Rational function: Difference between revisions

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{{Short description|Ratio of polynomial functions}}

{{About||the use in automata theory|Finite-state transducer|the use in monoid theory|Rational function (monoid)}}

{{Use American English|date = January 2019}}

{{More footnotes needed|date=September 2015}}

{{Short description|Ratio of polynomial functions}}

In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] that can be defined by a '''rational fraction''', which is an [[algebraic fraction]] such that both the [[numerator]] and the [[denominator]] are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s; they may be taken in any [[field (mathematics)|field]] ''{{mvar|K''}}. In this case, one speaks of a rational function and a rational fraction ''over {{mvar|K}}''. The values of the [[variable (mathematics)|variable]]s may be taken in any field ''{{mvar|L''}} containing ''{{mvar|K''}}. Then the [[___domain (function)|___domain]] of the function is the set of the values of the variables for which the denominator is not zero, and the [[codomain]] is ''{{mvar|L''}}.

{{More footnotes|date=September 2015}}

In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] that can be defined by a '''rational fraction''', which is an [[algebraic fraction]] such that both the [[numerator]] and the [[denominator]] are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s; they may be taken in any [[field (mathematics)|field]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)|variable]]s may be taken in any field ''L'' containing ''K''. Then the [[___domain (function)|___domain]] of the function is the set of the values of the variables for which the denominator is not zero, and the [[codomain]] is ''L''.

The set of rational functions over a field ''{{mvar|K''}} is a field, the [[field of fractions]] of the [[ring (mathematics)|ring]] of the [[polynomial function]]s over ''{{mvar|K''}}.

==Definitions==

A function <math>f(x)</math> is called a rational function if and only 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|page=267}}

</ref>

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

where <math>P\,</math> and <math>Q\,</math> are [[polynomial function]]s of <math>x\,</math> and <math>Q\,</math> is not the [[zero function]]. The [[___domain of a function|___domain]] of <math>f\,</math> is the set of all values of <math>x\,</math> for which the denominator <math>Q(x)\,</math> is not zero.

However, if <math>\textstyle P</math> and <math>\textstyle Q</math> have a non-constant [[polynomial greatest common divisor]] <math>\textstyle R</math>, then setting <math>\textstyle P=P_1R</math> and <math>\textstyle Q=Q_1R</math> produces a rational function

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:<math> f_1(x) = \frac{P_1(x)}{Q_1(x)}, </math>

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>\textstyle \frac{A(x)}{B(x)}</math> and <math>\textstyle \frac{C(x)}{D(x)}</math> are considered equivalent if <math>A(x)D(x)=B(x)C(x)</math>. In this case <math>\textstyle \frac{P(x)}{Q(x)}</math> is equivalent to <math>\textstyle \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 |page=163 |publisher=CRC Press |date=2003 |isbn=0203911377}}|{{cite book |first1=Malcolm W. |last1=Pownall |title=Functions and Graphs: Calculus Preparatory Mathematics |page=203 |publisher=Prentice-Hall |date=1983 |isbn=0133323048}}}}</ref>

===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]], i.e., a [[rational mapping]]. Iteration of rational functions on the Riemann sphere forms a [[discrete dynamical system]].<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>

InA the case of [[complex number|complex]] coefficients, a 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>

<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>

===Degree===

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:<math>f(z) = w \,</math>

has {{math|''d''}} distinct solutions in {{math|''z''}} except for certain values of {{math|''w''}}, called ''critical values'', where two or more solutions coincide or where some solution is rejected [[point at infinity|at infinity]] (that is, when the degree of the equation decreasedecreases after having [[clearing denominators|cleared the denominator]]).

The [[degree of an algebraic variety|degree]] of the [[graph of a function|graph]] of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.

In the case of [[complex number|complex]] coefficients, a rational function with degree one is a ''[[Möbius transformation]]''.

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. |authorlink = Nicolas Bourbaki|title=Algebra II |date=1990 |publisher=Springer |isbn=3-540-19375-8 |page=A.IV.20}}</ref>{{rp|at=Chapter IV}}

The [[degree of an algebraic variety|degree]] of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.

