# Lindelöf summation method

A semi-continuous method for summing series of numbers and functions (cf. 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  for the summation of power series.

The Lindelöf summation method is regular (see Regular summation methods) and is used as a tool for the 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), 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, 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.

How to Cite This Entry:
Lindelöf summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_summation_method&oldid=47643
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article