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One of the generalizations of the concept of a [[Derivative|derivative]]. Let there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719001.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719002.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719003.png" /> one has
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$#A+1 = 42 n = 0
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$#C+1 = 42 : ~/encyclopedia/old_files/data/P071/P.0701900 Peano derivative
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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/p/p071/p071900/p0719004.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719005.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719006.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719007.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719008.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p0719009.png" /> is called the generalized Peano derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190010.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190011.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190012.png" />. Symbol: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190013.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190016.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190018.png" />, also exists. If the finite ordinary two-sided derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190019.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190020.png" />. The converse is false for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190021.png" />: For the function
+
One of the generalizations of the concept of a [[Derivative|derivative]]. Let there exist a  $  \delta > 0 $
 +
such that for all  $  t $
 +
with  $  | t | < \delta $
 +
one has
  
<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/p/p071/p071900/p07190022.png" /></td> </tr></table>
+
$$
 +
f( x _ {0} + t)  = \alpha _ {0} + \alpha _ {1} t + \dots +
 +
\frac{\alpha _ {r} }{r!}
 +
t
 +
^ {r} + \gamma ( t) t  ^ {r} ,
 +
$$
  
one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190024.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190025.png" /> does not exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190026.png" /> (since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190027.png" /> is discontinuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190028.png" />). Consequently, the ordinary derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190029.png" /> does not exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190030.png" />.
+
where  $  \alpha _ {0} \dots \alpha _ {r} $
 +
are constants and  $  \gamma ( t) \rightarrow 0 $
 +
as  $  t \rightarrow 0 $;
 +
let  $  \gamma ( 0) = 0 $.  
 +
Then  $  \alpha _ {r} $
 +
is called the generalized Peano derivative of order  $  r $
 +
of the function  $  f $
 +
at the point  $  x _ {0} $.  
 +
Symbol:  $  f _ {(} r) ( x _ {0} ) = \alpha _ {r} $;
 +
in particular, $  \alpha _ {0} = f( x _ {0} ) $,
 +
$  \alpha _ {1} = f _ {(} 1) ( x _ {0} ) $.  
 +
If  $  f _ {(} r) ( x _ {0} ) $
 +
exists, then  $  f _ {(} r- 1) ( x _ {0} ) $,
 +
$  r \geq  1 $,
 +
also exists. If the finite ordinary two-sided derivative $  f ^ { ( r) } ( x _ {0} ) $
 +
exists, then  $  f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $.  
 +
The converse is false for  $  r > 1 $:  
 +
For the function
  
Infinite generalized Peano derivatives have also been introduced. Let for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190032.png" />,
+
$$
 +
f( x)  = \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/p/p071/p071900/p07190033.png" /></td> </tr></table>
+
one has  $  f _ {(} r) ( 0) = 0 $,
 +
$  r = 1, 2 \dots $
 +
but  $  f _ {(} 1) ( x) $
 +
does not exist for  $  x \neq 0 $(
 +
since  $  f( x) $
 +
is discontinuous for  $  x \neq 0 $).  
 +
Consequently, the ordinary derivative  $  f ^ { ( r) } ( 0) $
 +
does not exist for  $  r > 1 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190034.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190035.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190037.png" /> is a number or the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190038.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190039.png" /> is also called the Peano derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190040.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190041.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190042.png" />. It was introduced by G. Peano.
+
Infinite generalized Peano derivatives have also been introduced. Let for all  $  t $
 +
with  $  | t | < \delta $,
 +
 
 +
$$
 +
f( x _ {0} + t)  = \alpha _ {0} + \alpha _ {1} t + \dots
 +
+
 +
\frac{\alpha _ {r} ( t) }{r!}
 +
t  ^ {r} ,
 +
$$
 +
 
 +
where  $  \alpha _ {0} \dots \alpha _ {r-} 1 $
 +
are constants and $  \alpha _ {r} ( t) \rightarrow \alpha _ {r} $
 +
as $  t \rightarrow 0 $(
 +
$  \alpha _ {r} $
 +
is a number or the symbol $  \infty $).  
 +
Then $  \alpha _ {r} $
 +
is also called the Peano derivative of order $  r $
 +
of the function $  f $
 +
at the point $  x _ {0} $.  
 +
It was introduced by G. Peano.

Revision as of 08:05, 6 June 2020


One of the generalizations of the concept of a derivative. Let there exist a $ \delta > 0 $ such that for all $ t $ with $ | t | < \delta $ one has

$$ f( x _ {0} + t) = \alpha _ {0} + \alpha _ {1} t + \dots + \frac{\alpha _ {r} }{r!} t ^ {r} + \gamma ( t) t ^ {r} , $$

where $ \alpha _ {0} \dots \alpha _ {r} $ are constants and $ \gamma ( t) \rightarrow 0 $ as $ t \rightarrow 0 $; let $ \gamma ( 0) = 0 $. Then $ \alpha _ {r} $ is called the generalized Peano derivative of order $ r $ of the function $ f $ at the point $ x _ {0} $. Symbol: $ f _ {(} r) ( x _ {0} ) = \alpha _ {r} $; in particular, $ \alpha _ {0} = f( x _ {0} ) $, $ \alpha _ {1} = f _ {(} 1) ( x _ {0} ) $. If $ f _ {(} r) ( x _ {0} ) $ exists, then $ f _ {(} r- 1) ( x _ {0} ) $, $ r \geq 1 $, also exists. If the finite ordinary two-sided derivative $ f ^ { ( r) } ( x _ {0} ) $ exists, then $ f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $. The converse is false for $ r > 1 $: For the function

$$ f( x) = \left \{ one has $ f _ {(} r) ( 0) = 0 $, $ r = 1, 2 \dots $ but $ f _ {(} 1) ( x) $ does not exist for $ x \neq 0 $( since $ f( x) $ is discontinuous for $ x \neq 0 $). Consequently, the ordinary derivative $ f ^ { ( r) } ( 0) $ does not exist for $ r > 1 $. Infinite generalized Peano derivatives have also been introduced. Let for all $ t $ with $ | t | < \delta $, $$ f( x _ {0} + t) = \alpha _ {0} + \alpha _ {1} t + \dots + \frac{\alpha _ {r} ( t) }{r!}

t  ^ {r} ,

$$

where $ \alpha _ {0} \dots \alpha _ {r-} 1 $ are constants and $ \alpha _ {r} ( t) \rightarrow \alpha _ {r} $ as $ t \rightarrow 0 $( $ \alpha _ {r} $ is a number or the symbol $ \infty $). Then $ \alpha _ {r} $ is also called the Peano derivative of order $ r $ of the function $ f $ at the point $ x _ {0} $. It was introduced by G. Peano.

How to Cite This Entry:
Peano derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peano_derivative&oldid=48146
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article