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A method for summing series of functions, proposed by E. Borel [[#References|[1]]]. Suppose one is given a series of numbers
 
A method for summing series of functions, proposed by E. Borel [[#References|[1]]]. Suppose one is given a series of numbers
  
<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/b/b017/b017170/b0171701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\sum _ { k=0 } ^  \infty  a _ {k} ,
 +
$$
 +
 
 +
let  $  S _ {n} $
 +
be its partial sums and let  $  S $
 +
be a real number. The series (*) is summable by the Borel method ( $  B $-
 +
method) to the number  $  S $
 +
if
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171702.png" /> be its partial sums and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171703.png" /> be a real number. The series (*) is summable by the Borel method (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171704.png" />-method) to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171705.png" /> if
+
$$
 +
\lim\limits _ {x \rightarrow \infty } \
 +
e  ^ {-x}
 +
\sum _ { k=0 } ^  \infty 
  
<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/b/b017/b017170/b0171706.png" /></td> </tr></table>
+
\frac{x  ^ {k} }{k!}
 +
S _ {k}  = S .
 +
$$
  
There exists an integral summation method due to Borel. This is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171707.png" />-method: If
+
There exists an integral summation method due to Borel. This is the $  B ^ { \prime } $-
 +
method: If
  
<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/b/b017/b017170/b0171708.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
e  ^ {-u}
 +
\sum _ { k=0 } ^  \infty 
  
then one says that the series (*) is summable by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b0171709.png" />-method to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717010.png" />. For conditions under which the two methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717012.png" /> are equivalent, cf. [[#References|[2]]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717013.png" />-method originated in the context of analytic extension of a function regular at a point. Let
+
\frac{a _ {k} u  ^ {k} }{k!}
 +
  du  = S,
 +
$$
  
<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/b/b017/b017170/b01717014.png" /></td> </tr></table>
+
then one says that the series (*) is summable by the  $  B ^ { \prime } $-
 +
method to the number  $  S $.  
 +
For conditions under which the two methods  $  B $
 +
and  $  B ^ { \prime } $
 +
are equivalent, cf. [[#References|[2]]]. The  $  B $-
 +
method originated in the context of analytic extension of a function regular at a point. Let
  
be regular at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717015.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717016.png" /> be the set of all its singular points. Draw the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717017.png" /> and the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717018.png" /> normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717019.png" /> through any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717020.png" />. The set of points on the same side with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717021.png" /> for each straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717022.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717023.png" />; the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717024.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717025.png" /> is then called the Borel polygon of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717026.png" />, while the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717027.png" /> is called its interior domain. The following theorem is valid: The series
+
$$
 +
f(z)  = \sum _ { n=0 } ^  \infty  a _ {n} z  ^ {n}
 +
$$
  
<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/b/b017/b017170/b01717028.png" /></td> </tr></table>
+
be regular at the point  $  O $
 +
and let  $  C $
 +
be the set of all its singular points. Draw the segment  $  OP $
 +
and the straight line  $  L _ {P} $
 +
normal to  $  OP $
 +
through any point  $  P \in C $.  
 +
The set of points on the same side with  $  O $
 +
for each straight line  $  L _ {P} $
 +
is denoted by  $  \Pi $;
 +
the boundary  $  \Gamma $
 +
of the domain  $  \Pi $
 +
is then called the Borel polygon of the function  $  f(z) $,
 +
while the domain  $  \Pi $
 +
is called its interior domain. The following theorem is valid: The series
  
is summable by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717029.png" />-method in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717030.png" />, but not in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717031.png" /> which is the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017170/b01717032.png" /> [[#References|[2]]].
+
$$
 +
\sum _ { n=0 } ^  \infty 
 +
a _ {n} z  ^ {n}
 +
$$
 +
 
 +
is summable by the $  B ^ { \prime } $-
 +
method in $  \Pi $,  
 +
but not in the domain $  \Pi  ^ {*} $
 +
which is the complement of $  \Pi $[[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,  "Mémoire sur les séries divergentes"  ''Ann. Sci. École Norm. Sup. (3)'' , '''16'''  (1899)  pp. 9–131</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,  "Mémoire sur les séries divergentes"  ''Ann. Sci. École Norm. Sup. (3)'' , '''16'''  (1899)  pp. 9–131</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>

Latest revision as of 06:28, 30 May 2020


A method for summing series of functions, proposed by E. Borel [1]. Suppose one is given a series of numbers

$$ \tag{* } \sum _ { k=0 } ^ \infty a _ {k} , $$

let $ S _ {n} $ be its partial sums and let $ S $ be a real number. The series (*) is summable by the Borel method ( $ B $- method) to the number $ S $ if

$$ \lim\limits _ {x \rightarrow \infty } \ e ^ {-x} \sum _ { k=0 } ^ \infty \frac{x ^ {k} }{k!} S _ {k} = S . $$

There exists an integral summation method due to Borel. This is the $ B ^ { \prime } $- method: If

$$ \int\limits _ { 0 } ^ \infty e ^ {-u} \sum _ { k=0 } ^ \infty \frac{a _ {k} u ^ {k} }{k!} du = S, $$

then one says that the series (*) is summable by the $ B ^ { \prime } $- method to the number $ S $. For conditions under which the two methods $ B $ and $ B ^ { \prime } $ are equivalent, cf. [2]. The $ B $- method originated in the context of analytic extension of a function regular at a point. Let

$$ f(z) = \sum _ { n=0 } ^ \infty a _ {n} z ^ {n} $$

be regular at the point $ O $ and let $ C $ be the set of all its singular points. Draw the segment $ OP $ and the straight line $ L _ {P} $ normal to $ OP $ through any point $ P \in C $. The set of points on the same side with $ O $ for each straight line $ L _ {P} $ is denoted by $ \Pi $; the boundary $ \Gamma $ of the domain $ \Pi $ is then called the Borel polygon of the function $ f(z) $, while the domain $ \Pi $ is called its interior domain. The following theorem is valid: The series

$$ \sum _ { n=0 } ^ \infty a _ {n} z ^ {n} $$

is summable by the $ B ^ { \prime } $- method in $ \Pi $, but not in the domain $ \Pi ^ {*} $ which is the complement of $ \Pi $[2].

References

[1] E. Borel, "Mémoire sur les séries divergentes" Ann. Sci. École Norm. Sup. (3) , 16 (1899) pp. 9–131
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)

Comments

References

[a1] W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Borel summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_summation_method&oldid=46121
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article