Rational function

A rational function is a function , where is rational expression in , i.e. an expression obtained from an independent variable and some finite set of numbers (real or complex) by means of a finite number of arithmetical operations. A rational function can be written (non-uniquely) in the form

where , are polynomials, . The coefficients of these polynomials are called the coefficients of the rational function. The function is called irreducible when and have no common zeros (that is, and are relatively prime polynomials). Every rational function can be written as an irreducible fraction ; if has degree and has degree , then the degree of is either taken to be the pair or the number

A rational function of degree with , that is, a polynomial, is also called an entire rational function. Otherwise it is called a fractional-rational function. The degree of the rational function is not defined. When , the fraction is called proper, and it is called improper otherwise. An improper fraction can be uniquely written as

where is a polynomial, called the integral part of the fraction , and is a proper fraction. A proper fraction, , in irreducible form, where

admits a unique expansion as a sum of simple partial fractions

(1)

If is a proper rational function with real coefficients and

where are real numbers such that for , then can be uniquely written in the form

(2)

where all the coefficients are real. These coefficients, like the in (1), can be found by the method of indefinite coefficients (cf. Undetermined coefficients, method of).

A rational function of degree in irreducible form is defined and analytic in the extended complex plane (that is, the plane together with the point ), except at a finite number of singular points, poles: the zeros of its denominator and, when , also the point . Note that if , the sum of the multiplicities of the poles of is equal to its degree . Conversely, if is an analytic function whose only singular points in the extended complex plane are finitely many poles, then is a rational function.

The application of arithmetical operations (with the exception of division by ) to rational functions again gives a rational function, so that the set of all rational functions forms a field. In general, the rational functions with coefficients in a field form a field. If , are rational functions, then is also a rational function. The derivative of order of a rational function of degree is a rational function of degree at most . An indefinite integral (or primitive) of a rational function is the sum of a rational function and expressions of the form . If a rational function is real for all real , then the indefinite integral can be written as the sum of a rational function with real coefficients, expressions of the form

and an arbitrary constant (where , , , are the same as in (2), and , are real numbers). The function can be found by the Ostrogradski method, which avoids the need to expand into partial fractions (2).

For ease of computation, rational functions can be used to approximate a given function. Attention has also been paid to rational functions in several real or complex variables, where and are polynomials in these variables with , and to abstract rational functions

where are linearly independent functions on some compact space , and are numbers. See also Fractional-linear function; Zhukovskii function.

References

[1]	I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) MR0342680 MR0264037 MR0264036 MR0264038 MR0123686 MR0123685 MR0098843 Zbl 0177.33401 Zbl 0141.26003 Zbl 0141.26002 Zbl 0082.28802
[2]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) MR0945393 MR0926059 MR0778202 MR0759341 MR0628003 MR0384363 Zbl 0237.13001

Comments

For approximation results see Padé approximation.

References

[a1]	J.B. Conway, "Functions of one complex variable" , Springer (1973) MR0447532 Zbl 0277.30001
[a2]	S. Lang, "Algebra" , Addison-Wesley (1984) MR0783636 Zbl 0712.00001

Rational functions on an algebraic variety are a generalization of the classical concept of a rational function (see section 1)). A rational function on an irreducible algebraic variety is an equivalence class of pairs , where is a non-empty open subset of and is a regular function on . Two pairs and are said to be equivalent if on . The rational functions on form a field, denoted by .

In the case when is an irreducible affine variety, the field of rational functions on coincides with the field of fractions of the ring . The transcendence degree of over is called the dimension of the variety .

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Vik.S. Kulikov