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A semi-continuous method for summing series of numbers and functions (cf. [[Summation methods|Summation methods]]), defined by the system of functions

$$ 
g _ {0} ( \delta )  =  1 ,\ \ 
g _ {k} ( \delta )  =   \mathop{\rm exp} ( - \delta k   \mathop{\rm ln}  k ) ,\ \ 
\delta > 0 ,\  k = 1 , 2 , .  .  . .
$$

The series

$$ 
\sum _ { k= } 0 ^  \infty   u _ {k}  $$

is summable by the Lindelöf summation method to the sum  $  s $
if

$$ 
\lim\limits _ {\delta \rightarrow 0 } \ 
\left [
u _ {0} + \sum _ { k= } 0 ^  \infty    \mathop{\rm exp}
( - \delta k   \mathop{\rm ln}  k ) u _ {k} \right ]  =  s
$$

and the series under the limit sign converges. The method was introduced by E. Lindelöf [[#References|[1]]] for the summation of power series.

The Lindelöf summation method is regular (see [[Regular summation methods|Regular summation methods]]) and is used as a tool for the [[Analytic continuation|analytic continuation]] of functions. If  $  f ( z) $
is the principal branch of an analytic function, regular at the origin and representable by a series

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

for small  $  z $, 
then this series is summable by the Lindelöf summation method to  $  f ( z) $
in the whole star of the function  $  f ( z) $(
cf. [[Star of a function element|Star of a function element]]), and it is uniformly summable in every closed bounded domain contained in the interior of the star.

Of the summation methods determined by a transformation of a sequence into a sequence by semi-continuous matrices  $  a _ {k} ( \omega ) $
of type

$$ 
a _ {k} ( \omega )  = \ 

\frac{c _ {k+} 1 \omega  ^ {k+} 1 }{E ( \omega ) }
 ,
$$

where

$$ 
E ( \omega )  =  \sum _ { k= } 0 ^  \infty   c _ {k} \omega  ^ {k}
$$

is an [[Entire function|entire function]], Lindelöf considered the case when

$$ 
E ( \omega )  =  \sum _ { k= } 0 ^  \infty  
\left [

\frac \omega { \mathop{\rm ln} ( k + \beta ) }

\right ]  ^ {k} ,\ \ 
\beta > 1 .
$$

A matrix  $  \| a _ {k} ( \omega ) \| $
constructed from an entire function of this kind is called a Lindelöf matrix.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,   ''J. Math.'' , '''9'''  (1903)  pp. 213–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Lindelöf,   "Le calcul des résidus et ses applications à la théorie des fonctions" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>