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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}}
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}}.

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</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|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

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

which may have a larger domain than <math>f</math>, and is equal to <math>f</math> on the domain of <math>f.</math> It is a common usage to identify <math> f</math> and <math> f_1</math>, that is to extend "by continuity" the domain of <math>f</math> to that of <math>f_1.</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>

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>
<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===
There are several non equivalent definitions of the degree of a rational function.

Most commonly, the ''degree'' of a rational function is the maximum of the [[degree of a polynomial|degrees]] of its constituent polynomials {{math|''P''}} and {{math|''Q''}}, when the fraction is reduced to [[lowest terms]]. If the degree of {{math|''f''}} is {{math|''d''}}, then the equation

:<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 decreases 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 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}}

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=978-9048194438}}</ref>

==Examples==
{{multiple image
 | header = Examples of rational functions
 | align = right
 | direction = vertical
 | width = 300
 | image1 = RationalDegree3.svg
 | alt1 = Rational function of degree 3
 | caption1 = Rational function of degree 3, with a graph of [[degree of an algebraic variety|degree]]&nbsp;3: <math>y = \frac{x^3-2x}{2(x^2-5)}</math>
 | image2 = RationalDegree2byXedi.svg
 | alt2 = Rational function of degree 2
 | caption2 = Rational function of degree 2, with a graph of [[degree of an algebraic variety|degree]]&nbsp;3: <math>y = \frac{x^2-3x-2}{x^2-4}</math>
}}

The rational function

:<math>f(x) = \frac{x^3-2x}{2(x^2-5)}</math>

is not defined at

:<math>x^2=5 \Leftrightarrow x=\pm \sqrt{5}.</math>

It is asymptotic to <math>\tfrac{x}{2}</math> as <math>x\to \infty.</math>

The rational function

:<math>f(x) = \frac{x^2 + 2}{x^2 + 1}</math>

is defined for all [[real number]]s, but not for all [[complex number]]s, since if ''x'' were a square root of <math>-1</math> (i.e. the [[imaginary unit]] or its negative), then formal evaluation would lead to division by zero:

:<math>f(i) = \frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0},</math>

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.

==Taylor series==
The coefficients of a [[Taylor series]] of any rational function satisfy a [[Recurrence relation|linear recurrence relation]], which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting [[like terms]] after clearing the denominator.

For example,

:<math>\frac{1}{x^2 - x + 2} = \sum_{k=0}^{\infty} a_k x^k.</math>

Multiplying through by the denominator and distributing,

:<math>1 = (x^2 - x + 2) \sum_{k=0}^{\infty} a_k x^k</math>

:<math>1 = \sum_{k=0}^{\infty} a_k x^{k+2} - \sum_{k=0}^{\infty} a_k x^{k+1} + 2\sum_{k=0}^{\infty} a_k x^k.</math>

After adjusting the indices of the sums to get the same powers of ''x'', we get

:<math>1 = \sum_{k=2}^{\infty} a_{k-2} x^k - \sum_{k=1}^{\infty} a_{k-1} x^k + 2\sum_{k=0}^{\infty} a_k x^k.</math>

Combining like terms gives

:<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>

Then, since there are no powers of ''x'' on the left, all of the [[coefficient]]s on the right must be zero, from which it follows that

:<math>a_1 = \frac{1}{4}</math>

:<math>a_k = \frac{1}{2} (a_{k-1} - a_{k-2})\quad \text{for}\ k \ge 2.</math>

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== <!-- 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''.

===Notion of a rational function on an algebraic variety=== 
{{Main|Function field of an algebraic variety}}

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

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 approximant]]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==
* [[Partial fraction decomposition]]
* [[Partial fractions in integration]]
* [[Function field of an algebraic variety]]
* [[Algebraic fraction]]s{{snd}}a generalization of rational functions that allows taking integer roots

==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}}

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

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[[Category:Algebraic varieties]]
[[Category:Morphisms of schemes]]
[[Category:Meromorphic functions]]
[[Category:Rational functions| ]]