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|page=267}}

</ref>

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

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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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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]]).

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

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.

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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. |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=9048194431978-9048194438}}</ref>

==Examples==

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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]]s.

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

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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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==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 |url=http://nrbook.com/book.html |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==