Transseries

Informally speaking, exp-log transseries are well-based (i.e. reverse well-ordered) formal Hahn series of real powers of the positive infinite indeterminate $x$, exponentials, logarithms and their compositions, with real coefficients. Two important additional conditions are that the exponential and logarithmic depth of an exp-log transseries $f,$ that is the maximal numbers of iterations of exp and log occurring in $f,$ must be finite.

The following formal series are log-exp transseries:

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{e^\{x^\{\backslash frac\{1\}\{n\}\}\}\}\{n!\}\; +\; x^3\; +\; \backslash log\; x\; +\; \backslash log\backslash log\; x\; +\backslash sum\_\{n=0\}^\backslash infty\; x^\{-n\}\; +\; \backslash sum\; \_\{i=1\}^\backslash infty\; e^\{-\backslash sum\_\{j=1\}^\backslash infty\; e^\{ix^2-jx\}\}.$

:$\backslash sum\_\{m,n\; \backslash in\; \backslash N\}\; x^\{\backslash frac\{1\}\{m+1\}\}e^\{-(\backslash log\; x)^n\}.$

The following formal series are not log-exp transseries:

:$\backslash sum\_\{n\; \backslash in\; \backslash N\}\; x^n$ — this series is not well-based.

:$\backslash log\; x\; +\; \backslash log\; \backslash log\; x+\; \backslash log\; \backslash log\; \backslash log\; x+\; \backslash cdots$ — the logarithmic depth of this series is infinite

:$\backslash frac\{1\}\{2\}x+e^\{\backslash frac\{1\}\{2\}\backslash log\; x\}+e^\{e^\{\backslash frac\{1\}\{2\}\backslash log\; \backslash log\; x\}\}+\backslash cdots$ — 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 $\backslash mathbb\{T\}^\{EL\}$ and $\backslash R\backslash langle\backslash langle\; \backslash omega\; \backslash rangle\backslash rangle$ (see the paragraph Using surreal numbers below).

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 $f$ and $g$, we have $f\; \backslash leq\_\{\backslash infty\}\; g$ or $g\; \backslash leq\_\{\backslash infty\}\; f$, where $f\backslash leq\_\{\backslash infty\}g$ means $\backslash exists\; x.\; \backslash forall\; y>x.\; f(y)\backslash leq\; g(y)$. The equivalence class of $f$ under the relation $f\; \backslash leq\_\{\backslash infty\}\; g\; \backslash wedge\; g\; \backslash leq\_\{\backslash infty\}\; f$ is the asymptotic behavior of $f$, also called the germ of $f$ (or the germ of $f$ 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, $(\backslash sum\_\{k=0\}^n\; x^\{-k\})\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ is associated with $\backslash sum\_\{k=0\}^\backslash infty\; x^\{-k\}$ rather than $\backslash sum\_\{k=0\}^\backslash infty\; x^\{-k\}-e^\{-x\}$ because $e^\{-x\}$ 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 $\backslash sin\; x$). On the other hand, there are transseries such as $\backslash sum\; \_\{k\; \backslash in\; \backslash mathbb\{N\}\}\; k!e^\{x^\{-\backslash frac\{k\}\{k+1\}\}\}$ 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 $e^\{e^\{.^\{.^\{.^\{e^x\}\}\}\}\}$ of $e^x$, 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 $e\_\{\backslash omega\}$ of the Abel equation $e^\{e\_\{\backslash omega\}(x)\}=e\_\{\backslash omega\}(x+1)$.Schmeling, Michael, [http://www.texmacs.org/joris/schmeling/schmeling.pdf Corps de transséries], PhD thesis, 2001

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,

:$\backslash sum\; a\_i\; m\_i,$

with finite exponential depth, where each $a\_i$ is a nonzero real number and $m\_i$ is a monic transmonomial ($a\_i\; m\_i$ is a transmonomial but is not monic unless the coefficient $a\_i\; =\; 1$; each $m\_i$ 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 $m\_i$.

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, ..., e^{purely_large_transseries}.

: Note: Because $x^n\; =\; e^\{n\; \backslash log\; x\}$, we do not include it as a primitive, but many authors do; log-free transseries do not include $\backslash log$ but $x^n\; e^\backslash cdots$ 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 $x^a\; e^\backslash cdots$ and explicitly separates different stages.

