Namespaces
Variants
Actions

Difference between revisions of "Rational function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (fixing subscripts)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
A rational function is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775902.png" /> is rational expression in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775903.png" />, i.e. an expression obtained from an independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775904.png" /> 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
+
<!--
 +
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.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775905.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775907.png" /> are polynomials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775908.png" />. The coefficients of these polynomials are called the coefficients of the rational function. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r0775909.png" /> is called irreducible when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759011.png" /> have no common zeros (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759013.png" /> are relatively prime polynomials). Every rational function can be written as an irreducible fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759014.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759015.png" /> has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759017.png" /> has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759018.png" />, then the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759019.png" /> is either taken to be the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759020.png" /> or the number
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759021.png" /></td> </tr></table>
+
$$
 +
R ( z)  =
 +
\frac{P ( z) }{Q ( z) }
 +
,
 +
$$
  
A rational function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759023.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759024.png" /> is not defined. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759025.png" />, the fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759026.png" /> is called proper, and it is called improper otherwise. An improper fraction can be uniquely written as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759027.png" /></td> </tr></table>
+
$$
 +
= \max \{ m , n \} .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759028.png" /> is a polynomial, called the integral part of the fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759030.png" /> is a proper fraction. A proper fraction, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759031.png" />, in irreducible form, where
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759032.png" /></td> </tr></table>
+
$$
 +
 
 +
\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
 
admits a unique expansion as a sum of simple partial fractions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
R ( z)  = \sum _ { i= 1} ^ { l }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759034.png" /> is a proper rational function with real coefficients and
+
\frac{c _ {i _ {1}  } }{z - b _ {i} }
 +
+ \dots
 +
+
 +
\frac{c _ {i _ { n _ i }  } }{( z - b _ {i} ) ^ {n _ {i} } }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759035.png" /></td> </tr></table>
+
If  $  P ( x) / Q ( x) $
 +
is a proper rational function with real coefficients and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759036.png" /></td> </tr></table>
+
$$
 +
Q ( x) =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759037.png" /> are real numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759040.png" /> can be uniquely written in the form
+
$$
 +
= \
 +
b _ {0} ( x - b _ {1} ) ^ {l _ {1} } \dots ( x - b _ {r} ) ^ {l _ {r} } ( x  ^ {2} + p _ {1} x + q _ {1} ) ^ {t _ {1} } \dots
 +
( x  ^ {2} + p _ {s} x + q _ {s} ) ^ {t _ {s} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759042.png" /></td> </tr></table>
+
$$ \tag{2 }
  
where all the coefficients are real. These coefficients, like the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759043.png" /> in (1), can be found by the method of indefinite coefficients (cf. [[Undetermined coefficients, method of|Undetermined coefficients, method of]]).
+
\frac{P ( x) }{Q ( x) }
 +
  = \
 +
\sum _ { i= 1} ^ { r }
 +
\left [
  
A rational function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759044.png" /> in irreducible form is defined and analytic in the extended complex plane (that is, the plane together with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759045.png" />), except at a finite number of singular points, poles: the zeros of its denominator and, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759046.png" />, also the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759047.png" />. Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759048.png" />, the sum of the multiplicities of the poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759049.png" /> is equal to its degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759050.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759051.png" /> is an analytic function whose only singular points in the extended complex plane are finitely many poles, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759052.png" /> is a rational function.
+
\frac{c _ {i _ {1}  } }{x - b _ {i} }
 +
+ \dots
 +
+
 +
\frac{c _ {i _ { l _ i }  } }{( x - b _ {i} ) ^ {l _ {i} } }
  
The application of arithmetical operations (with the exception of division by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759053.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759055.png" /> are rational functions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759056.png" /> is also a rational function. The derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759057.png" /> of a rational function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759058.png" /> is a rational function of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759059.png" />. An indefinite integral (or primitive) of a rational function is the sum of a rational function and expressions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759060.png" />. If a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759061.png" /> is real for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759062.png" />, then the indefinite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759063.png" /> can be written as the sum of a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759064.png" /> with real coefficients, expressions of the form
+
\right ] +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759065.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ { j= 1} ^ { s }  \left [
 +
\frac{D _ {j _ {1}  } x + E _ {j _ {1}  } }{x  ^ {2} + p _ {j} x + q _ {j} }
 +
+ \dots +
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759066.png" /></td> </tr></table>
+
\frac{D _ {j _ { t _ j }  } x + E _ {j _ { t _ j }  } }{( x  ^ {2} + p _ {j} x + q _ {j} ) ^ {t _ {j} } }
 +
\right ] ,
 +
$$
  
and an arbitrary constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759067.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759071.png" /> are the same as in (2), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759073.png" /> are real numbers). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759074.png" /> can be found by the [[Ostrogradski method|Ostrogradski method]], which avoids the need to expand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759075.png" /> into partial fractions (2).
+
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]]).
  
For ease of computation, rational functions can be used to approximate a given function. Attention has also been paid to rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759076.png" /> in several real or complex variables, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759078.png" /> are polynomials in these variables with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759079.png" />, and to abstract rational functions
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759080.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759081.png" /> are linearly independent functions on some compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759083.png" /> are numbers. See also [[Fractional-linear function|Fractional-linear function]]; [[Zhukovskii function|Zhukovskii function]].
+
$$
 +
c _ {i _ {1}  }  \mathop{\rm log}  | x - b _ {i} | ,\ \
 +
M _ {j}  \mathop{\rm log} ( x  ^ {2} + p _ {j} x + q _ {j} ) ,
 +
$$
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow,   "Einführung in die Funktionentheorie" , '''1–3''' , Teubner (1958–1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table>
+
N _ {j}  \mathop{\rm arctan} 
 +
\frac{2 x + p _ {j} }{\sqrt {4 q _ {j} - p _ {j}  ^ {2} } }
 +
,\  i = 1 \dots r ; \  j = 1 \dots s ,
 +
$$
 +
 
 +
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====
 
====Comments====
Line 58: Line 200:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.B. Conway,   "Functions of one complex variable" , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang,   "Algebra" , Addison-Wesley (1984)</TD></TR></table>
+
<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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759084.png" /> is an equivalence class of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759086.png" /> is a non-empty open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759088.png" /> is a [[Regular function|regular function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759089.png" />. Two pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759091.png" /> are said to be equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759092.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759093.png" />. The rational functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759094.png" /> form a field, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759095.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759096.png" /> is an irreducible [[Affine variety|affine variety]], the field of rational functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759097.png" /> coincides with the field of fractions of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759098.png" />. The transcendence degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r07759099.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r077590100.png" /> is called the dimension of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077590/r077590101.png" />.
+
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR></table>
+
<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''
 
''Vik.S. Kulikov''

Latest revision as of 03:49, 4 March 2022


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

How to Cite This Entry:
Rational function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_function&oldid=17805
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article