Namespaces
Variants
Actions

Difference between revisions of "Symmetric difference of order n"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
−
''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''
+
<!--
 +
s0916302.png
 +
$#A+1 = 11 n = 0
 +
$#C+1 = 11 : ~/encyclopedia/old_files/data/S091/S.0901630 Symmetric difference of order \BMI n\EMI
 +
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}}
 +
 
 +
''at a point  $  x $
 +
of a function $  f $
 +
of a real variable''
  
 
The expression
 
The expression
  
−
<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>
+
$$
 +
\Delta _ {s}  ^ {n} f ( x, h)  = \
 +
\sum _ {k = 0 } ^ { n }
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) (- 1)  ^ {k}
 +
f \left ( x + {
 +
\frac{n - 2k }{2}
 +
} h \right ) .
 +
$$
  
 
The following expression is often also referred to as a symmetric difference:
 
The following expression is often also referred to as a symmetric difference:
  
−
<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>
+
$$
−
 
+
\sum _ {k = 0 } ^ { n }
−
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
+
\left ( \begin{array}{c}
−
 
+
n \\
−
<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>
+
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====
 
====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=12454
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Symmetric_difference_of_order_n&oldid=48924"