In some[[network contexts,synthesis]] such as inand [[asymptoticNetwork analysis (electrical circuits)|network analysis]], the ''degree'' of a rational function isof thedegree differencetwo between(that is, the degreesratio of thetwo numeratorpolynomials andof thedegree denominator.<ref>{{citeat bookmost |last1=Bourlestwo) |first1=Henriis |title=Linearoften Systemscalled |date=2010a '''{{vanchor|publisher=Wileybiquadratic |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 2022function}}</ref>{{rp|at=§13'''.6.1}}<ref>{{cite book |last1=BourbakiGlisson |first1=NTildon H. |title=AlgebraIntroduction IIto |date=1990Circuit Analysis and Design |publisher=Springer |isbndate=3-540-19375-82011 |pageisbn=A.IV.20978-9048194438}}</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=9048194431}}</ref>

==Examples==

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which is undefined.

A [[constant function]] such as {{math|''f''(''x'') {{=}} π}} is a rational function since constants are polynomials. The function itself is rational, even though the [[value (mathematics)|value]] of {{math|''f''(''x'')}} is irrational for all ''{{mvar|x''}}.

Every [[polynomial function]] <math>f(x) = P(x)</math> is a rational function with <math>Q(x) = 1.</math> A function that cannot be written in this form, such as <math>f(x) = \sin(x),</math> is not a rational function. However, the adjective "irrational" is '''not''' generally used for functions.

Every [[Laurent polynomial]] can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a [[subring]] of the rational functions.

The rational function <math>f(x) = \tfrac{x}{x}</math> is equal to 1 for all ''x'' except 0, where there is a [[removable singularity]]. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since ''x''/''x'' is equivalent to 1/1.

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:<math>1 = 2a_0 + (2a_1 - a_0)x + \sum_{k=2}^{\infty} (a_{k-2} - a_{k-1} + 2a_k) x^k.</math>

Since this holds true for all ''x'' in the [[radius of convergence]] of the original Taylor series, we can compute as follows. Since the [[constant term]] on the left must equal the constant term on the right it follows that

:<math>a_0 = \frac{1}{2}.</math>

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Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using [[partial fraction|partial fraction decomposition]] we can write any proper rational function as a sum of factors of the form {{nowrap|1 / (''ax'' + ''b'')}} and expand these as [[geometric series]], giving an explicit formula for the Taylor coefficients; this is the method of [[generating functions]].

==Abstract algebra and geometric notion== <!-- Rational expression redirects here -->

In [[abstract algebra]] the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any [[field (mathematics)|field]]. In this setting, given a field ''F'' and some indeterminate ''X'', a '''rational expression''' (also known as a '''rational fraction''' or, in [[algebraic geometry]], a '''rational function''') is any element of the [[field of fractions]] of the [[polynomial ring]] ''F''[''X'']. Any rational expression can be written as the quotient of two polynomials ''P''/''Q'' with ''Q'' ≠ 0, although this representation isn't unique. ''P''/''Q'' is equivalent to ''R''/''S'', for polynomials ''P'', ''Q'', ''R'', and ''S'', when ''PS'' = ''QR''. However, since ''F''[''X''] is a [[unique factorization ___domain]], there is a [[irreducible fraction|unique representation]] for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be [[monic polynomial|monic]]. This is similar to how a [[Fraction (mathematics)|fraction]] of integers can always be written uniquely in lowest terms by canceling out common factors.

The field of rational expressions is denoted ''F''(''X''). This field is said to be generated (as a field) over ''F'' by (a [[transcendental element]]) ''X'', because ''F''(''X'') does not contain any proper subfield containing both ''F'' and the element ''X''.

===Complex rational functions===

<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>

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]]).

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 |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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Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''₁,..., ''X''_''n'', by taking the field of fractions of ''F''[''X''₁,..., ''X''_''n''], which is denoted by ''F''(''X''₁,..., ''X''_''n'').

An extended version of the abstract idea of rational function is used in algebraic geometry. There the [[function field of an algebraic variety]] ''V'' is formed as the field of fractions of the [[coordinate ring]] of ''V'' (more accurately said, of a [[Zariski topology|Zariski]]-[[dense subset|dense]] affine open set in ''V''). Its elements ''f'' are considered as regular functions in the sense of algebraic geometry on non-empty open sets ''U'', and also may be seen as morphisms to the [[projective line]].

==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 fields[[field (physics)|field]]s and forces[[force]]s in physics, [[spectroscopy]] in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, [[wave functionsfunction]]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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==References==

{{Reflist}}

==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 }}

==External links==

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions Dynamic visualization of rational functions with JSXGraph]

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