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''of an expansion of a function''
 
''of an expansion of a function''
  
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The approximating formulas alluded to include the [[Taylor formula|Taylor formula]], interpolation formulas, asymptotic formulas, formulas for the approximate evaluation of some quantity, etc. Thus, in the Taylor formula
 
The approximating formulas alluded to include the [[Taylor formula|Taylor formula]], interpolation formulas, asymptotic formulas, formulas for the approximate evaluation of some quantity, etc. Thus, in the Taylor formula
  
<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/r081/r081200/r0812001.png" /></td> </tr></table>
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$$
 
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f( x)  = \
the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812002.png" /> is called the remainder (in Peano's form). Given the asymptotic expansion
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\sum _ {k=0} ^ { n } 
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\frac{f ^ { ( k) } ( x _ {0} ) }{k!}
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( x - x _ {0} )  ^ {k} +
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o(( x - x _ {0} ) ^ {n} ),\  {\textrm{ as }  } x \rightarrow x _ {0} ,
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$$
  
<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/r081/r081200/r0812003.png" /></td> </tr></table>
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the term  $  o(( x - x _ {0} )  ^ {n} ) $
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is called the remainder (in Peano's form). Given the asymptotic expansion
  
of a function, the remainder is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812004.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812005.png" />. In the [[Stirling formula|Stirling formula]], which gives an asymptotic expansion of the Euler [[Gamma-function|gamma-function]],
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$$
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f( x)  =  a _ {0} +
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\frac{a _ {1} }{x}
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+ \dots +
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\frac{a _ {n} }{x  ^ {n} }
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+ O \left (
  
<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/r081/r081200/r0812006.png" /></td> </tr></table>
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\frac{1}{x  ^ {n+} 1 }
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\right ) ,\  {\textrm{ as }  } x \rightarrow + \infty ,
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$$
  
the remainder is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812007.png" />.
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of a function, the remainder is $  O( x  ^ {-} n- 1 ) $,
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as  $  x \rightarrow \infty $.  
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In the [[Stirling formula|Stirling formula]], which gives an asymptotic expansion of the Euler [[Gamma-function|gamma-function]],
  
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$$
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\Gamma ( s + 1 )  =  \sqrt {2 \pi s } \left (
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\frac{s}{e}
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\right )  ^ {s} + O \left ( e  ^ {-} s s ^ {s - 1 / 2 } \right ) ,\  {\textrm{ as }  } s \rightarrow + \infty ,
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$$
  
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the remainder is  $  O( e  ^ {-} s s  ^ {s-} 1/2 ) $.
  
 
====Comments====
 
====Comments====
The remainder of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812008.png" /> upon division by a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r0812009.png" /> is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r08120010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r08120011.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r08120012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081200/r08120013.png" /> an integer. See also [[Remainder of an integer|Remainder of an integer]].
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The remainder of an integer $  a $
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upon division by a natural number $  b $
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is the number $  c $,
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0 \leq  c < b $,  
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for which $  a= kb+ c $
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with $  k $
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an integer. See also [[Remainder of an integer]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bleistein,  R.A. Handelsman,  "Asymptotic expansions of integrals" , Dover, reprint  (1986)  pp. Chapts. 1, 3, 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Spivak,  "Calculus" , Benjamin  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bleistein,  R.A. Handelsman,  "Asymptotic expansions of integrals" , Dover, reprint  (1986)  pp. Chapts. 1, 3, 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Spivak,  "Calculus" , Benjamin  (1967)</TD></TR></table>

Latest revision as of 09:03, 6 January 2024


of an expansion of a function

An additive term in a formula approximating a function by another, simpler, function. The remainder equals the difference between the given function and its approximating function, and an estimate of it is therefore an estimate of the accuracy of the approximation.

The approximating formulas alluded to include the Taylor formula, interpolation formulas, asymptotic formulas, formulas for the approximate evaluation of some quantity, etc. Thus, in the Taylor formula

$$ f( x) = \ \sum _ {k=0} ^ { n } \frac{f ^ { ( k) } ( x _ {0} ) }{k!} ( x - x _ {0} ) ^ {k} + o(( x - x _ {0} ) ^ {n} ),\ {\textrm{ as } } x \rightarrow x _ {0} , $$

the term $ o(( x - x _ {0} ) ^ {n} ) $ is called the remainder (in Peano's form). Given the asymptotic expansion

$$ f( x) = a _ {0} + \frac{a _ {1} }{x} + \dots + \frac{a _ {n} }{x ^ {n} } + O \left ( \frac{1}{x ^ {n+} 1 } \right ) ,\ {\textrm{ as } } x \rightarrow + \infty , $$

of a function, the remainder is $ O( x ^ {-} n- 1 ) $, as $ x \rightarrow \infty $. In the Stirling formula, which gives an asymptotic expansion of the Euler gamma-function,

$$ \Gamma ( s + 1 ) = \sqrt {2 \pi s } \left ( \frac{s}{e} \right ) ^ {s} + O \left ( e ^ {-} s s ^ {s - 1 / 2 } \right ) ,\ {\textrm{ as } } s \rightarrow + \infty , $$

the remainder is $ O( e ^ {-} s s ^ {s-} 1/2 ) $.

Comments

The remainder of an integer $ a $ upon division by a natural number $ b $ is the number $ c $, $ 0 \leq c < b $, for which $ a= kb+ c $ with $ k $ an integer. See also Remainder of an integer.

References

[a1] N. Bleistein, R.A. Handelsman, "Asymptotic expansions of integrals" , Dover, reprint (1986) pp. Chapts. 1, 3, 5
[a2] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a3] M. Spivak, "Calculus" , Benjamin (1967)
How to Cite This Entry:
Remainder. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Remainder&oldid=15868
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article