The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent to the field of rational functions

Question

I’d like to prove the following statement from my lecture notes:

The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent to the field of rational functions, with the convention $1/0=\infty$.

In particular, I want to see that any rational function $f(z)=p(z)/q(z)$ is meromorphic. I’ve been told as a hint that the proof should begin by looking at $f(1/z) = p(1/z)/q(1/z)$.

I’m trying to do that. Let’s say $p(z):=a_pz^p+\ldots+a_1z+a_0$, $q(z):=b_qz^q+\ldots+b_1z+b_0$, and $b_q, a_p \neq 0$.

Then if $p<q$, we get

$f(1/z)=\frac{a_pz^{q-p}+\ldots+a_1z^{q-1}+a_0z^q}{b_q+\ldots+b_1z^{q-1}+b_0z^q}$

…which is holomorphic when $z \rightarrow 0$ since the denominator is not zero, right?

If $p=q$, we get

$$f(1/z)=\frac{a_p+\ldots+a_0z^q}{b_q+\ldots+b_0z^q}$$

…which is holomorphic on $\mathbb{C}$.

But I’m not sure what to do for $p>q$. Any help would be welcome.

(I’ve looked for similar questions but I haven’t found this particular proof anywhere and I’d like to understand the statement by my own means).

Answer 1

I would use the fundamental theorem of algebra to write $p(z)$ and $q(z)$ as products of linear factors. Thus, any non-zero rational function $f(z)$ can be written in the form

$$ f(z) = A \times z^l \times \frac{\prod_{i=1}^M(z - a_i)^{p_j}}{\prod_{j=1}^N(z - b_j)^{q_j}},$$ where the $a_i$'s and $b_j$'s are distinct points in $\mathbb C \setminus \{0\}$, and the exponents $p_j, q_j$ are positive integers. The exponent $l$ is an integer that can be positive, negative or zero. $A$ is a non-zero constant.

Now observe that

$$f(1/w) = A \times w^{(\sum_{j=1}^N q_j-\sum_{i=1}^M p_i-l)}\times \frac{\prod_{i=1}^M(1 - a_i w)^{p_i}}{\prod_{j=1}^N(1 - b_j w)^{q_j}}.$$

The function $w \mapsto f(1/w)$ is evidently a meromorphic function on $\mathbb C$: it has a pole of order $q_j$ at $w = 1/b_j$ for each $j \in \{ 1, \dots, N\}$. Additionally, if $\sum_{i=1}^M p_i- \sum_{j=1}^N q_j+ l > 0$, then this function has a pole of order $\sum_{i=1}^M p_i - \sum_{j=1}^N q_j + l$ at $w =0$.

Let me state our findings once more, using a different choice of words. Let $f(z) = p(z) / q(z)$, where $p(z)$ and $q(z)$ are polynomials.

• If $\text{deg }p > \text{deg }q$, then $f$ has a pole of order $\text{deg }p - \text{deg } q$ at the point at infinity.
• If $\text{deg }p < \text{deg }q$, then $f$ has a zero of order $\text{deg }q - \text{deg }p$ at the point at infinity.