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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

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

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. [[#References|[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 $[[#References|[2]]].

====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>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,   K. Zeller,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>