# 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 [1] 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.

#### References

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