# Borel summation method

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)