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''(algebraic) polynomial''
 
''(algebraic) polynomial''
  
 
A function of the form
 
A function of 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/e/e035/e035730/e0357301.png" /></td> </tr></table>
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$$
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= P _ {n} ( z)  = \
 +
a _ {0} z  ^ {n} + a _ {1} z ^ {n - 1 } + \dots + a _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357302.png" /> is a non-negative integer, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357303.png" /> are real or complex numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357304.png" /> is a real or complex variable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357305.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357306.png" /> is called the degree of the polynomial; the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357307.png" /> does not have a degree. The simplest non-constant entire rational function is the linear function
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where $  n $
 +
is a non-negative integer, the coefficients $  a _ {0} \dots a _ {n} $
 +
are real or complex numbers, and $  z $
 +
is a real or complex variable. If $  a _ {0} \neq 0 $,  
 +
then $  n $
 +
is called the degree of the polynomial; the polynomial $  P ( z) \equiv 0 $
 +
does not have a degree. The simplest non-constant entire rational function is the linear 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/e/e035/e035730/e0357308.png" /></td> </tr></table>
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$$
 +
= az + b,\ \
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a  \neq  0.
 +
$$
  
An entire rational function is analytic in the whole plane, that is, it is an [[Entire function|entire function]] of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e0357309.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573010.png" /> is a pole of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573012.png" />. (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573014.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573015.png" />; conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573016.png" /> is an entire function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573017.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035730/e03573019.png" /> is an entire rational function.) Polynomials in several real or complex variables also play an important role in mathematical analysis. Entire rational functions are used for the approximate representation of more complicated functions because they are most convenient for computations.
+
An entire rational function is analytic in the whole plane, that is, it is an [[Entire function|entire function]] of the complex variable $  z $,  
 +
and $  \infty $
 +
is a pole of order $  n $
 +
for $  P _ {n} ( z) $.  
 +
( $  P _ {n} ( z) \rightarrow \infty $
 +
for $  n \geq  1 $,  
 +
as $  z \rightarrow \infty $;  
 +
conversely, if $  f ( z) $
 +
is an entire function and $  f ( z) \rightarrow \infty $
 +
as $  z \rightarrow \infty $,  
 +
then $  f ( z) $
 +
is an entire rational function.) Polynomials in several real or complex variables also play an important role in mathematical analysis. Entire rational functions are used for the approximate representation of more complicated functions because they are most convenient for computations.
  
 
See also [[Polynomial|Polynomial]].
 
See also [[Polynomial|Polynomial]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,  "Introduction to the theory of functions of a complex variable" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,  "Introduction to the theory of functions of a complex variable" , Moscow  (1977)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:37, 5 June 2020


(algebraic) polynomial

A function of the form

$$ w = P _ {n} ( z) = \ a _ {0} z ^ {n} + a _ {1} z ^ {n - 1 } + \dots + a _ {n} , $$

where $ n $ is a non-negative integer, the coefficients $ a _ {0} \dots a _ {n} $ are real or complex numbers, and $ z $ is a real or complex variable. If $ a _ {0} \neq 0 $, then $ n $ is called the degree of the polynomial; the polynomial $ P ( z) \equiv 0 $ does not have a degree. The simplest non-constant entire rational function is the linear function

$$ w = az + b,\ \ a \neq 0. $$

An entire rational function is analytic in the whole plane, that is, it is an entire function of the complex variable $ z $, and $ \infty $ is a pole of order $ n $ for $ P _ {n} ( z) $. ( $ P _ {n} ( z) \rightarrow \infty $ for $ n \geq 1 $, as $ z \rightarrow \infty $; conversely, if $ f ( z) $ is an entire function and $ f ( z) \rightarrow \infty $ as $ z \rightarrow \infty $, then $ f ( z) $ is an entire rational function.) Polynomials in several real or complex variables also play an important role in mathematical analysis. Entire rational functions are used for the approximate representation of more complicated functions because they are most convenient for computations.

See also Polynomial.

References

[1] I.I. Privalov, "Introduction to the theory of functions of a complex variable" , Moscow (1977) (In Russian)

Comments

In non-Soviet literature the phrase "entire rational function" is not used.

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 24–26
[a2] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Entire rational function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Entire_rational_function&oldid=17457
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article