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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
 
A semi-continuous method for summing series of numbers and functions (cf. [[Summation methods|Summation methods]]), defined by the system of functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589901.png" /></td> </tr></table>
+
$$
 +
g _ {0} ( \delta )  = 1 ,\ \
 +
g _ {k} ( \delta )  =   \mathop{\rm exp} ( - \delta k  \mathop{\rm ln}  k ) ,\ \
 +
\delta > 0 ,\  k = 1 , 2 , . . . .
 +
$$
  
 
The series
 
The series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589902.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 0 ^  \infty  u _ {k}  $$
  
is summable by the Lindelöf summation method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589903.png" /> if
+
is summable by the Lindelöf summation method to the sum $  s $
 +
if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589904.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\delta \rightarrow 0 } \
 +
\left [
 +
u _ {0} + \sum _ { k= } 0 ^  \infty    \mathop{\rm exp}
 +
( - \delta k  \mathop{\rm ln}  k ) u _ {k} \right ]  = s
 +
$$
  
 
and the series under the limit sign converges. The method was introduced by E. Lindelöf [[#References|[1]]] for the summation of power series.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589905.png" /> is the principal branch of an analytic function, regular at the origin and representable by a 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589906.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 0 ^  \infty  a _ {k} z  ^ {k}
 +
$$
  
for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589907.png" />, then this series is summable by the Lindelöf summation method to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589908.png" /> in the whole star of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l0589909.png" /> (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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l05899010.png" /> of type
+
Of the summation methods determined by a transformation of a sequence into a sequence by semi-continuous matrices $  a _ {k} ( \omega ) $
 +
of type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l05899011.png" /></td> </tr></table>
+
$$
 +
a _ {k} ( \omega )  = \
 +
 
 +
\frac{c _ {k+} 1 \omega  ^ {k+} 1 }{E ( \omega ) }
 +
,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l05899012.png" /></td> </tr></table>
+
$$
 +
E ( \omega )  = \sum _ { k= } 0 ^  \infty  c _ {k} \omega  ^ {k}
 +
$$
  
 
is an [[Entire function|entire function]], Lindelöf considered the case when
 
is an [[Entire function|entire function]], Lindelöf considered the case when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l05899013.png" /></td> </tr></table>
+
$$
 +
E ( \omega )  = \sum _ { k= } 0 ^  \infty 
 +
\left [
 +
 
 +
\frac \omega { \mathop{\rm ln} ( k + \beta ) }
 +
 
 +
\right ]  ^ {k} ,\ \
 +
\beta > 1 .
 +
$$
  
A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058990/l05899014.png" /> constructed from an entire function of this kind is called a Lindelöf matrix.
+
A matrix $  \| a _ {k} ( \omega ) \| $
 +
constructed from an entire function of this kind is called a Lindelöf matrix.
  
 
====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 22:16, 5 June 2020


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=23394
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article