I wonder whether hyperreal numbers isomorphic with formal Laurent series?


It seems that any hyperreal number can be represented in the form of Laurent series over $\omega$. For instance,


$e^{\omega}=\frac{\omega^0}{0!}+\frac {\omega^1}{1!}+\frac{\omega^2}{2!}+...+\frac{\omega^n}{n!}+...$


If so, it follows that any analytic function corresponds to a hyperreal number.


$f(x)=x$ corresponds to $\omega$


$f(x)=x^2$ corresponds to $\omega^2$


etc.


Thus operators on analytic functions are isomorphic to functions of hyperreals. Am I correct?



Are hyperreal numbers isomorphic to formal power series?
One problem is that the set of formal Laurent series is not a [real closed field](http://en.wikipedia.org/wiki/Real_closed_field) (an ordered field where every positive element has a square root, and every polynomial of odd degree has a root). That particular problem would be fixed if one considered instead the real closure of the field of Laurent series, which is the field of [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_series) (Laurent series involving powers of $x^{1/n}$ where $n$ is allowed to vary). So the question would become: is the field of hypperreals isomorphic to the field of Puiseux series? 


The answer is "certainly not"; the hyperreals are much "bigger", or it would be better to say "much more saturated" than Puiseux series. As you probably know, the hyperreals do not form a complete ordered field (only the standard real numbers do that), but they do have properties that approximate completeness. Namely, there is the $\eta\_1$ property: given *countable* sets $L$ and $U$ such that every element of $L$ is less than every element of $U$, there exists an element that is an upper bound of all elements of $L$ and a lower bound of all elements of $U$. So for example: in the field of Puiseux series (and taking $x$ to be an infinitesimal element), there is a pretty big gap between all the rational constants (forming an $L$) and the $x^{-1/n}$ (forming a $U$). The hyperreals fill in such gaps. 


(Added on request.) The hyperreals thus form an ordered field extension of the field of Puiseux series, and admits elements such as $\sin \varepsilon$ where $\varepsilon$ is an infinitesimal. 


