Rational function

A rational function is a function $ w = R ( z) $, where $ R ( z) $ is rational expression in $ z $, i.e. an expression obtained from an independent variable $ z $ 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

$$ R ( z) = \frac{P ( z) }{Q ( z) } , $$

where $ P $, $ Q $ are polynomials, $ Q ( z) \not\equiv 0 $. The coefficients of these polynomials are called the coefficients of the rational function. The function $ P / Q $ is called irreducible when $ P $ and $ Q $ have no common zeros (that is, $ P $ and $ Q $ are relatively prime polynomials). Every rational function can be written as an irreducible fraction $ R ( z) = P ( z) / Q ( z) $; if $ P $ has degree $ m $ and $ Q $ has degree $ n $, then the degree of $ R ( z) $ is either taken to be the pair $ ( m , n ) $ or the number

$$ N = \max \{ m , n \} . $$

A rational function of degree $ ( m , n ) $ with $ n = 0 $, 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 $ R ( z) \equiv 0 $ is not defined. When $ m < n $, the fraction $ P / Q $ is called proper, and it is called improper otherwise. An improper fraction can be uniquely written as

$$ \frac{P}{Q} = P _ {1} + \frac{P _ {2} }{Q} , $$

where $ P _ {1} $ is a polynomial, called the integral part of the fraction $ P / Q $, and $ P _ {2} / Q $ is a proper fraction. A proper fraction, $ R ( z) = P ( z) / Q ( z) $, in irreducible form, where

$$ Q ( z) = b _ {0} ( z - b _ {1} ) ^ {n _ {1} } \dots ( z - b _ {l} ) ^ {n _ {l} } , $$

admits a unique expansion as a sum of simple partial fractions

$$ \tag{1 } R ( z) = \sum _ { i= 1} ^ { l } \frac{c _ {i _ {1} } }{z - b _ {i} } + \dots + \frac{c _ {i _ { n _ i } } }{( z - b _ {i} ) ^ {n _ {i} } } . $$

If $ P ( x) / Q ( x) $ is a proper rational function with real coefficients and

$$ Q ( x) = $$

$$ = \ b _ {0} ( x - b _ {1} ) ^ {l _ {1} } \dots ( x - b _ {r} ) ^ {l _ {r} } ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {t _ {1} } \dots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {t _ {s} } , $$

where $ b _ {0} \dots b _ {r} , p _ {1} , q _ {1} \dots p _ {s} , q _ {s} $ are real numbers such that $ p _ {j} ^ {2} - 4 q _ {j} < 0 $ for $ j = 1 \dots s $, then $ P ( x) / Q ( x) $ can be uniquely written in the form

$$ \tag{2 } \frac{P ( x) }{Q ( x) } = \ \sum _ { i= 1} ^ { r } \left [ \frac{c _ {i _ {1} } }{x - b _ {i} } + \dots + \frac{c _ {i _ { l _ i } } }{( x - b _ {i} ) ^ {l _ {i} } } \right ] + $$

$$ + \sum _ { j= 1} ^ { s } \left [ \frac{D _ {j _ {1} } x + E _ {j _ {1} } }{x ^ {2} + p _ {j} x + q _ {j} } + \dots + \frac{D _ {j _ { t _ j } } x + E _ {j _ { t _ j } } }{( x ^ {2} + p _ {j} x + q _ {j} ) ^ {t _ {j} } } \right ] , $$

where all the coefficients are real. These coefficients, like the $ c _ {ij} $ in (1), can be found by the method of indefinite coefficients (cf. Undetermined coefficients, method of).

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

The application of arithmetical operations (with the exception of division by $ R ( z) \equiv 0 $) 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 $ R _ {1} ( z) $, $ R _ {2} ( z) $ are rational functions, then $ R _ {1} ( R _ {2} ( z) ) $ is also a rational function. The derivative of order $ p $ of a rational function of degree $ N $ is a rational function of degree at most $ ( p + 1 ) N $. An indefinite integral (or primitive) of a rational function is the sum of a rational function and expressions of the form $ c _ {r} \mathop{\rm log} ( z - b _ {r} ) $. If a rational function $ R ( x) $ is real for all real $ x $, then the indefinite integral $ \int R ( x) d x $ can be written as the sum of a rational function $ R _ {0} ( x) $ with real coefficients, expressions of the form

$$ c _ {i _ {1} } \mathop{\rm log} | x - b _ {i} | ,\ \ M _ {j} \mathop{\rm log} ( x ^ {2} + p _ {j} x + q _ {j} ) , $$

$$ N _ {j} \mathop{\rm arctan} \frac{2 x + p _ {j} }{\sqrt {4 q _ {j} - p _ {j} ^ {2} } } ,\ i = 1 \dots r ; \ j = 1 \dots s , $$

and an arbitrary constant $ C $ (where $ c _ {i _ {1} } $, $ b _ {i} $, $ p _ {j} $, $ q _ {j} $ are the same as in (2), and $ M _ {j} $, $ N _ {j} $ are real numbers). The function $ R _ {0} ( x) $ can be found by the Ostrogradski method, which avoids the need to expand $ R ( x) $ 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 $ R = P / Q $ in several real or complex variables, where $ P $ and $ Q $ are polynomials in these variables with $ Q \not\equiv 0 $, and to abstract rational functions

$$ R = \ \frac{A _ {1} \Phi _ {1} + \dots + A _ {m} \Phi _ {m} }{B _ {1} \Phi _ {1} + \dots + B _ {n} \Phi _ {n} } , $$

where $ \Phi _ {1} , \Phi _ {2} \dots $ are linearly independent functions on some compact space $ X $, and $ A _ {1} \dots A _ {m} , B _ {1} \dots B _ {n} $ 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 $ X $ is an equivalence class of pairs $ ( U , f ) $, where $ U $ is a non-empty open subset of $ X $ and $ f $ is a regular function on $ U $. Two pairs $ ( U , f ) $ and $ ( V , g ) $ are said to be equivalent if $ f = g $ on $ U \cap V $. The rational functions on $ X $ form a field, denoted by $ k ( X) $.

In the case when $ X = \mathop{\rm spec} R $ is an irreducible affine variety, the field of rational functions on $ X $ coincides with the field of fractions of the ring $ R $. The transcendence degree of $ k ( X) $ over $ k $ is called the dimension of the variety $ X $.

References

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

Vik.S. Kulikov