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 $\omega$

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

etc.

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

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?