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 (or in every case if using special semantics such as through infinite surreal numbers), 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.
The field was introduced independently by Dahn-GöringDahn, Bernd and Göring, Peter, Notes on exponential-logarithmic terms, Fundamenta Mathematicae, 1987 and EcalleEcalle, Jean, Introduction aux fonctions analyzables et preuve constructive de la conjecture de Dulac, Actualités mathématiques (Paris), Hermann, 1992 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. It constitutes a formal object, extending the field of exp-log functions of Hardy and the field of accelerando-summable series of Ecalle.
The field 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.
Examples and counter-examples
Informally speaking, exp-log transseries are well-based (i.e. reverse well-ordered) formal Hahn series of real powers of the positive infinite indeterminate , exponentials, logarithms and their compositions, with real coefficients. 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 must be finite.
The following formal series are log-exp transseries:
The following formal series are not log-exp transseries:
— this series is not well-based. — the logarithmic depth of this series is infinite — the exponential and logarithmic depths of this series are infinite
It is possible to define differential fields of transseries containing the two last series; they belong respectively to and (see the paragraph Using surreal numbers below).
Introduction
A remarkable fact is that asymptotic growth rates of elementary nontrigonometric functions and even all functions definable in the model theoretic structure of the ordered exponential field of real numbers are all comparable: For all such and , we have or , where means . The equivalence class of under the relation is the asymptotic behavior of , also called the germ of (or the germ of at infinity).
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. For example, is associated with rather than because decays too quickly, and if we identify fast decay with complexity, it has greater complexity than necessary (also, because we care only about asymptotic behavior, pointwise convergence is not dispositive).
Because of the comparability, transseries do not include oscillatory growth rates (such as ). On the other hand, there are transseries 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 of , thereby excluding tetration and other transexponential functions, i.e. functions which grow faster than any tower of exponentials. There are ways to construct fields of generalized transseries including formal transexponential terms, for instance formal solutions of the Abel equation .Schmeling, Michael, Corps de transséries, PhD thesis, 2001
Formal construction
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.
Transseries can be formalized in several equivalent ways; we use one of the simplest ones here.
A transseries is a well-based sum,
with finite exponential depth, where each is a nonzero real number and is a monic transmonomial ( is a transmonomial but is not monic unless the coefficient ; each is different; the order of the summands is irrelevant).
The sum might be infinite or transfinite; it is usually written in the order of decreasing .
Here, well-based means that there is no infinite ascending sequence (see well-ordering).
A monic transmonomial is one of 1, x, log x, log log x, ..., epurely_large_transseries.
Note: Because , we do not include it as a primitive, but many authors do; log-free transseries do not include but is permitted. Also, circularity in the definition is avoided because the purely_large_transseries (above) will have lower exponential depth; the definition works by recursion on the exponential depth. See "Log-exp transseries as iterated Hahn series" (below) for a construction that uses and explicitly separates different stages.
A purely large transseries is a nonempty transseries with every .
Transseries have finite exponential depth, where each level of nesting of e or log increases depth by 1 (so we cannot have x + log x + log log x + ...).
Addition of transseries is termwise: (absence of a term is equated with a zero coefficient).
Comparison:
The most significant term of is for the largest (because the sum is well-based, this exists for nonzero transseries). is positive iff the coefficient of the most significant term is positive (this is why we used 'purely large' above). X > Y iff X − Y is positive.
Comparison of monic transmonomials:
– these are the only equalities in our construction.
iff (also ).
Multiplication:
This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.
Differentiation:
(division is defined using multiplication).
With these definitions, transseries is an ordered differential field. Transseries is also a valued field, with the valuation given by the leading monic transmonomial, and the corresponding asymptotic relation defined for by if (where is the absolute value).
Other constructions
Log-exp transseries as iterated Hahn series Log-free transseries
We first define the subfield of of so-called log-free transseries. Those are transseries which exclude any logarithmic term.
Inductive definition:
For we will define a linearly ordered multiplicative group of monomials . We then let denote the field of well-based series . This is the set of maps with well-based (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series). In , we distinguish the (non-unital) subring of purely large transseries, which are series whose support contains only monomials lying strictly above .
We start with equipped with the product and the order .
If is such that , and thus and are defined, we let denote the set of formal expressions where and . This forms a linearly ordered commutative group under the product and the lexicographic order if and only if or ( and ).
The natural inclusion of into given by identifying and inductively provides a natural embedding of into , and thus a natural embedding of into . We may then define the linearly ordered commutative group and the ordered field which is the field of log-free transseries.
The field is a proper subfield of the field of well-based series with real coefficients and monomials in . Indeed, every series in has a bounded exponential depth, i.e. the least positive integer such that , whereas the series
has no such bound.
Exponentiation on :
The field of log-free transseries is equipped with an exponential function which is a specific morphism . Let be a log-free transseries and let be the exponential depth of , so . Write as the sum in where , is a real number and is infinitesimal (any of them could be zero). Then the formal Hahn sum
converges in , and we define where is the value of the real exponential function at .
Right-composition with :
A right composition with the series can be defined by induction on the exponential depth by
with . It follows inductively that monomials are preserved by so at each inductive step the sums are well-based and thus well defined.
Log-exp transseries
Definition:
The function defined above is not onto so the logarithm is only partially defined on : for instance the series has no logarithm. Moreover, every positive infinite log-free transseries is greater than some positive power of . In order to move from to , one can simply "plug" into the variable of series formal iterated logarithms which will behave like the formal reciprocal of the -fold iterated exponential term denoted .
For let denote the set of formal expressions where . We turn this into an ordered group by defining , and defining when . We define . If and we embed into by identifying an element with the term
We then obtain as the directed union
On the right-composition with is naturally defined by
Exponential and logarithm:
Exponentiation can be defined on in a similar way as for log-free transseries, but here also has a reciprocal on . Indeed, for a strictly positive series , write where is the dominant monomial of (largest element of its support), is the corresponding positive real coefficient, and is infinitesimal. The formal Hahn sum
converges in . Write where itself has the form where and . We define . We finally set
Using surreal numbers
Direct construction of log-exp transseries
One may also define the field of log-exp transseries as a subfield of the ordered field of surreal numbers.Berarducci, Alessandro and Mantova, Vincenzo, Transseries as germs of surreal functions, Transactions of the American Mathematical Society, 2017 The field is equipped with Gonshor-Kruskal's exponential and logarithm functionsGonshor, Harry, An Introduction to the Theory of Surreal Numbers, 'Cambridge University Press', 1986 and with its natural structure of field of well-based series under Conway normal form.Conway, John, Horton, On numbers and games, Academic Press, London, 1976
Define , the subfield of generated by and the simplest positive infinite surreal number (which corresponds naturally to the ordinal , and as a transseries to the series ). Then, for , define as the field generated by , exponentials of elements of and logarithms of strictly positive elements of , as well as (Hahn) sums of summable families in . The union is naturally isomorphic to . In fact, there is a unique such isomorphism which sends to and commutes with exponentiation and sums of summable families in lying in .
Other fields of transseries
Continuing this process by transfinite induction on beyond , taking unions at limit ordinals, one obtains a proper class-sized field canonically equipped with a derivation and a composition extending that of (see Operations on transseries below).