# Example of a field of functions containing ln(x)

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!

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 $$\log$$.
• That makes sense, would the field of meromorphic functions also include sin(z) and cos(z)? (I haven't taken complex analysis yet so my knowledge of this stuff is a little shaky). Is the set of holomorphic functions on $\Omega$ also a field? Jan 4, 2019 at 17:46
• $\sin$ and $\cos$ would also be included. The holomorphic functions are not a field, because if you take $1/(z-i)$ (for instance), we can see that it isn't holomorphic: it has a singularity at $z=i$. Jan 4, 2019 at 17:47