Example of a field of functions containing ln(x)

Question

Up until now, I've mainly worked with the polynomial ring $\mathbb{R}[x_1,...,x_n]$ or the corresponding field of fractions. But I can't think of an example of a field of functions that contain non-polynomial functions. What examples are there of a field of functions that contain the set of real rational functions and the natural logarithm function $ln(x)$? Thanks!

Answer 1

You can definitely study the field of functions $\mathbb{R}(x_1,\ldots, x_n)=\text{Frac}(\mathbb{R}[x_1,\ldots, x_n]).$ This is the space of rational functions in the variables $x_1,\ldots, x_n$. More abstractly, for any integral domain $A$, one can study $A[x_1,\ldots, x_n]$ and $A(x_1,\ldots, x_n)$.

As far as an example of a field containing $\log$ or something like that, we can take $\Omega=\mathbb{C}\setminus [0,\infty)$ and study $$\mathcal{M}(\Omega)=\{\text{meromorphic functions on}\:\Omega\}.$$ Among these functions is $\log(z)$ where we use the branch cut $\text{arg}(z)\in (0,2\pi)$. It can be shown that the ring $\mathcal{M}(\Omega)$ is actually a field, containing $\log$.