Rational function

A rational function is a quotient $p/q$ of polynomials, with $q$ nonzero. After cancelling common factors, it defines a holomorphic map from the Riemann sphere to itself.

Over a field in which the denominator splits, a proper rational function is a sum of terms $c/(x-a{)}^k$. The coefficient at a simple pole $a$ is obtained by multiplying by $x-a$ and evaluating at $a$.

Let $P$ be a monic polynomial of positive degree $m\ge 1$ with distinct roots of a polynomial $r_j$. The partial fraction decompositionfollows by matching the simple pole residues; the difference is entire and vanishes at infinity. Comparing its expansion there gives ${\sum }_j{r}_j^{\ell }/P^{\prime }(r_j)=0$ for $0\le \ell <m-1$ and ${\sum }_j{r}_j^{m-1}/P^{\prime }(r_j)=1$. These identities give the continuity and derivative jumps needed for constant-coefficient Green functions.

The order of vanishing of a nonzero rational function $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.