Hardy field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that is closed under differentiation. They are named after the English mathematician G. H. Hardy.
Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection H of functions that are defined for all large real numbers, that is functions f that map (u,∞) to the real numbers R, where u is some real number depending on f. Here and in the rest of the article we say a function has a property "eventually" if it has the property for all sufficiently large x, so for example we say a function f in H is eventually zero if there is some real number U such that f(x) = 0 for all x ≥ U. We can form an equivalence relation on H by saying f is equivalent to g if and only if f − g is eventually zero. The equivalence classes of this relation are called germs at infinity.
If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field.
Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f.
If F is a subfield of R then we can consider it as a Hardy field by considering the elements of F as constant functions, that is by considering the number α in F as the constant function fα that maps every x in R to α. This is a field since F is, and since the derivative of every function in this field is 0 which must be in F it is a Hardy field.
...
Wikipedia