How do i define 'complex rational function'?

Question

http://en.wikipedia.org/wiki/Rational_function

I don't get the definition in wikipedia.

It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely extended complex plane. So that $\frac{0}{\infty}=0$, $\frac{\infty}{0}=\infty$ and such of these forms are well defined except $\frac{0}{0}$ and $\frac{\infty}{\infty}$

Here's what i figured out to be a right definition:

=My guess for the definition of complex rational function=

Let $\mathbb{C}[X]$ be the ring of complex polynomials.

Define $G=\{(F(X),G(X))\in \mathbb{C}[X]\times \mathbb{C}[X]\setminus\{0\}: \text { $F(X),G(X)$ are relatively prime } \}$

Let $(F(X),G(X))\in G$

Define $r(z)=\frac{F(z)}{G(z)}, \forall z\in\overline{\mathbb{C}}$

Then, $r:\overline{\mathbb{C}}\rightarrow \overline{\mathbb{C}}$ is said to be a rational function.

I think this is what wikipedia describes but since $\frac{\infty}{\infty}$ is not defined, $r$ may not be defined.

Is there any way to resolve this? Or if this is not the right definition, what it would be?

Answer 1

$\infty$ is not really a first-class arithmetic citizen in this case.

What one usually does is to consider $r(z)=\infty$ the encoding of "$r$ has a pole at $z$".

When doing arithmetic on rational functions such as $r/s$ you have a choice between

1. Do it symbolically, following the usual fraction rules, and canceling common factors.

2. Do pointwise arithmetic on all points where the operands are both finite (and, in the case of a denominator, nonzero). There will be finitely many such points because a nonzero polynomial has only finintely many roots. Afterwards fill in the value of any removable singularities.

The function determined by the symbolic result of (1) is the same as the one that results from (2).

---

Your definition of $G$ is slightly off. Usually you would require that $F$ and $G$ are coprime and $G$ is monic -- otherwise there will be many representations of the same function that differ by a common factor above and below the fraction bar.

Alternatively start with $\mathbb C[X]\times(\mathbb C[X]\setminus\{0\})$ and quotient out the relation $$ F/G \sim F'/G' \iff FG'=F'G $$ This is how a field of fractions is defined abstractly, because it doesn't depend on having a canonical choice of representative such as "take the monic one" that needs to look "inside" the structure of the ring elements.

Answer 2

Both the arguments and the values of these rational functions should be viewed as being in $\mathbb C\cup\{\infty\}$. At points where the denominator is $0$ and the numerator is a non-zero complex number or $\infty$, the value of the function is $\infty$. Where the argument to the function is $\infty$, the value of the function is a limit as $z$ approaches $\infty$. This is either some finite complex number or $\infty$. With some functions, like $e^z$, there is no well defined value at $\infty$ in this sense, but with rational functions, there always is. The expressions $0/0$ and $\infty/\infty$ don't arise with rational functions except when there is a well defined limit that is either a complex number or $\infty$.