Rational function: Difference between revisions

Revision as of 14:17, 23 November 2022

Revision as of 11:46, 8 January 2023

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which may have a larger ___domain than <math> f(x)</math>, and is equal to <math> f(x)</math> on the ___domain of <math> f(x).</math> It is a common usage to identify <math> f(x)</math> and <math> f_1(x)</math>, that is to extend "by continuity" the ___domain of <math> f(x)</math> to that of <math> f_1(x).</math> Indeed, one can define a rational fraction as an [[equivalence class]] of fractions of polynomials, where two fractions <math>\frac{A(x)}{B(x)}</math> and <math>\frac{C(x)}{D(x)}</math> are considered equivalent if <math>A(x)D(x)=B(x)C(x)</math>. In this case <math>\frac{P(x)}{Q(x)}</math> is equivalent to <math>\frac{P_1(x)}{Q_1(x)}</math>.

A '''proper rational function''' is a rational function in which the [[Degree of a polynomial|degree]] of <math>P(x)</math> is less than the degree of <math>Q(x)</math> and both are [[real polynomial]]s, named by analogy to a [[fraction#Proper and improper fractions|proper fraction]] in <math>\mathbb{Q}</math>.<ref>{{multiref|{{cite book |first1=Martin J. |last1=Corless, |first2=Art |last2=Frazho, ''|title=Linear Systems and Control'', p. |page=163, |publisher=CRC Press, |date=2003 {{|isbn|=0203911377}}.|{{cite book |first1=Malcolm W. |last1=Pownall, ''|title=Functions and Graphs: Calculus Preparatory Mathematics'', p. |page=203, |publisher=Prentice-Hall, |date=1983 {{|isbn|=0133323048}}.}}</ref>

===Degree===

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In some contexts, such as in [[asymptotic analysis]], the ''degree'' of a rational function is the difference between the degrees of the numerator and the denominator.<ref>{{cite book |last1=Bourles |first1=Henri |title=Linear Systems |date=2010 |publisher=Wiley |isbn=978-1-84821-162-9 |page=515 |doi=10.1002/9781118619988 |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781118619988 |access-date=5 November 2022}}</ref>{{rp|at=§13.6.1}}<ref>{{cite book |last1=Bourbaki |first1=N. |title=Algebra II |date=1990 |publisher=Springer |isbn=3-540-19375-8 |page=A.IV.20}}</ref>{{rp|at=Chapter IV}}

In [[network synthesis]] and [[Network analysis (electrical circuits)|network analysis]], a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a '''{{vanchor|biquadratic function}}'''.<ref>{{cite book |last1=Glisson, |first1=Tildon H., ''|title=Introduction to Circuit Analysis and Design'', |publisher=Springer, |date=2011 {{ISBN|isbn=9048194431}}.</ref>

==Examples==

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Rational functions are representative examples of [[meromorphic function]]s.

Iteration of rational functions (maps)<ref>[{{cite web |url=https://www.matem.unam.mx/~omar/no-wandering-domains.pdf |title=Iteration of Rational Functions by |first=Omar Antolín |last=Camarena]}}</ref> on the [[Riemann sphere]] creates [[Discrete dynamical system|discrete dynamical systems]].

===Notion of a rational function on an algebraic variety===

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{{Reflist}}

*{{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| publication-place=New York|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}}

==External links==