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Rational function

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Ratio of polynomial functions

For the use in automata theory, seeFinite-state transducer. For the use in monoid theory, seeRational function (monoid).

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Inmathematics, arational function is anyfunction that can be defined by arational fraction, which is analgebraic fraction such that both thenumerator and thedenominator arepolynomials. Thecoefficients of the polynomials need not berational numbers; they may be taken in anyfieldK. In this case, one speaks of a rational function and a rational fractionoverK. The values of thevariables may be taken in any fieldL containingK. Then thedomain of the function is the set of the values of the variables for which the denominator is not zero, and thecodomain isL.

The set of rational functions over a fieldK is a field, thefield of fractions of thering of thepolynomial functions overK.

Definitions

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A function $f {\displaystyle f}$ is called a rational function if it can be written in the form^[1]

f(x)={\frac {P(x)}{Q(x)}}

where $P {\displaystyle P}$ and $Q {\displaystyle Q}$ arepolynomial functions of $x {\displaystyle x}$ and $Q {\displaystyle Q}$ is not thezero function. Thedomain of $f {\displaystyle f}$ is the set of all values of $x {\displaystyle x}$ for which the denominator $Q(x)$ is not zero.

However, if $\textstyle P$ and $\textstyle Q$ have a non-constantpolynomial greatest common divisor $\textstyle R$ , then setting $\textstyle P=P_{1}R$ and $\textstyle Q=Q_{1}R$ produces a rational function

f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},

which may have a larger domain than $f {\displaystyle f}$ , and is equal to $f {\displaystyle f}$ on the domain of $f . {\displaystyle f.}$ It is a common usage to identify $f {\displaystyle f}$ and $f_{1}$ , that is to extend "by continuity" the domain of $f {\displaystyle f}$ to that of $f_{1}.$ Indeed, one can define a rational fraction as anequivalence class of fractions of polynomials, where two fractions $\textstyle {\frac {A(x)}{B(x)}}$ and $\textstyle {\frac {C(x)}{D(x)}}$ are considered equivalent if $A(x)D(x)=B(x)C(x)$ . In this case $\textstyle {\frac {P(x)}{Q(x)}}$ is equivalent to $\textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.$

Aproper rational function is a rational function in which thedegree of $P(x)$ is less than the degree of $Q(x)$ and both arereal polynomials, named by analogy to aproper fraction in $\mathbb {Q} .$ ^[2]

Complex rational functions

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Incomplex analysis, a rational function

f(z)={\frac {P(z)}{Q(z)}}

is the ratio of two polynomials with complex coefficients, whereQ is not the zero polynomial andP andQ have no common factor (this avoidsf taking the indeterminate value 0/0).

The domain off is the set of complex numbers such that $Q(z)\neq 0$ .Every rational function can be naturally extended to a function whose domain and range are the wholeRiemann sphere, i.e., arational mapping. Iteration of rational functions on the Riemann sphere forms adiscrete dynamical system.^[3]

A complex rational function with degree one is aMöbius transformation.

Rational functions are representative examples ofmeromorphic functions.^[4]

Julia sets for rational maps
${\frac {1}{az^{5}+z^{3}+bz}}$
${\frac {1}{z^{3}+z(-3-3i)}}$
${\frac {z^{2}-0.2+0.7i}{z^{2}+0.917}}$
${\frac {z^{2}}{z^{9}-z+0.025}}$

Degree

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

Most commonly, thedegree of a rational function is the maximum of thedegrees of its constituent polynomialsP andQ, when the fraction is reduced tolowest terms. If the degree off isd, then the equation

f(z)=w\,

hasd distinct solutions inz except for certain values ofw, calledcritical values, where two or more solutions coincide or where some solution is rejectedat infinity (that is, when the degree of the equation decreases after havingcleared the denominator).

Thedegree of thegraph 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 inasymptotic analysis, thedegree of a rational function is the difference between the degrees of the numerator and the denominator.^[5]^: §13.6.1^[6]^{: Chapter IV}

Innetwork synthesis andnetwork analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called abiquadratic function.^[7]

Examples

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Examples of rational functions

Rational function of degree 3, with a graph ofdegree 3:

y={\frac {x^{3}-2x}{2(x^{2}-5)}}

Rational function of degree 2, with a graph ofdegree 3:

y={\frac {x^{2}-3x-2}{x^{2}-4}}

The rational function

f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}

is not defined at

x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.

It is asymptotic to ${\tfrac {x}{2}}$ as $x\to \infty .$

The rational function

f(x)={\frac {x^{2}+2}{x^{2}+1}}

is defined for allreal numbers, but not for allcomplex numbers, since ifx were a square root of $-1$ (i.e. theimaginary unit or its negative), then formal evaluation would lead to division by zero:

f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},

which is undefined.

Aconstant function such asf(x) = π is a rational function since constants are polynomials. The function itself is rational, even though thevalue off(x) is irrational for allx.

Everypolynomial function $f(x)=P(x)$ is a rational function with $Q(x)=1.$ A function that cannot be written in this form, such as $f(x)=\sin(x),$ is not a rational function. However, the adjective "irrational" isnot generally used for functions.

