Namespaces
Variants
Actions

Difference between revisions of "Lindelöf summation method"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(latex details)
Line 11: Line 11:
 
{{TEX|done}}
 
{{TEX|done}}
  
A semi-continuous method for summing series of numbers and functions (cf. [[Summation methods|Summation methods]]), defined by the system of functions
+
A semi-continuous method for summing series of numbers and functions (cf. [[Summation methods]]), defined by the system of functions
  
 
$$  
 
$$  
Line 22: Line 22:
  
 
$$  
 
$$  
\sum _ { k= } 0 ^  \infty  u _ {k}  $$
+
\sum _ {k=0} ^  \infty  u _ {k}  $$
  
 
is summable by the Lindelöf summation method to the sum  $  s $
 
is summable by the Lindelöf summation method to the sum  $  s $
Line 30: Line 30:
 
\lim\limits _ {\delta \rightarrow 0 } \  
 
\lim\limits _ {\delta \rightarrow 0 } \  
 
\left [
 
\left [
u _ {0} + \sum _ { k= } 0 ^  \infty    \mathop{\rm exp}
+
u _ {0} + \sum _ {k=0} ^  \infty    \mathop{\rm exp}
 
( - \delta k  \mathop{\rm ln}  k ) u _ {k} \right ]  =  s
 
( - \delta k  \mathop{\rm ln}  k ) u _ {k} \right ]  =  s
 
$$
 
$$
Line 36: Line 36:
 
and the series under the limit sign converges. The method was introduced by E. Lindelöf [[#References|[1]]] for the summation of power series.
 
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) $
+
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
 
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}
+
\sum _ {k=0}^  \infty  a _ {k} z  ^ {k}
 
$$
 
$$
  
Line 46: Line 46:
 
then this series is summable by the Lindelöf summation method to  $  f ( z) $
 
then this series is summable by the Lindelöf summation method to  $  f ( z) $
 
in the whole star of the function  $  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.
+
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 the summation methods determined by a transformation of a sequence into a sequence by semi-continuous matrices  $  a _ {k} ( \omega ) $
Line 54: Line 54:
 
a _ {k} ( \omega )  = \  
 
a _ {k} ( \omega )  = \  
  
\frac{c _ {k+} 1 \omega  ^ {k+} 1 }{E ( \omega ) }
+
\frac{c _ {k+1} \omega  ^ {k+1} {E ( \omega ) }
 
  ,
 
  ,
 
$$
 
$$
Line 61: Line 61:
  
 
$$  
 
$$  
E ( \omega )  =  \sum _ { k= } 0 ^  \infty  c _ {k} \omega  ^ {k}
+
E ( \omega )  =  \sum _ {k=0} ^  \infty  c _ {k} \omega  ^ {k}
 
$$
 
$$
  
is an [[Entire function|entire function]], Lindelöf considered the case when
+
is an [[entire function]], Lindelöf considered the case when
  
 
$$  
 
$$  
E ( \omega )  =  \sum _ { k= } 0 ^  \infty   
+
E ( \omega )  =  \sum _ {k=0}^  \infty   
 
\left [
 
\left [
  
Line 80: Line 80:
  
 
====References====
 
====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>
+
<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>

Revision as of 08:15, 6 January 2024


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=54829
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article