A purely large transseries is a nonempty transseries $\backslash sum\; a\_i\; m\_i$ with every $m\_i>1$.

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: $\backslash sum\; a\_i\; m\_i\; +\; \backslash sum\; b\_i\; m\_i\; =\; \backslash sum(a\_i\; +\; b\_i)\; m\_i$ (absence of a term is equated with a zero coefficient).

Comparison:

The most significant term of $\backslash sum\; a\_i\; m\_i$ is $a\_i\; m\_i$ for the largest $m\_i$ (because the sum is well-based, this exists for nonzero transseries). $\backslash sum\; a\_i\; m\_i$ 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:

: $x\; =\; e^\{\backslash log\; x\},\; \backslash log\; x\; =\; e^\{\backslash log\; \backslash log\; x\},\; \backslash ldots$ – these are the only equalities in our construction.

: $x\; >\; \backslash log\; x\; >\; \backslash log\; \backslash log\; x\; >\; \backslash cdots\; >1\; >0.$

: iff (also $e^0\; =\; 1$).

Multiplication:

: $e^a\; e^b\; =\; e^\{a+b\}$

: $\backslash left(\backslash sum\; a\_i\; x\_i\backslash right)\; \backslash left(\backslash sum\; b\_j\; y\_j\backslash right)\; =\; \backslash sum\_k\; \backslash left(\; \backslash sum\_\{i,j\backslash ,:\backslash ,z\_k=x\_i\; y\_j\}\; a\_i\; b\_j\backslash right)\; z\_k.$

This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.

Differentiation:

: $\backslash left(\backslash sum\; a\_i\; x\_i\backslash right)\text{'}\; =\; \backslash sum\; a\_i\; x\_i\text{'}$

: $1\text{'}\; =\; 0,\; x\text{'}\; =\; 1$

: $(e^y)\text{'}\; =\; y\text{'}\; e^y$

: $(\backslash log\; y)\text{'}\; =\; y\text{'}/y$ (division is defined using multiplication).

With these definitions, transseries is an ordered differential field. Transseries is also a valued field, with the valuation $\backslash nu$ given by the leading monic transmonomial, and the corresponding asymptotic relation defined for $0\backslash neq\; f,g\; \backslash in\; \backslash mathbb\{T\}^\{LE\}$ by $f\; \backslash prec\; g$ if

We first define the subfield $\backslash mathbb\{T\}^\{E\}$ of $\backslash mathbb\{T\}^\{LE\}$ of so-called log-free transseries. Those are transseries which exclude any logarithmic term.

Inductive definition:

For $n\; \backslash in\; \backslash N,$ we will define a linearly ordered multiplicative group of monomials $\backslash mathfrak\{M\}\_n$. We then let $\backslash mathbb\{T\}^E\_n$ denote the field of well-based series $\backslash R\backslash mathfrak\{M\}\_n$. This is the set of maps $\backslash R\backslash to\; \backslash mathfrak\{M\}\_n$ with well-based (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series). In $\backslash mathbb\{T\}^E\_n$, we distinguish the (non-unital) subring $\backslash mathbb\{T\}^E\_\{n,\backslash succ\}$ of purely large transseries, which are series whose support contains only monomials lying strictly above $1$.

:We start with $\backslash mathfrak\{M\}\_0=x^\{\backslash R\}$ equipped with the product $x^a\; x^b:=x^\{a+b\}$ and the order

:If $n\backslash in\; \backslash N$ is such that $\backslash mathfrak\{M\}\_n$, and thus $\backslash mathbb\{T\}^E\_n$ and $\backslash mathbb\{T\}^E\_\{n,\backslash succ\}$ are defined, we let $\backslash mathfrak\{M\}\_\{n+1\}$ denote the set of formal expressions $x^a\; e^\{\backslash theta\}$ where $a\; \backslash in\; \backslash R$ and $\backslash theta\; \backslash in\; \backslash mathbb\{T\}^E\_\{n,\backslash succ\}$. This forms a linearly ordered commutative group under the product $(x^a\; e^\{\backslash theta\})(x^\{a\text{'}\}\; e^\{\backslash theta\text{'}\})=(x^\{a+a\text{'}\})\; e^\{\backslash theta+\backslash theta\text{'}\}$ and the lexicographic order $x^a\; e^\{\backslash theta\}\; \backslash prec\; x^\{a\text{'}\}\; e^\{\backslash theta\text{'}\}$ if and only if or ($\backslash theta=\backslash theta\text{'}$ and

