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.

A Padé approximant is a rational function with numerator degree at most $m$ and denominator degree at most $n$, normalized by $Q_n(0)=1$, such that $Q_n(z)f(z)-P_m(z)=O(z^{m+n+1})$ near zero, when such an approximant exists. For the exponential function, its $[1/1]$ approximant is $(1+z/2)/(1-z/2)$. Applying this rational function to a matrix gives the Crank-Nicolson method. Diagonal and subdiagonal approximants also give stability functions of Gauss collocation methods and Radau IIA methods respectively.

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 $a$.

For $A\ne 0$ and $m\ne n$, combining the right-hand fractions gives numerator $A(m-n)$ and proves the identity. Integration on an interval avoiding the poles gives $\ln |(A+mx)/(A+nx)|/[A(m-n)]$ plus a constant. The overall $1/A$ cannot be dropped unless the variable has already been normalized to $A=1$.

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.

The local multiplicity in zeros and poles is positive for a zero and negative for a pole; it is zero at a regular nonvanishing point.