Rational function - Knowlpedia

A rational function is a quotient

R(z)=\frac{P(z)}{Q(z)}

of complex polynomials $P,Q\in\mathbb C[z]$ with $Q\ne0$ , where two quotients represent the same function when they agree after cancellation. It defines a meromorphic function on $\mathbb C$ and extends uniquely to a holomorphic map

R:\widehat{\mathbb C}\longrightarrow\widehat{\mathbb C}

of the Riemann sphere.

Value at infinity

The zero rational function has $R(\infty)=0$ . For nonzero $R$ , choose coprime $P\ne0$ and $Q$ . Then $R(\infty)$ is $0$ if $\deg P<\deg Q$ , the ratio of leading coefficients if the degrees agree, and $\infty$ if $\deg P>\deg Q$ . This is exactly the value obtained in the coordinate $w=1/z$ near infinity.

Degree

For nonconstant $R=P/Q$ in lowest terms,

\deg R=\max(\deg P,\deg Q).

Every value of the sphere has $\deg R$ preimages counted with multiplicity. Degree $1$ rational functions are precisely Möbius transformations; higher-degree maps are branched coverings rather than automorphisms.

Characterization

Every nonconstant holomorphic map $\widehat{\mathbb C}\to\widehat{\mathbb C}$ is rational. More generally, every such map that is not identically $\infty$ is represented by a quotient $P/Q$ ; the constant map with value $\infty$ is represented projectively by $[1:0]$ . Equivalently, the field of meromorphic functions on the projective line is $\mathbb C(z)$ .

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§8–9.