The natural inclusion of $\backslash mathfrak\{M\}\_0$ into $\backslash mathfrak\{M\}\_1$ given by identifying $x^a$ and $x^a\; e^0$ inductively provides a natural embedding of $\backslash mathfrak\{M\}\_n$ into $\backslash mathfrak\{M\}\_\{n+1\}$, and thus a natural embedding of $\backslash mathbb\{T\}^E\_n$ into $\backslash mathbb\{T\}^E\_\{n+1\}$. We may then define the linearly ordered commutative group $\backslash mathfrak\{M\}=\backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; \backslash mathfrak\{M\}\_n$ and the ordered field $\backslash mathbb\{T\}^E=\backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; \backslash mathbb\{T\}^E\_n$ which is the field of log-free transseries.

The field $\backslash mathbb\{T\}^E$ is a proper subfield of the field $\backslash R\backslash mathfrak\{M\}$ of well-based series with real coefficients and monomials in $\backslash mathfrak\{M\}$. Indeed, every series $f$ in $\backslash mathbb\{T\}^E$ has a bounded exponential depth, i.e. the least positive integer $n$ such that $f\; \backslash in\; \backslash mathbb\{T\}^E\_n$, whereas the series

:$e^\{-x\}+e^\{-e^x\}+e^\{-e^\{e^x\}\}+\; \backslash cdots\; \backslash in\; \backslash R\backslash mathfrak\{M\}$

has no such bound.

Exponentiation on $\backslash mathbb\{T\}^E$:

The field of log-free transseries is equipped with an exponential function which is a specific morphism $\backslash exp:(\backslash mathbb\{T\}^E,+)\backslash to(\backslash mathbb\{T\}^\{E,>\},\; \backslash times)$. Let $f$ be a log-free transseries and let $n\; \backslash in\; \backslash N$ be the exponential depth of $f$, so $f\; \backslash in\; \backslash mathbb\{T\}^E\_n$. Write $f$ as the sum $f=\backslash theta+r+\backslash varepsilon$ in $\backslash mathbb\{T\}^E\_n,$ where $\backslash theta\; \backslash in\; \backslash mathbb\{T\}^E\_\{n,\backslash succ\}$, $r$ is a real number and $\backslash varepsilon$ is infinitesimal (any of them could be zero). Then the formal Hahn sum

:$E(\backslash varepsilon):=\backslash sum\_\{k\; \backslash in\; \backslash N\}\; \backslash frac\{\backslash varepsilon^k\}\{k!\}$

converges in $\backslash mathbb\{T\}^E\_n$, and we define $\backslash exp(f)=e^\{\backslash theta\}\backslash exp(r)\; E(\backslash varepsilon)\; \backslash in\; \backslash mathbb\{T\}^E\_\{n+1\}$ where $\backslash exp(r)$ is the value of the real exponential function at $r$.

Right-composition with $e^x$:

A right composition $\backslash circ\_\{e^x\}$ with the series $e^x$ can be defined by induction on the exponential depth by

:$\backslash left\; (\backslash sum\; f\_\{\backslash mathfrak\{m\}\}\; \backslash mathfrak\{m\}\; \backslash right\; )\; \backslash circ\; e^x:=\backslash sum\; f\_\{\backslash mathfrak\{m\}\}\; (\backslash mathfrak\{m\}\; \backslash circ\; e^x),$

with $x^r\; \backslash circ\; e^x:=e^\{rx\}$. It follows inductively that monomials are preserved by $\backslash circ\_\{e^x\},$ so at each inductive step the sums are well-based and thus well defined.

Definition:

The function $\backslash exp$ defined above is not onto $\backslash mathbb\{T\}^\{E,>\}$ so the logarithm is only partially defined on $\backslash mathbb\{T\}^E$: for instance the series $x$ has no logarithm. Moreover, every positive infinite log-free transseries is greater than some positive power of $x$. In order to move from $\backslash mathbb\{T\}^E$ to $\backslash mathbb\{T\}^\{LE\}$, one can simply "plug" into the variable $x$ of series formal iterated logarithms $\backslash ell\_n,n\; \backslash in\; \backslash N$ which will behave like the formal reciprocal of the $n$-fold iterated exponential term denoted $e\_n$.

