Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions.

Definitions

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

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

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

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. 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

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}.

The rational function f(x) = frac{x^2 + 2}{x^2 + 1} is defined for all real numbers, but not for all complex numbers, since if "x" were the square root of -1 (i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: frac{i^2 + 2}{i^2 + 1} = frac{-1 + 2}{-1 + 1} = frac{1}{0}, which is undefined.

The limit of the rational function f(x) = frac{x^3-2x}{2(x^2-5)} as x approaches infinity is frac{x}{2}.

A constant function such as "f"("x") = π 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 of any rational function satisfy a linear recurrence relation, which can be found by setting the rational function equal to its Taylor series and collecting like terms.

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} + 2sum_{k=0}^{infty} a_k x^k.

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

:1 = sum_{k=2}^{infty} a_{k-2} x^k - sum_{k=1}^{infty} a_{k-1} x^k + 2sum_{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 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 = frac{1}{2}.

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{1}{4}

:a_{k} = frac{1}{2} (a_{k-1} - a_{k-2})quad for 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 we can write any rational function as a sum of factors of the form "1 / (ax + b)" and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

Complex analysis

In complex 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 and range of "f" are usually taken to be the Riemann sphere, which avoids any need for special treatment at the poles of the function (where "Q"("z") is 0).

The "degree" of a rational function is the maximum of the degrees of its constituent polynomials "P" and "Q". 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", called "critical values", where two or more solutions coincide. "f" can therefore be thought of as a "d"-fold covering of the "w"-sphere by the "z"-sphere.

Rational functions with degree 1 are called "Möbius transformations" and are automorphisms of the Riemann sphere. Rational functions are representative examples of meromorphic functions.

Abstract algebra

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. In this setting, a rational expression is a class representative of an equivalence class 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, especially in the construction of field extensions. They also provide an example of a "nonarchimedean field" (see Archimedean property).

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.

ee also

*Partial fraction decomposition
*Partial fractions in integration


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • rational function — noun Date: 1859 a function that is the quotient of two polynomials; also polynomial …   New Collegiate Dictionary

  • rational function — Math. a function that can be written as the quotient of two polynomials with integral coefficients. [1880 85] * * * …   Universalium

  • rational function — noun Any function whose value can be expressed as the quotient of two polynomials (except division by zero) …   Wiktionary

  • rational function — noun 1. : polynomial 2. : the quotient of two polynomials * * * Math. a function that can be written as the quotient of two polynomials with integral coefficients. [1880 85] …   Useful english dictionary

  • Polynomial and rational function modeling — In statistical modeling (especially process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.Polynomial function modelsA polynomial function is one that has the form:y = a… …   Wikipedia

  • Trigonometric rational function — In mathematics, a trigonometric rational function is a rational function in the functions sin theta; and cos theta;. Equivalently, it is a ratio of trigonometric polynomials. The simplest examples (besides sin theta; and cos theta; themselves)… …   Wikipedia

  • integral rational function — noun : polynomial …   Useful english dictionary

  • Rational trigonometry — is a recently introduced approach to trigonometry that eschews all transcendental functions (such as sine and cosine) and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity… …   Wikipedia

  • Rational — may refer to: * Rationality, a concept of reason * Rational number, a number that can be expressed as a ratio of two integers * Rational function, a mathematical function which can be written as the ratio of two polynomial functions * Rational… …   Wikipedia

  • Rational mapping — In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties. In this article we use the convention that varieties are irreducible.DefinitionA first attemptSuppose we… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.