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Rational Function

 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a rational function is any function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 which can be written as the ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 of two polynomial functions.
he case of one variable, x, a rational function is a function of the form

where P and Q are polynomial function in x and Q is not the zero polynomial.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a rational function is any function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 which can be written as the ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 of two polynomial functions.

Definitions

Rationaldegree2byxedi
In the case of one variable, x, a rational function is a function of the form

where P and Q are polynomial function in x and Q is not the zero polynomial. The domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 of f is the set of all points x for which the denominator Q(x) is not zero.

An irrational function is a function that is not rational. That is: it cannot be expressed as a ratio of two polynomials.

If x is not variable, but rather an indeterminate, one talks about rational expressions instead of rational functions. The distinction between the two notions is important only in abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
.

A rational equation is an equation in which two rational expressions are set equal to each other. These expressions obey the same rules as fractions
Fraction (mathematics)

A fraction is a number that can represent part of a whole.The earliest fractions were reciprocals of integers, symbols representing one half, one third, one quarter, and so on....
. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.

Examples


Rationaldegree3byxedi
The rational function is not defined at .

The rational function is defined for all real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, but not for all complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, since if x were the square root of (i.e. the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
) or its negative, then formal evaluation would lead to division by zero: , which is undefined.

The limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
 of the rational function as x approaches infinity is .

A constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 such as f(x) = p is a rational function since constants are polynomials. Although f(x) is irrational for all x, note that what is rational is the function, not necessarily the values of the function.

Taylor series

The coefficients of a Taylor series
Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
 of any rational function satisfy a linear recurrence relation
Recurrence relation

In mathematics, a recurrence relation is an equation that defines a sequence recursion: each term of the sequence is defined as a Function of the preceding terms....
, which can be found by setting the rational function equal to its Taylor series and collecting like terms.

For example,

Multiplying through by the denominator and distributing,

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

Combining like terms gives

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
Constant term

In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomialover the variable X, the constant term is 3....
 on the left must equal the constant term on the right it follows that

Then, since there are no powers of x on the left, all of the coefficient
Coefficient

In mathematics, a coefficient is a constant multiplication factor of a certain object. For example, in the expression 9x2, the coefficient of x2 is 9....
s on the right must be zero, from which it follows that

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 we can write any rational function as a sum of factors of the form 1 / (ax + b) and expand these as geometric series
Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive term . For example, the seriesis geometric, because each term is equal to half of the previous term....
, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

Complex analysis


In complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
, a rational function

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 and range of f are usually taken to be the Riemann sphere
Riemann sphere

In mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as...
, which avoids any need for special treatment at the poles
Pole (mathematics)

In mathematics, a pole may refer to:* pole , a singularity of a meromorphic function* pole , the dual concept to a polar line...
 of the function (where Q(z) is 0).

The degree of a rational function is the maximum of the degrees
Degree of a polynomial

When a polynomial is expressed as a sum or difference of term s , the exponent of the term with the highest exponent is the degree of the polynomial....
 of its constituent polynomials P and Q. If the degree of f is d then the equation

has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide. f can therefore be thought of as a d-fold covering
Covering map

File:PSTricks-Cubriente.pngIn mathematics, more specifically algebraic topology, a covering map is a continuous function surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p....
 of the w-sphere by the z-sphere.

Rational functions with degree 1 are called Möbius transformation
Möbius transformation

In geometry, a M?bius transformation is a rational function of the form:where z, a, b, c, d are complex numbers satisfying ad − bc ? 0....
s and are automorphism
Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map the object to itself while preserving all of its structure....
s of the Riemann sphere. Rational functions are representative examples of meromorphic function
Meromorphic function

In complex analysis, a meromorphic function on an open set D of the complex plane is a function that is holomorphic function on all D except a set of isolated points, which are pole s for the function....
s.

Abstract algebra


In abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
 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 (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
. In this setting, a rational expression is a class representative of an equivalence class
Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
 of formal quotients of polynomials, where P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR.

Applications


These objects are first encountered in school algebra. In more advanced mathematics they play an important role in ring theory
Ring theory

In mathematics, ring theory is the study of ring , algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers....
, especially in the construction of field extension
Field extension

In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory . The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties....
s. They also provide an example of a nonarchimedean field (see Archimedean property
Archimedean property

In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, is a property held by some group , field , and other algebraic structures....
).

Rational functions are used in numerical analysis
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
 for interpolation
Interpolation

In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
 and approximation
Approximation

An approximation is an Accuracy and precision representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as Function , shapes, and physical laws....
 of functions, for example the Padé approximations introduced by Henri Padé
Henri Padé

Henri Eug?ne Pad? was a France mathematician, who is now remembered mainly for his development of approximation techniques for functions using rational functions....
. Approximations in terms of rational functions are well suited for computer algebra system
Computer algebra system

A computer algebra system is a Application software that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form....
s and other numerical software. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

See also

  • Partial fraction decomposition
  • Partial fractions in integration
    Partial fractions in integration

    In integral calculus, the use of partial fractions is required to integrate the general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of partial fractions....


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