2
$\begingroup$

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!

$\endgroup$
0
$\begingroup$

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 $\log$.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – ChrisWong Jan 4 at 17:46
  • $\begingroup$ $\sin$ 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 $z=i$. $\endgroup$ – Antonios-Alexandros Robotis Jan 4 at 17:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.