Rational function

Inmathematics,arational functionis anyfunctionthat can be defined by arational fraction,which is analgebraic fractionsuch that both thenumeratorand thedenominatorarepolynomials.Thecoefficientsof the polynomials need not berational numbers;they may be taken in anyfield $K$ .In this case, one speaks of a rational function and a rational fractionover $K$ .The values of thevariablesmay be taken in any field $L$ containing $K$ .Then thedomainof the function is the set of the values of the variables for which the denominator is not zero, and thecodomainis $L$ .

The set of rational functions over a field $K$ is a field, thefield of fractionsof theringof thepolynomial functionsover $K$ .

Definitions

A function $f$ is called a rational function if it can be written in the form

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

where $P$ and $Q$ arepolynomial functionsof $x$ and $Q$ is not thezero function.Thedomainof $f$ is the set of all values of $x$ 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 $f$ ,and is equal to $f$ on the domain of $f.$ It is a common usage to identify $f$ and $f_{1}$ ,that is to extend "by continuity" the domain of $f$ to that of $f_{1}.$ Indeed, one can define a rational fraction as anequivalence classof 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 functionis a rational function in which thedegreeof $P(x)$ is less than the degree of $Q(x)$ and both arereal polynomials,named by analogy to aproper fractionin $\mathbb {Q}.$ ^[1]

Degree

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

Most commonly, thedegreeof a rational function is the maximum of thedegreesof its constituent polynomials $P$ and $Q$ ,when the fraction is reduced tolowest terms.If the degree of $f$ is $d$ ,then the equation

f(z)=w\,

has $d$ distinct solutions in $z$ except for certain values of $w$ ,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).

In the case ofcomplexcoefficients, a rational function with degree one is aMöbius transformation.

Thedegreeof thegraphof 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,thedegreeof a rational function is the difference between the degrees of the numerator and the denominator.^[2]^: §13.6.1^[3]^{: Chapter IV}

Innetwork synthesisandnetwork analysis,a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called abiquadratic function.^[4]

Examples

Examples of rational functions

Rational function of degree 3, with a graph ofdegree3:

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

Rational function of degree 2, with a graph ofdegree3:

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 ifxwere a square root of $-1$ (i.e. theimaginary unitor 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 functionsuch as $f (x) = π$ is a rational function since constants are polynomials. The function itself is rational, even though thevalueof $f (x)$ is irrational for all $x$ .

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" isnotgenerally used for functions.

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

The rational function $f(x)={\tfrac {x}{x}}$ is equal to 1 for allxexcept 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/xis equivalent to 1/1.

Taylor series

The coefficients of aTaylor seriesof 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 termsafter 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 allxin theradius of convergenceof the original Taylor series, we can compute as follows. Since theconstant termon the left must equal the constant term on the right it follows that

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

Then, since there are no powers ofxon the left, all of thecoefficientson 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 decompositionwe 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 and geometric notion

Inabstract algebrathe 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 fieldFand some indeterminateX,arational expression(also known as arational fractionor, inalgebraic geometry,arational function) is any element of thefield of fractionsof thepolynomial ringF[X]. Any rational expression can be written as the quotient of two polynomialsP/QwithQ≠ 0, although this representation isn't unique.P/Qis equivalent toR/S,for polynomialsP,Q,R,andS,whenPS=QR.However, sinceF[X] is aunique factorization domain,there is aunique representationfor any rational expressionP/QwithPandQpolynomials of lowest degree andQchosen to bemonic.This is similar to how afractionof integers can always be written uniquely in lowest terms by canceling out common factors.

The field of rational expressions is denotedF(X). This field is said to be generated (as a field) overFby (atranscendental element)X,becauseF(X) does not contain any proper subfield containing bothFand the elementX.

Complex rational functions

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

Incomplex analysis,a rational function

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

is the ratio of two polynomials with complex coefficients, where $Q$ is not the zero polynomial and $P$ and $Q$ have no common factor (this avoids $f$ taking the indeterminate value 0/0).

The domain of $f$ 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(complex projective line).

Rational functions are representative examples ofmeromorphic functions.

Iteration of rational functions (maps)^[5]on theRiemann spherecreatesdiscrete dynamical systems.

Notion of a rational function on an algebraic variety

Likepolynomials,rational expressions can also be generalized tonindeterminatesX₁,...,X_n,by taking the field of fractions ofF[X₁,...,X_n], which is denoted byF(X₁,...,X_n).

An extended version of the abstract idea of rational function is used in algebraic geometry. There thefunction field of an algebraic varietyVis formed as the field of fractions of thecoordinate ringofV(more accurately said, of aZariski-denseaffine open set inV). Its elementsfare considered as regular functions in the sense of algebraic geometry on non-empty open setsU,and also may be seen as morphisms to theprojective line.

Applications

Rational functions are used innumerical analysisforinterpolationandapproximationof functions, for example thePadé approximationsintroduced byHenri Padé.Approximations in terms of rational functions are well suited forcomputer algebra systemsand 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 includingfieldsandforcesin physics,spectroscopyin analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo,wave functionsfor 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 responseof commonly-usedlinear time-invariant systems(filters) withinfinite impulse responseare rational functions over complex numbers.

References

^
- 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.
^Bourles, Henri (2010).Linear Systems.Wiley. p. 515.doi:10.1002/9781118619988.ISBN 978-1-84821-162-9.Retrieved5 November2022.
^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 9048194431.
^Camarena, Omar Antolín."Iteration of Rational Functions"(PDF).

"Rational function",Encyclopedia of Mathematics,EMS Press,2001 [1994]
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007),"Section 3.4. Rational Function Interpolation and Extrapolation",Numerical Recipes: The Art of Scientific Computing(3rd ed.), Cambridge University Press,ISBN 978-0-521-88068-8

External links

Dynamic visualization of rational functions with JSXGraph

[1] 
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.

[2] Corless, Martin J.; Frazho, Art (2003).Linear Systems and Control.CRC Press. p. 163.ISBN 0203911377.

[3] Pownall, Malcolm W. (1983).Functions and Graphs: Calculus Preparatory Mathematics.Prentice-Hall. p. 203.ISBN 0133323048.

[2] Bourles, Henri (2010).Linear Systems.Wiley. p. 515.doi:10.1002/9781118619988.ISBN 978-1-84821-162-9.Retrieved5 November2022.

[3] Bourbaki, N.(1990).Algebra II.Springer. p. A.IV.20.ISBN 3-540-19375-8.

[4] Glisson, Tildon H. (2011).Introduction to Circuit Analysis and Design.Springer.ISBN 9048194431.

[5] Camarena, Omar Antolín."Iteration of Rational Functions"(PDF).

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