Transseries

The differential field of transseries $\mathbb{T}$ extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each transseries represents a formal asymptotic growth rate, 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 an actual growth rate.

Here is an example of transseries: $\sum_{k=1}^\infty \frac{e^{x^{1/k}}}{k!} + x^3 + \log x + \log \log x +\sum_{k=0}^\infty x^{-k} + \sum _{i=1}^\infty e^{-\sum_{j=1}^\infty e^{ix^2-jx}}$.

Introduction

A remarkable fact is that asymptotic growth rates of elementary nontrigonometric functions are all comparable: For all such f and g, f≼g or g≼f holds where f≼g means ∃x ∀y>x f(y)≤g(y). The equivalence class of f under ≼ is the asymptotic growth rate of f, also called the germ of f.

The field of transseries can be intuitively viewed as a completion of these growth rates. A complication is that growth rates are non-Archimedean and there is no unconditional least upper bound. We can address this by associating a sequence with the least upper bound of minimal complexity, analogously to construction of surreal numbers. For example, $(\sum_{k=0}^n x^{-k})_{n \in \mathbb{N}}$ is associated with $\sum_{k=0}^\infty x^{-k}$ rather than $\sum_{k=0}^\infty x^{-k}-e^{-x}$ 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 growth, pointwise convergence is not dispositive). An analogy is that (even for surreal numbers) $(1/n)_{n\in \mathbb{N}}$ converges to 0 rather than to an infinitesimal ε.

A limitation of transseries is that they exclude tetration and related functions. We do not know whether tetration can have a natural rate of growth, or whether a formal tetration can be added in a natural way to transseries.

Formal construction

The treatment of transseries as formal objects involves defining the syntax for transseries, comparison of transseries, arithmetical operations on transseries, and even differentiation. Appropriate transseries can then be assigned to corresponding asymptotic growth rates, though there are subtleties involving convergence. Transseries can be formalized in several equivalent ways; we use one of the simplest ones.

A transseries is a well-based sum $\sum a_i m_i$ 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 $m_{i_1} < m_{i_2} < m_{i_3} < ...$ (see well-ordering).
A monic transmonomial is one of 1, x, log x, log log x, ..., e^{purely_large_transseries}.
Note: Because $x^n = e^{n \log x}$, we do not include it as a primitive, but many authors do.
A purely large transseries is a nonempty transseries $\sum a_i m_i$ 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: $\sum a_i m_i + \sum b_i m_i = \sum(a_i + b_i) m_i$ (absence of a term is equated with a zero coefficient).

Comparison:
The most significant term of $\sum a_i m_i$ is $a_i m_i$ for the largest $m_i$ (because the sum is well-based, this exists for nonzero transseries). $\sum a_i m_i$ 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:
$x = e^{\log x}$, $\log x = e^{\log \log x}$, ... -- these are the only equalities in our construction.
x > log x > log log x > ... > 1 > 0.
$e^a < e^b$ iff $a < b$ (also $e^0 = 1$).

Multiplication:
$e^a e^b = e^{a+b}$
$(\sum a_i x_i) (\sum b_j y_j) = \sum_k (\sum_{i,j\,:\,z_k=m_i m_j} a_i b_j) z_k$. This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.

Differentiation:
$(\sum a_i x_i)' = \sum a_i x_i'$
$1' = 0$, $x' = 1$
$(e^y)' = y' e^y$
$(\log y)' = y'/y$ (division is defined using multiplication).

With these definitions, transseries is an ordered differential field. Transseries have very strong closure properties, and many other operations can be defined, including composition F(G(x)) if G(x) is ω(1).

See also

• Hardy field
• Formal power series
• Hahn series

References

• "Transseries for beginners" by G. A. Edgar, Real Analysis Exchange 35 (2010) 253-310, [1].
• "On Numbers, Germs, and Transseries" by Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, 2017, [2].