algebra (Lecture Notes in Mathematics Book 1888)Joris van der Hoeven · 2006- 📕Équations différentielles et singularitésFelipe Cano Torres · 2009
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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