For $m,n\; \backslash in\; \backslash N,$ let $\backslash mathfrak\{M\}\_\{m,n\}$ denote the set of formal expressions $\backslash mathfrak\{u\}\; \backslash circ\; \backslash ell\_n$ where $\backslash mathfrak\{u\}\; \backslash in\; \backslash mathfrak\{M\}\_m$. We turn this into an ordered group by defining $(\backslash mathfrak\{u\}\; \backslash circ\; \backslash ell\_n)(\backslash mathfrak\{v\}\; \backslash circ\; \backslash ell\_n(x)):=(\backslash mathfrak\{u\}\backslash mathfrak\{v\})\; \backslash circ\; \backslash ell\_n$, and defining $\backslash mathfrak\{u\}\; \backslash circ\; \backslash ell\_n\backslash prec\; \backslash mathfrak\{v\}\backslash circ\; \backslash ell\_n$ when $\backslash mathfrak\{u\}\backslash prec\; \backslash mathfrak\{v\}$. We define $\backslash mathbb\{T\}^\{LE\}\_\{m,n\}:=\backslash R\backslash mathfrak\{M\}\_\{m,n\}$. If $n\text{'}>\; n$ and $m\text{'}\; \backslash geq\; m+(n\text{'}-n),$ we embed $\backslash mathfrak\{M\}\_\{m,n\}$ into $\backslash mathfrak\{M\}\_\{m\text{'},n\text{'}\}$ by identifying an element $\backslash mathfrak\{u\}\; \backslash circ\; \backslash ell\_n$ with the term

:$\backslash left\; (\backslash mathfrak\{u\}\; \backslash circ\; \backslash overbrace\{e^x\; \backslash circ\; \backslash cdots\; \backslash circ\; e^x\}^\{n\text{'}-n\}\; \backslash right\; )\; \backslash circ\; \backslash ell\_\{n\text{'}\}.$

We then obtain $\backslash mathbb\{T\}^\{LE\}$ as the directed union

:$\backslash mathbb\{T\}^\{LE\}=\backslash bigcup\_\{m,n\; \backslash in\; \backslash N\}\; \backslash mathbb\{T\}^\{LE\}\_\{m,n\}.$

On $\backslash mathbb\{T\}^\{LE\},$ the right-composition $\backslash circ\_\{\backslash ell\}$ with $\backslash ell$ is naturally defined by

:$\backslash mathbb\{T\}^\{LE\}\_\{m,n\}\; \backslash ni\; \backslash left\; (\backslash sum\; f\_\{\backslash mathfrak\{m\}\; \backslash circ\; \backslash ell\_n\}\; \backslash mathfrak\{m\}\; \backslash circ\; \backslash ell\_n\; \backslash right\; )\; \backslash circ\; \backslash ell:=\; \backslash sum\; f\_\{\backslash mathfrak\{m\}\; \backslash circ\; \backslash ell\_n\}\; \backslash mathfrak\{m\}\; \backslash circ\; \backslash ell\_\{n+1\}\backslash in\; \backslash mathbb\{T\}^\{LE\}\_\{m,n+1\}.$

Exponential and logarithm:

Exponentiation can be defined on $\backslash mathbb\{T\}^\{LE\}$ in a similar way as for log-free transseries, but here also $\backslash exp$ has a reciprocal $\backslash log$ on $\backslash mathbb\{T\}^\{LE,>\}$. Indeed, for a strictly positive series $f\; \backslash in\; \backslash mathbb\{T\}^\{LE,>\}\_\{m,n\}$, write $f=\backslash mathfrak\{m\}\; r(1+\backslash varepsilon)$ where $\backslash mathfrak\{m\}$ is the dominant monomial of $f$ (largest element of its support), $r$ is the corresponding positive real coefficient, and $\backslash varepsilon:=\backslash frac\{f\}\{\backslash mathfrak\{m\}\; r\}-1$ is infinitesimal. The formal Hahn sum

