Content deleted Content added
No edit summary |
Rescuing 0 sources and tagging 1 as dead.) #IABot (v2.0.9.5 |
||
| (29 intermediate revisions by 17 users not shown) | |||
Line 8:
==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>
Line 21 ⟶ 22:
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>
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===
Line 30 ⟶ 51:
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.
Line 37 ⟶ 56:
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=
==Examples==
Line 76 ⟶ 95:
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.
Line 112 ⟶ 133:
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
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>.
===Notion of a rational function on an algebraic variety===
Line 145 ⟶ 146:
==Applications==
Rational functions are used in [[numerical analysis]] for [[interpolation]] and [[approximation]] of functions, for example the [[Padé
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
==See also==
* [[Partial fraction decomposition]]
* [[Partial fractions in integration]]
Line 160:
==References==
{{Reflist}}
==Further reading==
*{{springer|id=Rational_function&oldid=17805|title=Rational function}}
*{{Citation
==External links==
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions Dynamic visualization of rational functions with JSXGraph]
{{Functions navbox}}
{{Authority control}}
| |||