Rational function: Difference between revisions

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==Definitions==

A function <math>f</math> is called a rational function if it can be written in the form<ref>{{cite book | last=Rudin | first=Walter |author-link=Walter Rudin | title=Real and Complex Analysis | publisher=McGraw-Hill Education | publication-place=New York, NY | date=1987 | isbn=978-0-07-100276-9|ppage=267}}

</ref>

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The ___domain of {{mvar|f}} is the set of complex numbers such that <math>Q(z)\ne 0</math>.

Iteration ofEvery rational functionsfunction oncan be naturally extended to a function whose ___domain and range are the whole [[Riemann sphere]], (i.e., a [[rational mapping]]). createsIteration of rational functions on the Riemann sphere forms a [[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>

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 | lastlast1=Ablowitz | firstfirst1=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|ppage=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>

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==Applications==

Rational functions are used in [[numerical analysis]] for [[interpolation]] and [[approximation]] of functions, for example the [[Padé approximationapproximant]]s introduced by [[Henri Padé]]. Approximations in terms of rational functions are well suited for [[computer algebra system]]s and other numerical [[software]]. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials. <!-- Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation. -->

Rational functions are used to approximate or model more complex equations in science and engineering including [[field (physics)|field]]s and [[force]]s in physics, [[spectroscopy]] in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, [[wave function]]s for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.{{Citation needed|date=April 2017}}

In [[signal processing]], the [[Laplace transform]] (for continuous systems) or the [[z-transform]] (for discrete-time systems) of the [[impulse response]] of commonly- used [[linear time-invariant system]]s (filters) with [[infinite impulse response]] are rational functions over complex numbers.

==See also==

* [[Field of fractions]]

* [[Partial fraction decomposition]]

* [[Partial fractions in integration]]

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==Further reading==

*{{springer|id=Rational_function&oldid=17805|title=Rational function}}

*{{Citation |last1=Press|first1=W.H.|last2=Teukolsky|first2=S.A.|last3=Vetterling|first3=W.T.|last4=Flannery|first4=B.P.|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press |isbn=978-0-521-88068-8|chapter=Section 3.4. Rational Function Interpolation and Extrapolation|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=124}}{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }}

* J. L. Walsh: ''Interpolation and Approximation by Rational Functions in the Complex Domain'', AMS (Colloquium Publications 20), ISBN 0-8218-1020-0 (1960).

==External links==