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Difference between revisions of "Symmetric difference of order n"

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''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916302.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916303.png" /> of a real variable''
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$#A+1 = 11 n = 0
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$#C+1 = 11 : ~/encyclopedia/old_files/data/S091/S.0901630 Symmetric difference of order \BMI n\EMI
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''at a point  $  x $
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of a function $  f $
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of a real variable''
  
 
The expression
 
The expression
  
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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/s/s091/s091630/s0916304.png" /></td> </tr></table>
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$$
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\Delta _ {s}  ^ {n} f ( x, h)  = \
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\sum _ {k = 0 } ^ { n }
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\left ( \begin{array}{c}
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n \\
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k
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\end{array}
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\right ) (- 1)  ^ {k}
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f \left ( x + {
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\frac{n - 2k }{2}
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} h \right ) .
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$$
  
 
The following expression is often also referred to as a symmetric difference:
 
The following expression is often also referred to as a symmetric difference:
  
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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/s/s091/s091630/s0916305.png" /></td> </tr></table>
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$$
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\sum _ {k = 0 } ^ { n }
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It is obtained from the above by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916306.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916307.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916308.png" /> has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s0916309.png" />-th order derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s09163010.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s09163011.png" />, then
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\left ( \begin{array}{c}
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n \\
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091630/s09163012.png" /></td> </tr></table>
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k
 +
\end{array}
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\right ) (- 1)  ^ {k}
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f ( x + ( n - 2k) h).
 +
$$
  
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It is obtained from the above by substituting  $  2h $
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for  $  h $.
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If  $  f ( x) $
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has an  $  n $-
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th order derivative  $  f ^ { ( n) } ( x) $
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at  $  x $,
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then
  
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$$
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\Delta _ {s}  ^ {n} f ( x, h)  = \
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f ^ { ( n) } ( x) h  ^ {n} + o ( h  ^ {n} ).
 +
$$
  
 
====Comments====
 
====Comments====
−
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Meschkowski,  "Differenzengleichungen" , Vandenhoeck &amp; Ruprecht  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.N. Milne-Thomson,  "The calculus of finite differences" , Chelsea, reprint  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Chelsea, reprint  (1954)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Meschkowski,  "Differenzengleichungen" , Vandenhoeck &amp; Ruprecht  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.N. Milne-Thomson,  "The calculus of finite differences" , Chelsea, reprint  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Chelsea, reprint  (1954)</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


at a point $ x $ of a function $ f $ of a real variable

The expression

$$ \Delta _ {s} ^ {n} f ( x, h) = \ \sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f \left ( x + { \frac{n - 2k }{2} } h \right ) . $$

The following expression is often also referred to as a symmetric difference:

$$ \sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f ( x + ( n - 2k) h). $$

It is obtained from the above by substituting $ 2h $ for $ h $. If $ f ( x) $ has an $ n $- th order derivative $ f ^ { ( n) } ( x) $ at $ x $, then

$$ \Delta _ {s} ^ {n} f ( x, h) = \ f ^ { ( n) } ( x) h ^ {n} + o ( h ^ {n} ). $$

Comments

References

[a1] H. Meschkowski, "Differenzengleichungen" , Vandenhoeck & Ruprecht (1959)
[a2] L.N. Milne-Thomson, "The calculus of finite differences" , Chelsea, reprint (1981)
[a3] N.E. Nörlund, "Volesungen über Differenzenrechnung" , Chelsea, reprint (1954)
How to Cite This Entry:
Symmetric difference of order n. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_difference_of_order_n&oldid=48924
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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