Editing Rational function (section)

===Complex rational functions===
In [[complex analysis]], a rational function

:<math>f(z) = \frac{P(z)}{Q(z)}</math>

is the ratio of two polynomials with complex coefficients, where {{math|''Q''}} is not the zero polynomial and {{math|''P''}} and {{math|''Q''}} have no common factor (this avoids {{math|''f''}} taking the indeterminate value 0/0).

The domain of {{mvar|f}} is the set of complex numbers such that <math>Q(z)\ne 0</math>.
Every rational function can be naturally extended to a function whose domain and range are the whole [[Riemann sphere]] ([[complex projective line]]).

A complex rational function with degree one is a [[Möbius transformation]].

Rational functions are representative examples of [[meromorphic function]]s.<ref>{{cite book | last1=Ablowitz | first1=Mark J. | author1-link = Mark Ablowitz | last2=Fokas | first2=Athanassios S. | author2-link=Athanassios Fokas | title=Complex Variables | publisher=Cambridge University Press | date=2003 | isbn=978-0-521-53429-1|page=150}}</ref>

Iteration of rational functions on the [[Riemann sphere]] (i.e. a [[rational mapping]]) creates [[discrete dynamical system]]s.<ref>{{cite journal | last=Blanchard | first=Paul | title=Complex analytic dynamics on the Riemann sphere | journal=Bulletin of the American Mathematical Society | volume=11 | issue=1 | date=1984 | issn=0273-0979 | doi=10.1090/S0273-0979-1984-15240-6 | doi-access=free | pages=85–141|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-11/issue-1/Complex-analytic-dynamics-on-the-Riemann-sphere/bams/1183551835.full}} p. 87</ref>
<gallery caption = "[[Julia set]]s for rational maps ">
Julia set f(z)=1 over az5+z3+bz.png| <math>\frac{1}{ az^5+z^3+bz}</math>
Julia set f(z)=1 over z3+z*(-3-3*I).png|<math>\frac{1}{z^3+z(-3-3i)}</math>
Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917.png|<math>\frac{z^2 - 0.2 + 0.7i}{z^2 + 0.917}</math>
Julia set for f(z)=z2 over (z9-z+0.025).png| <math>\frac{z^2}{z^9 - z + 0.025}</math>
</gallery>