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In mathematics, a rational function is any
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that can be defined by a rational fraction, which is an
algebraic fraction In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmet ...
such that both the
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
and the
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s. The coefficients of the polynomials need not be
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s; they may be taken in any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s may be taken in any field ''L'' containing ''K''. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
is ''L''. The set of rational functions over a field ''K'' is a field, the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of the polynomial functions over ''K''.


Definitions

A function f(x) is called a rational function if and only if it can be written in the form : f(x) = \frac where P\, and Q\, are polynomial functions of x\, and Q\, is not the
zero function 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
. The domain of 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-constant
polynomial greatest common divisor In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common di ...
\textstyle R, then setting \textstyle P=P_1R and \textstyle Q=Q_1R produces a rational function : f_1(x) = \frac, which may have a larger domain than f(x), and is equal to f(x) on the domain of f(x). It is a common usage to identify f(x) and f_1(x), that is to extend "by continuity" the domain of f(x) to that of f_1(x). Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions \frac and \frac are considered equivalent if A(x)D(x)=B(x)C(x). In this case \frac is equivalent to \frac. A proper rational function is a rational function in which the degree of P(x) is less than the degree of Q(x) and both are real polynomials, named by analogy to a
proper fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
in \mathbb.


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 degrees of its constituent polynomials and , when the fraction is reduced to
lowest terms An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). I ...
. If the degree of is , then the equation :f(z) = w \, has distinct solutions in except for certain values of , called ''critical values'', where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator). In the case of
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coefficients, a rational function with degree one is a '' Möbius transformation''. The degree of the 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. In
network synthesis Network synthesis is a design technique for linear circuit, linear electrical circuits. Synthesis starts from a prescribed electrical impedance, impedance function of frequency or frequency response and then determines the possible networks that ...
and
network analysis Network analysis can refer to: * Network theory, the analysis of relations through mathematical graphs ** Social network analysis, network theory applied to social relations * Network analysis (electrical circuits) See also *Network planning and ...
, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a .


Examples

The rational function :f(x) = \frac is not defined at :x^2=5 \Leftrightarrow x=\pm \sqrt. It is asymptotic to \tfrac as x\to \infty. The rational function :f(x) = \frac is defined for all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, but not for all
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, since if ''x'' were a square root of -1 (i.e. the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
or its negative), then formal evaluation would lead to division by zero: :f(i) = \frac = \frac = \frac, which is undefined. A constant function such as ''f''(''x'') = π is a rational function since constants are polynomials. The function itself is rational, even though the value of ''f''(''x'') is irrational for all ''x''. Every polynomial 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" is not generally used for functions. The rational function f(x) = \tfrac is equal to 1 for all ''x'' except 0, where there is a
removable singularity In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbour ...
. 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of any rational function satisfy a
linear recurrence relation In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is line ...
, 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, :\frac = \sum_^ a_k x^k. Multiplying through by the denominator and distributing, :1 = (x^2 - x + 2) \sum_^ a_k x^k :1 = \sum_^ a_k x^ - \sum_^ a_k x^ + 2\sum_^ a_k x^k. After adjusting the indices of the sums to get the same powers of ''x'', we get :1 = \sum_^ a_ x^k - \sum_^ a_ x^k + 2\sum_^ a_k x^k. Combining like terms gives :1 = 2a_0 + (2a_1 - a_0)x + \sum_^ (a_ - a_ + 2a_k) x^k. 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 In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial :x^2 + 2x + 3,\ the 3 is a constant term. After like terms are com ...
on the left must equal the constant term on the right it follows that :a_0 = \frac. Then, since there are no powers of ''x'' on the left, all of the coefficients on the right must be zero, from which it follows that :a_1 = \frac :a_k = \frac (a_ - a_)\quad \text\ k \ge 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 using
partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
we can write any proper rational function as a sum of factors of the form and expand these as
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
, giving an explicit formula for the Taylor coefficients; this is the method of
generating functions In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
.


Abstract algebra and geometric notion

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. In this setting given a field ''F'' and some indeterminate ''X'', a rational expression is any element of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''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 In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
, there is a unique representation for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be monic. This is similar to how a 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 In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic over ...
) ''X'', because ''F''(''X'') does not contain any proper subfield containing both ''F'' and the element ''X''.


Complex rational functions

Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
s for rational maps "> Julia set f(z)=1 over az5+z3+bz.png, \frac Julia set f(z)=1 over z3+z*(-3-3*I).png, \frac Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917.png, \frac Julia set for f(z)=z2 over (z9-z+0.025).png, \frac In complex analysis, a rational function :f(z) = \frac is the ratio of two polynomials with complex coefficients, where is not the zero polynomial and and have no common factor (this avoids taking the indeterminate value 0/0). The domain of is the set of complex numbers such that Q(z)\ne 0. Every rational function can be naturally extended to a function whose domain and range are the whole
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
(
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers p ...
). Rational functions are representative examples of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s. Iteration of rational functions (maps)Iteration of Rational Functions by Omar Antolín Camarena
/ref> on the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
creates discrete dynamical systems.


Notion of a rational function on an algebraic variety

Like polynomials, rational expressions can also be generalized to ''n'' indeterminates ''X''1,..., ''X''''n'', by taking the field of fractions of ''F'' 'X''1,..., ''X''''n'' which is denoted by ''F''(''X''1,..., ''X''''n''). An extended version of the abstract idea of rational function is used in algebraic geometry. There the
function field of an algebraic variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these ...
''V'' is formed as the field of fractions of the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
of ''V'' (more accurately said, of a Zariski-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 In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
.


Applications

Rational functions are used in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
for interpolation and approximation of functions, for example the Padé approximations introduced by
Henri Padé Henri Eugène Padé (; 17 December 1863 – 9 July 1953) was a French mathematician, who is now remembered mainly for his development of Padé approximation techniques for functions using rational functions. Education and career Padà ...
. Approximations in terms of rational functions are well suited for
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The d ...
s and other numerical
software Software is a set of computer programs and associated software documentation, documentation and data (computing), data. This is in contrast to Computer hardware, hardware, from which the system is built and which actually performs the work. ...
. 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 including fields and forces 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. In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
(for continuous systems) or the
z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
(for discrete-time systems) of the impulse response of commonly-used linear time-invariant systems (filters) with
infinite impulse response Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response h(t) which does not become exactly zero past a certain point, but continues indefinitely. This is in ...
are rational functions over complex numbers.


See also

*
Field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
*
Partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
*
Partial fractions in integration In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
*
Function field of an algebraic variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these ...
*
Algebraic fraction In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmet ...
sa generalization of rational functions that allows taking integer roots


References

* *{{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, publication-place=New York, 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


Dynamic visualization of rational functions with JSXGraph
Algebraic varieties Morphisms of schemes Meromorphic functions