Difference between revisions of "Lindelöf summation method"
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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 | ||
| − | + | $$ | |
| + | g _ {0} ( \delta ) = 1 ,\ \ | ||
| + | g _ {k} ( \delta ) = \mathop{\rm exp} ( - \delta k \mathop{\rm ln} k ) ,\ \ | ||
| + | \delta > 0 ,\ k = 1 , 2 , . . . . | ||
| + | $$ | ||
The series | The series | ||
| − | + | $$ | |
| + | \sum _ { k= } 0 ^ \infty u _ {k} $$ | ||
| − | is summable by the Lindelöf summation method to the sum | + | 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. | 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 | + | 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 | + | 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 | + | 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 | where | ||
| − | + | $$ | |
| + | E ( \omega ) = \sum _ { k= } 0 ^ \infty c _ {k} \omega ^ {k} | ||
| + | $$ | ||
is an [[Entire function|entire function]], Lindelöf considered the case when | is an [[Entire function|entire function]], Lindelöf considered the case when | ||
| − | + | $$ | |
| + | E ( \omega ) = \sum _ { k= } 0 ^ \infty | ||
| + | \left [ | ||
| + | |||
| + | \frac \omega { \mathop{\rm ln} ( k + \beta ) } | ||
| + | |||
| + | \right ] ^ {k} ,\ \ | ||
| + | \beta > 1 . | ||
| + | $$ | ||
| − | A 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=47643
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