:$L(1+\backslash varepsilon):=\backslash sum\_\{k\; \backslash in\; \backslash N\}\backslash frac\{(-\backslash varepsilon)^k\}\{k+1\}$

converges in $\backslash mathbb\{T\}^\{LE\}\_\{m,n\}$. Write $\backslash mathfrak\{m\}=\backslash mathfrak\{u\}\backslash circ\; \backslash ell\_n$ where $\backslash mathfrak\{u\}\; \backslash in\; \backslash mathfrak\{M\}\_m$ itself has the form $\backslash mathfrak\{u\}=x^ae^\{\backslash theta\}$ where $\backslash theta\; \backslash in\; \backslash mathbb\{T\}^E\_\{m,\backslash succ\}$ and $a\; \backslash in\; \backslash R$. We define $\backslash ell(\backslash mathfrak\{m\}):=a\; \backslash ell\_\{n+1\}\; +\backslash theta\; \backslash circ\; \backslash ell\_n$. We finally set

:$\backslash log(f):=\backslash ell(\backslash mathfrak\{m\})+\backslash log(c)+L(1+\backslash varepsilon)\; \backslash in\; \backslash mathbb\{T\}^\{LE\}\_\{m,n+1\}.$

One may also define the field of log-exp transseries as a subfield of the ordered field $\backslash mathbf\{No\}$ of surreal numbers.Berarducci, Alessandro and Mantova, Vincenzo, [https://arxiv.org/abs/1703.019, Transseries as germs of surreal functions], Transactions of the American Mathematical Society, 2017 The field $\backslash mathbf\{No\}$ 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, [https://keybase.pub/procspero/%5BMath%5D%20Probability%20and%20Statistics/Combinatorial%20Game%20Theory/On%20Numbers%20and%20Games,%20John%20Horton%20Conway.pdf, On numbers and games], Academic Press, London, 1976

Define $F^\{LE\}\_0=\backslash R(\backslash omega)$, the subfield of $\backslash mathbf\{No\}$ generated by $\backslash R$ and the simplest positive infinite surreal number $\backslash omega$ (which corresponds naturally to the ordinal $\backslash omega$, and as a transseries to the series $x$). Then, for $n\; \backslash in\; \backslash N$, define $F^\{LE\}\_\{n+1\}$ as the field generated by $F^\{LE\}\_n$, exponentials of elements of $F^\{LE\}\_n$ and logarithms of strictly positive elements of $F^\{LE\}\_n$, as well as (Hahn) sums of summable families in $F^\{LE\}\_n$. The union $F^\{LE\}\_\{\backslash omega\}=\backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; F^\{LE\}\_n$ is naturally isomorphic to $\backslash mathbb\{T\}^\{LE\}$. In fact, there is a unique such isomorphism which sends $\backslash omega$ to $x$ and commutes with exponentiation and sums of summable families in $F^\{LE\}\_\{\backslash omega\}$ lying in $F\_\{\backslash omega\}$.

• Continuing this process by transfinite induction on $\backslash mathbf\{Ord\}$ beyond $F^\{LE\}\_\{\backslash omega\}$, taking unions at limit ordinals, one obtains a proper class-sized field $\backslash R\backslash langle\backslash langle\backslash omega\backslash rangle\backslash rangle$ canonically equipped with a derivation and a composition extending that of $\backslash mathbb\{T\}^\{LE\}$ (see Operations on transseries below).

• If instead of $F^\{LE\}\_0$ one starts with the subfield $F^\{EL\}\_0:=\backslash R(\backslash omega,\backslash log\; \backslash omega,\; \backslash log\; \backslash log\; \backslash omega,\; \backslash ldots)$ generated by $\backslash R$ and all finite iterates of $\backslash log$ at $\backslash omega$, and for $n\backslash in\; \backslash N,\; F^\{EL\}\_\{n+1\}$ is the subfield generated by $F^\{EL\}\_n$, exponentials of elements of $F^\{EL\}\_n$ and sums of summable families in $F^\{EL\}\_n$, then one obtains an isomorphic copy the field $\backslash mathbb\{T\}^\{EL\}$ of exponential-logarithmic transseries, which is a proper extension of $\backslash mathbb\{T\}^\{LE\}$ equipped with a total exponential function.Kuhlmann, Salma and Tressl, Marcus, [https://arxiv.org/abs/1112.4189, Comparison of exponential-logarithmic and logarithmic-exponential series], Mathematical Logic Quarterly, 2012

