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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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p0719001.png
 
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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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One of the generalizations of the concept of a [[Derivative|derivative]]. Let there exist a  $  \delta > 0 $
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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
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such that for all  $  t $
 
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with  $  | t | < \delta $
 
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one has
 
  
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$$
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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/p07190022.png" /></td> </tr></table>
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f( x _ {0} + t)  = \alpha _ {0} + \alpha _ {1} t + \dots +
 
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\frac{\alpha _ {r} }{r!}
 
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t
 
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^ {r} + \gamma ( t) t  ^ {r} ,
 
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$$
 
  
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where  $  \alpha _ {0} \dots \alpha _ {r} $
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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" />.
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are constants and  $  \gamma ( t) \rightarrow 0 $
 
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as  $  t \rightarrow 0 $;
 
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let  $  \gamma ( 0) = 0 $.  
 
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Then  $  \alpha _ {r} $
 
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is called the generalized Peano derivative of order  $  r $
 
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of the function  $  f $
 
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at the point  $  x _ {0} $.  
 
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Symbol:  $  f _ {(} r) ( x _ {0} ) = \alpha _ {r} $;
 
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in particular,  $  \alpha _ {0} = f( x _ {0} ) $,
 
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$  \alpha _ {1} = f _ {(} 1) ( x _ {0} ) $.  
 
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If  $  f _ {(} r) ( x _ {0} ) $
 
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exists, then  $  f _ {(} r- 1) ( x _ {0} ) $,
 
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$  r \geq  1 $,  
 
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also exists. If the finite ordinary two-sided derivative $  f ^ { ( r) } ( x _ {0} ) $
 
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exists, then  $  f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $.  
 
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The converse is false for  $  r > 1 $:
 
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For the function
 
  
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$$
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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" />,
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f( x)  = \left \{
 
  
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one has  $  f _ {(} r) ( 0) = 0 $,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071900/p07190033.png" /></td> </tr></table>
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$  r = 1, 2 \dots $
 
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but  $  f _ {(} 1) ( x) $
 
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does not exist for  $  x \neq 0 $(
 
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since  $  f( x) $
 
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is discontinuous for  $  x \neq 0 $).  
 
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Consequently, the ordinary derivative  $  f ^ { ( r) } ( 0) $
 
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does not exist for  $  r > 1 $.
 
  
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Infinite generalized Peano derivatives have also been introduced. Let for all  $  t $
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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.
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with  $  | t | < \delta $,
 
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$$
 
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f( x _ {0} + t)  = \alpha _ {0} + \alpha _ {1} t + \dots
 
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+
 
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\frac{\alpha _ {r} ( t) }{r!}
 
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t  ^ {r} ,
 
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$$
 
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where  $  \alpha _ {0} \dots \alpha _ {r-} 1 $
 
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are constants and $  \alpha _ {r} ( t) \rightarrow \alpha _ {r} $
 
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as $  t \rightarrow 0 $(
 
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$  \alpha _ {r} $
 
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is a number or the symbol $  \infty $).  
 
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Then $  \alpha _ {r} $
 
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is also called the Peano derivative of order $  r $
 
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of the function $  f $
 
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at the point $  x _ {0} $.  
 
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It was introduced by G. Peano.
 

Revision as of 14:52, 7 June 2020

One of the generalizations of the concept of a derivative. Let there exist a such that for all with one has

where are constants and as ; let . Then is called the generalized Peano derivative of order of the function at the point . Symbol: ; in particular, , . If exists, then , , also exists. If the finite ordinary two-sided derivative exists, then . The converse is false for : For the function

one has , but does not exist for (since is discontinuous for ). Consequently, the ordinary derivative does not exist for .

Infinite generalized Peano derivatives have also been introduced. Let for all with ,

where are constants and as ( is a number or the symbol ). Then is also called the Peano derivative of order of the function at the point . 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=49361
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Peano_derivative&oldid=49361"