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

The rational function $f(x)={\tfrac {x}{x}}$ is equal to 1 for allx except 0, where there is aremovable 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, sincex/x is equivalent to 1/1.

Taylor series

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The coefficients of aTaylor series of any rational function satisfy alinear recurrence relation, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collectinglike terms after clearing the denominator.

For example,

{\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.

Multiplying through by the denominator and distributing,

1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}

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}.

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

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}.

Combining like terms gives

1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.

Since this holds true for allx in theradius of convergence of the original Taylor series, we can compute as follows. Since theconstant term on the left must equal the constant term on the right it follows that

a_{0}={\frac {1}{2}}.

Then, since there are no powers ofx on the left, all of thecoefficients on the right must be zero, from which it follows that

a_{1}={\frac {1}{4}}

a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.

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 usingpartial fraction decomposition we can write any proper rational function as a sum of factors of the form1 / (ax +b) and expand these asgeometric series, giving an explicit formula for the Taylor coefficients; this is the method ofgenerating functions.

Abstract algebra

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Inabstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from anyfield. In this setting, given a field $F {\displaystyle F}$ and some indeterminate $X {\displaystyle X}$ , arational expression (also known as arational fraction or, inalgebraic geometry, arational function) is any element of thefield of fractions of thepolynomial ring $F[X]$ . Any rational expression can be written as the quotient of two polynomials $P/Q$ with $Q\neq 0$ , although this representation isn't unique. $P/Q$ is equivalent to $R/S$ , for polynomials $P {\displaystyle P}$ , $Q {\displaystyle Q}$ , $R {\displaystyle R}$ , and $S {\displaystyle S}$ , when $PS=QR$ . However, since $F[X]$ is aunique factorization domain, there is aunique representation for any rational expression $P/Q$ with $P {\displaystyle P}$ and $Q {\displaystyle Q}$ polynomials of lowest degree and $Q {\displaystyle Q}$ chosen to bemonic. This is similar to how afraction 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 $F {\displaystyle F}$ by (atranscendental element) $X {\displaystyle X}$ , because $F(X)$ does not contain any proper subfield containing both $F {\displaystyle F}$ and the element $X {\displaystyle X}$ .

Notion of a rational function on an algebraic variety

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Main article:Function field of an algebraic variety

Likepolynomials, rational expressions can also be generalized to $n {\displaystyle n}$ indeterminates $X_{1},\ldots ,X_{n}$ , by taking the field of fractions of $F[X_{1},\ldots ,X_{n}]$ , which is denoted by $F(X_{1},\ldots ,X_{n})$ .

An extended version of the abstract idea of rational function is used in algebraic geometry. There thefunction field of an algebraic variety $V {\displaystyle V}$ is formed as the field of fractions of thecoordinate ring of $V {\displaystyle V}$ (more accurately said, of aZariski-dense affine open set in $V {\displaystyle V}$ ). Its elements $f {\displaystyle f}$ are considered as regular functions in the sense of algebraic geometry on non-empty open sets $U {\displaystyle U}$ , and also may be seen as morphisms to theprojective line.

Applications

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Rational functions are used innumerical analysis forinterpolation andapproximation of functions, for example thePadé approximants introduced byHenri Padé. Approximations in terms of rational functions are well suited forcomputer algebra systems and other numericalsoftware. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

Rational functions are used to approximate or model more complex equations in science and engineering includingfields andforces in physics,spectroscopy in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo,wave functions for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.^{[citation needed]}

Insignal processing, theLaplace transform (for continuous systems) or thez-transform (for discrete-time systems) of theimpulse response of commonly usedlinear time-invariant systems (filters) withinfinite impulse response are rational functions over complex numbers.

References

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^Rudin, Walter (1987).Real and Complex Analysis. New York, NY: McGraw-Hill Education. p. 267.ISBN 978-0-07-100276-9.
^
- Corless, Martin J.; Frazho, Art (2003).Linear Systems and Control. CRC Press. p. 163.ISBN 0203911377.
- Pownall, Malcolm W. (1983).Functions and Graphs: Calculus Preparatory Mathematics. Prentice-Hall. p. 203.ISBN 0133323048.
^Blanchard, Paul (1984)."Complex analytic dynamics on the Riemann sphere".Bulletin of the American Mathematical Society.11 (1):85–141.doi:10.1090/S0273-0979-1984-15240-6.ISSN 0273-0979. p. 87
^Ablowitz, Mark J.;Fokas, Athanassios S. (2003).Complex Variables. Cambridge University Press. p. 150.ISBN 978-0-521-53429-1.
^Bourles, Henri (2010).Linear Systems. Wiley. p. 515.doi:10.1002/9781118619988.ISBN 978-1-84821-162-9. Retrieved5 November 2022.
^Bourbaki, N. (1990).Algebra II. Springer. p. A.IV.20.ISBN 3-540-19375-8.
^Glisson, Tildon H. (2011).Introduction to Circuit Analysis and Design. Springer.ISBN 978-9048194438.

External links

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Dynamic visualization of rational functions with JSXGraph

v t e Function
History List of specific functions
Types by domain and codomain	X → 𝔹 𝔹 → X 𝔹ⁿ → 𝔹 X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
Category