The Berarducci-Mantova derivationBerarducci, Alessandro and Mantova, Vincenzo, [https://arxiv.org/abs/1503.00315 Surreal numbers, derivations and transseries], European Mathematical Society, 2015 on $\backslash mathbf\{No\}$ coincides on $\backslash mathbb\{T\}^\{LE\}$ with its natural derivation, and is unique to satisfy compatibility relations with the exponential ordered field structure and generalized series field structure of $\backslash mathbb\{T\}^\{EL\}$ and $\backslash R\backslash langle\backslash langle\backslash omega\backslash rangle\backslash rangle.$

Contrary to $\backslash mathbb\{T\}^\{LE\},$ the derivation in $\backslash mathbb\{T\}^\{EL\}$ and $\backslash R\; \backslash langle\backslash langle\backslash omega\backslash rangle\backslash rangle$ is not surjective: for instance the series

:$\backslash frac\{1\}\{\backslash omega\; \backslash log\; \backslash omega\; \backslash log\; \backslash log\; \backslash omega\; \backslash cdots\}:=\backslash exp(-(\backslash log\; \backslash omega+\backslash log\; \backslash log\backslash omega+\backslash log\; \backslash log\; \backslash log\; \backslash omega+\; \backslash cdots))\; \backslash in\; \backslash mathbb\{T\}^\{EL\}$

doesn't have an antiderivative in $\backslash mathbb\{T\}^\{EL\}$ or $\backslash R\; \backslash langle\; \backslash langle\backslash omega\; \backslash rangle\backslash rangle$ (this is linked to the fact that those fields contain no transexponential function).

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:

::$\backslash exp(x^\{-1\})\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{n!\}x^\{-n\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash log(x+\backslash ell)=\backslash ell+\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{(x^\{-1\}\backslash ell)^n\}\{n+1\}.$

• Logarithm is defined for positive arguments.
• Log-exp transseries are real-closed.
• Integration: every log-exp transseries $f$ has a unique antiderivative with zero constant term $F\; \backslash in\; \backslash mathbb\{T\}^\{LE\}$, $F\text{'}=f$ and $F\_1=0$.
• Logarithmic antiderivative: for $f\backslash in\; \backslash mathbb\{T\}^\{LE\}$, there is $h\backslash in\; \backslash mathbb\{T\}^\{LE\}$ with $f\text{'}=f\; h\text{'}$.

Note 1. The last two properties mean that $\backslash mathbb\{T\}^\{LE\}$ is Liouville closed.

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

:$\backslash exists\; k,n\; \backslash in\; \backslash N:\; \backslash quad\; \backslash ell\_\{n-k\}\; -1\backslash leq\; \backslash ell\_n\; \backslash circ\; f\; \backslash leq\; \backslash ell\_\{n-k\}+1.$

The number $k$ is unique, it is called the exponentiality of $f$.

An original property of $\backslash mathbb\{T\}^\{LE\}$ is that it admits a composition $\backslash circ\; :\backslash mathbb\{T\}^\{LE\}\; \backslash times\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}\; \backslash to\; \backslash mathbb\{T\}^\{LE\}$ (where $\backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ is the set of positive infinite log-exp transseries) which enables us to see each log-exp transseries $f$ as a function on $\backslash mathbb\{T\}^\{LE,>,\backslash succ\}$. Informally speaking, for $g\backslash in\backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ and $f\backslash in\; \backslash mathbb\{T\}^\{LE\}$, the series $f\; \backslash circ\; g$ is obtained by replacing each occurrence of the variable $x$ in $f$ by $g$.

• Associativity: for $f\; \backslash in\; \backslash mathbb\{T\}^\{LE\}$ and $g,h\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$, we have $g\backslash circ\; h\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ and $f\; \backslash circ\; (g\backslash circ\; h)=(f\; \backslash circ\; g)\; \backslash circ\; h$.
• Compatibility of right-compositions: For $g\backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$, the function $\backslash circ\_g:f\backslash mapsto\; f\; \backslash circ\; g$ is a field automorphism of $\backslash mathbb\{T\}^\{LE\}$ which commutes with formal sums, sends $x$ onto $g$, $e^x$ onto $\backslash exp(g)$ and $\backslash ell$ onto $\backslash log(g)$. We also have $\backslash circ\_x=\backslash operatorname\{id\}\_\{\backslash mathbb\{T\}^\{LE\}\}$.
• Unicity: the composition is unique to satisfy the two previous properties.
• Monotonicity: for $f\backslash in\; \backslash mathbb\{T\}^\{LE\}$, the function $g\backslash mapsto\; f\; \backslash circ\; g$ is constant or strictly monotonous on $\backslash mathbb\{T\}^\{LE,>,\backslash succ\}$. The monotony depends on the sign of $f\text{'}$.
• Chain rule: for $f\; \backslash in\; \backslash mathbb\{T\}^\{LE\}\backslash times$ and $g\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$, we have $(f\; \backslash circ\; g)\text{'}=g\text{'}f\text{'}\; \backslash circ\; g$.
• Functional inverse: for $g\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$, there is a unique series $h\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ with $g\; \backslash circ\; h=\; h\; \backslash circ\; g=\; x$.
• Taylor expansions: each log-exp transseries $f$ has a Taylor expansion around every point in the sense that for every $g\backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ and for sufficiently small $\backslash varepsilon\; \backslash in\; \backslash mathbb\{T\}^\{LE\}$, we have

