In

Julia set f(z)=1 over az5+z3+bz.png, $\backslash frac$
Julia set f(z)=1 over z3+z*(-3-3*I).png, $\backslash frac$
Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917.png, $\backslash frac$
Julia set for f(z)=z2 over (z9-z+0.025).png, $\backslash frac$
In complex analysis, a rational function
:$f(z)\; =\; \backslash frac$
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 $Q(z)\backslash ne\; 0,$ and its range is the set of the complex numbers such that $P(z)\backslash ne\; wQ(z).$
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 functions.
Iteration of rational functions (maps)Iteration of Rational Functions by Omar Antolín Camarena

/ref> on the on the Riemann sphere creates Discrete dynamical system, discrete dynamical systems.

_{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 ''V'' is formed as the field of fractions of the coordinate ring 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.

Dynamic visualization of rational functions with JSXGraph

Algebraic varieties Morphisms of schemes Meromorphic functions Rational functions,

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a rational function is any function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

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 arithmeti ...

such that both the numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

and the denominator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...

are polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s. The coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of the polynomials need not be rational numbers; they may be taken in any field (mathematics), field ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variable (mathematics), variables may be taken in any field ''L'' containing ''K''. Then the domain (function), domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is ''L''.
The set of rational functions over a field ''K'' is a field, the field of fractions of the ring (mathematics), ring 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 :$f(x)\; =\; \backslash frac$ where $P\backslash ,$ and $Q\backslash ,$ are polynomial functions of $x\backslash ,$ and $Q\backslash ,$ is not the zero function. The domain of a function, domain of $f\backslash ,$ is the set of all values of $x\backslash ,$ for which the denominator $Q(x)\backslash ,$ is not zero. However, if $\backslash textstyle\; P$ and $\backslash textstyle\; Q$ have a non-constant polynomial greatest common divisor $\backslash textstyle\; R$, then setting $\backslash textstyle\; P=P\_1R$ and $\backslash textstyle\; Q=Q\_1R$ produces a rational function :$f\_1(x)\; =\; \backslash frac,$ 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 $\backslash frac$ and $\backslash frac$ are considered equivalent if $A(x)D(x)=B(x)C(x)$. In this case $\backslash frac$ is equivalent to $\backslash frac$. A proper rational function is a rational function in which the Degree of a polynomial, degree of $P(x)$ is less than the degree of $Q(x)$ and both are real polynomials, named by analogy to a fraction#Proper and improper fractions, proper fraction in $\backslash mathbb$.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 degree of a polynomial, degrees of its constituent polynomials and , when the fraction is reduced to lowest terms. If the degree of is , then the equation :$f(z)\; =\; w\; \backslash ,$ has distinct solutions in except for certain values of , 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 decrease 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 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 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 .Examples

The rational function :$f(x)\; =\; \backslash frac$ is not defined at :$x^2=5\; \backslash Leftrightarrow\; x=\backslash pm\; \backslash sqrt.$ It is asymptotic to $\backslash tfrac$ as $x\backslash to\; \backslash infty.$ The rational function :$f(x)\; =\; \backslash frac$ is defined for all real numbers, but not for all complex numbers, since if ''x'' were a square root of $-1$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero: :$f(i)\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac,$ 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 (mathematics), 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 $f(x)\; =\; \backslash sin(x),$ is not a rational function. However, the adjective "irrational" is not generally used for functions. The rational function $f(x)\; =\; \backslash tfrac$ 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.Taylor series

The coefficients of a Taylor series of any rational function satisfy a Recurrence relation, linear recurrence relation, 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, :$\backslash frac\; =\; \backslash sum\_^\; a\_k\; x^k.$ Multiplying through by the denominator and distributing, :$1\; =\; (x^2\; -\; x\; +\; 2)\; \backslash sum\_^\; a\_k\; x^k$ :$1\; =\; \backslash sum\_^\; a\_k\; x^\; -\; \backslash sum\_^\; a\_k\; x^\; +\; 2\backslash sum\_^\; a\_k\; x^k.$ After adjusting the indices of the sums to get the same powers of ''x'', we get :$1\; =\; \backslash sum\_^\; a\_\; x^k\; -\; \backslash sum\_^\; a\_\; x^k\; +\; 2\backslash sum\_^\; a\_k\; x^k.$ Combining like terms gives :$1\; =\; 2a\_0\; +\; (2a\_1\; -\; a\_0)x\; +\; \backslash sum\_^\; (a\_\; -\; a\_\; +\; 2a\_k)\; x^k.$ 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 on the left must equal the constant term on the right it follows that :$a\_0\; =\; \backslash frac.$ Then, since there are no powers of ''x'' on the left, all of thecoefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s on the right must be zero, from which it follows that
:$a\_1\; =\; \backslash frac$
:$a\_k\; =\; \backslash frac\; (a\_\; -\; a\_)\backslash quad\; \backslash text\backslash \; k\; \backslash ge\; 2.$
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, partial fraction decomposition we can write any proper rational function as a sum of factors of the form 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

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 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

/ref> on the on the Riemann sphere creates Discrete dynamical system, discrete dynamical systems.

Notion of a rational function on an algebraic variety

Like Polynomial ring#The polynomial ring in several variables, polynomials, rational expressions can also be generalized to ''n'' indeterminates ''X''Applications

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. 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, the Laplace transform (for continuous systems) or the z-transform (for discrete-time systems) of the impulse response of commonly-used linear time-invariant systems (filters) with infinite impulse response are rational functions over complex numbers.See also

* Field of fractions * Partial fraction decomposition * Partial fractions in integration * Function field of an algebraic variety * Algebraic fractionsa generalization of rational functions that allows taking integer rootsReferences

* *{{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=124External links

Dynamic visualization of rational functions with JSXGraph

Algebraic varieties Morphisms of schemes Meromorphic functions Rational functions,