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 (or in every case if using special semantics such as through infinite surreal numbers), corresponds to actual behavior.
Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity (
was introduced independently by Dahn-Göring[1] and Ecalle[2] in the respective contexts of model theory or exponential fields and of the study of analytic singularity and proof by Ecalle of the Dulac conjectures.
enjoys a rich structure: an ordered field with a notion of generalized series and sums, with a compatible derivation with distinguished antiderivation, compatible exponential and logarithm functions and a notion of formal composition of series.
Informally speaking, exp-log transseries are well-based (i.e. reverse well-ordered) formal Hahn series of real powers of the positive infinite indeterminate
Two important additional conditions are that the exponential and logarithmic depth of an exp-log transseries
that is the maximal numbers of iterations of exp and log occurring in
of the ordered exponential field of real numbers are all comparable: For all such
The field of transseries can be intuitively viewed as a formal generalization of these growth rates: In addition to the elementary operations, transseries are closed under "limits" for appropriate sequences with bounded exponential and logarithmic depth.
However, a complication is that growth rates are non-Archimedean and hence do not have the least upper bound property.
We can address this by associating a sequence with the least upper bound of minimal complexity, analogously to construction of surreal numbers.
Because of the comparability, transseries do not include oscillatory growth rates (such as
that do not directly correspond to convergent series or real valued functions.
Another limitation of transseries is that each of them is bounded by a tower of exponentials, i.e. a finite iteration
[3] Transseries can be defined as formal (potentially infinite) expressions, with rules defining which expressions are valid, comparison of transseries, arithmetic operations, and even differentiation.
Appropriate transseries can then be assigned to corresponding functions or germs, but there are subtleties involving convergence.
Even transseries that diverge can often be meaningfully (and uniquely) assigned actual growth rates (that agree with the formal operations on transseries) using accelero-summation, which is a generalization of Borel summation.
The sum might be infinite or transfinite; it is usually written in the order of decreasing
Comparison of monic transmonomials: Multiplication: This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.
given by the leading monic transmonomial, and the corresponding asymptotic relation defined for
we will define a linearly ordered multiplicative group of monomials
with well-based (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series).
of purely large transseries, which are series whose support contains only monomials lying strictly above
: The field of log-free transseries is equipped with an exponential function which is a specific morphism
is equipped with Gonshor-Kruskal's exponential and logarithm functions[5] and with its natural structure of field of well-based series under Conway normal form.
(this is linked to the fact that those fields contain no transexponential function).
is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.): In this theory, exponentiation is essentially defined for functions (using differentiation) but not constants; in fact, every definable subset of
(This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of
[10] Logarithmic-transseries do not themselves correspond to a maximal Hardy field for not every transseries corresponds to a real function, and maximal Hardy fields always contain transexponential functions.