::$f\backslash circ\; (g+\backslash varepsilon)=\backslash sum\_\{k\; \backslash in\; \backslash N\}\; \backslash frac\{f^\{(k)\}\backslash circ\; g\}\{k!\}\backslash varepsilon^k$

:where the sum is a formal Hahn sum of a summable family.

• Fractional iteration: for $f\; \backslash in\; \backslash mathbb\{T\}^\{LE,>,\backslash succ\}$ with exponentiality $0$ and any real number $a$, the fractional iterate $f^a$ of $f$ is defined.{{Citation | last = Edgar | first = G. A. | title = Fractional Iteration of Series and Transseries | date = 2010 | arxiv = 1002.2378 | bibcode = 2010arXiv1002.2378E }}

The theory of $\backslash mathbb\{T\}^\{LE\}$ is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.):

• $\backslash mathbb\{T\}^\{LE\}$ is an ordered valued differential field.
• $f\; >\; 0\; \backslash wedge\; f\; \backslash succ\; 1\; \backslash Longrightarrow\; f\text{'}\; >\; 0$
• $f\; \backslash prec\; 1\; \backslash Longrightarrow\; f\text{'}\; \backslash prec\; 1$
• $\backslash forall\; f\; \backslash exists\; g:\; \backslash quad\; g\text{'}\; =\; f$
• $\backslash forall\; f\; \backslash exists\; h:\; \backslash quad\; h\text{'}\; =\; fh$
• Intermediate value property (IVP):

::

:where P is a differential polynomial, i.e. a polynomial in $f,\; f\text{'},\; f\text{'}\text{'},\; \backslash ldots,\; f^\{(k)\}.$

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

The theory of $\backslash mathbb\{T\}^\{LE\}$ is that of the exponential real ordered exponential field , which is model complete by Wilkie's theorem.

$\backslash mathbb\{T\}\_\backslash mathrm\{as\}$ 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 $\backslash mathbb\{T\}$. (This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of $\backslash mathbb\{T\}$ are permitted.) $\backslash mathbb\{T\}\_\backslash mathrm\{as\}$ is conjectured to satisfy the above axioms of $\backslash mathbb\{T\}$. 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 $\backslash mathbb\{T\}^\{LE\}$.Aschenbrenner, Matthias, and van den Dries, Lou and van der Hoeven, Joris, [https://arxiv.org/abs/1711.06936 On Numbers, Germs, and Transseries], In Proc. Int. Cong. of Math., vol. 1, pp. 1–24, 2018 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.Boshernitzan, Michael, Hardy fields and existence of transexponential functions, In aequationes mathematicae, vol. 30, issue 1, pp. 258–280, 1986.

• Formal power series
• Hahn series
• Exponentially closed field
• Hardy field

{{Reflist}}

• {{Citation | last = Edgar | first = G. A. | title = Transseries for beginners | journal = Real Analysis Exchange | volume = 35 | pages = 253–310 | date = 2010 | issue = 2 | doi = 10.14321/realanalexch.35.2.0253 | arxiv = 0801.4877 | s2cid = 14290638 }}.
• {{Citation | last1 = Aschenbrenner | first1 = Matthias | last2 = Dries | first2 = Lou van den| last3 = Hoeven | first3 = Joris van der | title = On Numbers, Germs, and Transseries | date = 2017 | arxiv = 1711.06936 | bibcode = 2017arXiv171106936A }}.

Category:Asymptotic analysis

Category:Series (mathematics)

Category:Exponentials

Category:Logarithms

Category:Real closed field