Transseries
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-Göring and Ecalle 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 .

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. The field  is equipped with Gonshor-Kruskal's exponential and logarithm functions and with its natural structure of field of well-based series under Conway normal form.

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).

If instead of  one starts with the subfield  generated by  and all finite iterates of  at , and for  is the subfield generated by , exponentials of elements of  and sums of summable families in , then one obtains an isomorphic copy the field  of exponential-logarithmic transseries, which is a proper extension of  equipped with a total exponential function.

The Berarducci-Mantova derivation on  coincides on  with its natural derivation, and is unique to satisfy compatibility relations with the exponential ordered field structure and generalized series field structure of  and 

Contrary to  the derivation in  and  is not surjective: for instance the series 

 

doesn't have an antiderivative in  or  (this is linked to the fact that those fields contain no transexponential function).

Additional properties

Operations on transseries

Operations on the differential exponential ordered field

Transseries have very strong closure properties, and many operations can be defined on transseries:
 Log-exp transseries form an exponentially closed ordered field: the exponential and logarithmic functions are total. For example:

 Logarithm is defined for positive arguments.
 Log-exp transseries are real-closed.
 Integration: every log-exp transseries  has a unique antiderivative with zero constant term ,  and .
 Logarithmic antiderivative: for , there is  with .

Note 1. The last two properties mean that  is Liouville closed.

Note 2. Just like an elementary nontrigonometric function, each positive infinite transseries  has integral exponentiality, even in this strong sense: 

The number  is unique, it is called the exponentiality of .

Composition of transseries

An original property of  is that it admits a composition  (where  is the set of positive infinite log-exp transseries) which enables us to see each log-exp transseries  as a function on . Informally speaking, for  and , the series  is obtained by replacing each occurrence of the variable  in  by .

Properties
 Associativity: for  and , we have  and . 
 Compatibility of right-compositions: For , the function  is a field automorphism of  which commutes with formal sums, sends  onto ,  onto  and  onto . We also have . 
 Unicity: the composition is unique to satisfy the two previous properties.
 Monotonicity: for , the function  is constant or strictly monotonous on . The monotony depends on the sign of .
 Chain rule: for  and , we have .
 Functional inverse: for , there is a unique series  with .
 Taylor expansions: each log-exp transseries  has a Taylor expansion around every point in the sense that for every  and for sufficiently small , we have 
 
where the sum is a formal Hahn sum of a summable family. 
 Fractional iteration: for  with exponentiality  and any real number , the fractional iterate  of  is defined.

Decidability and model theory

Theory of differential ordered valued differential field
The  theory of  is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.):
  is an ordered valued differential field.
 
 
 
 
 Intermediate value property (IVP): 

where P is a differential polynomial, i.e. a polynomial in 

In this theory, exponentiation is essentially defined for functions (using differentiation) but not constants; in fact, every definable subset of  is semialgebraic.

Theory of ordered exponential field
The  theory of  is that of the exponential real ordered exponential field , which is model complete by Wilkie's theorem.

Hardy fields 

 is the field of accelero-summable transseries, and using accelero-summation, we have the corresponding Hardy field, which is conjectured to be the maximal Hardy field corresponding to a subfield of . (This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of  are permitted.)  is conjectured to satisfy the above axioms of . Without defining accelero-summation, we note that when operations on convergent transseries produce a divergent one while the same operations on the corresponding germs produce a valid germ, we can then associate the divergent transseries with that germ.

A Hardy field is said maximal if it is properly contained in no Hardy field. By an application of Zorn's lemma, every Hardy field is contained in a maximal Hardy field. It is conjectured that all maximal Hardy fields are elementary equivalent as differential fields, and indeed have the same first order theory as . 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 transsexponential functions.

See also 

 Formal power series
 Hahn series
 Exponentially closed field
 Hardy field

References 

 .
 .

Asymptotic analysis
Mathematical series
Exponentials
Logarithms
Real closed field