View source for Rational function

<!--
r0775901.png
$#A+1 = 101 n = 0
$#C+1 = 101 : ~/encyclopedia/old_files/data/R077/R.0707590 Rational function
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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|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|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|Fractional-linear function]]; [[Zhukovskii function|Zhukovskii function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , '''1–3''' , Teubner (1958–1959) (Translated from Russian) {{MR|0342680}} {{MR|0264037}} {{MR|0264036}} {{MR|0264038}} {{MR|0123686}} {{MR|0123685}} {{MR|0098843}} {{ZBL|0177.33401}} {{ZBL|0141.26003}} {{ZBL|0141.26002}} {{ZBL|0082.28802}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) {{MR|0945393}} {{MR|0926059}} {{MR|0778202}} {{MR|0759341}} {{MR|0628003}} {{MR|0384363}} {{ZBL|0237.13001}} </TD></TR></table>

====Comments====
For approximation results see [[Padé approximation|Padé approximation]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1973) {{MR|0447532}} {{ZBL|0277.30001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR></table>

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|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>

''Vik.S. Kulikov''