en>Tony1 at 02:51, 25 December 2013

2013-12-25T02:51:05Z

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In mathematics, '''Mittag-Leffler summation''' is any of several variations of the [[Borel summation]] method for summing possibly [[divergent series|divergent]] [[formal power series]], introduced by {{harvs|txt|last=Mittag-Leffler|authorlink=Gösta Mittag-Leffler|year=1908}}

==Definition==

Let
:<math>y(z) = \sum_{k = 0}^\infty y_kz^k</math>

be a [[formal power series]] in ''z''.

Define the transform <math>\scriptstyle \mathcal{B}_\alpha y</math> of <math>\scriptstyle y</math> by
:<math>\mathcal{B}_\alpha y(t) \equiv \sum_{k=0}^\infty \frac{y_k}{\Gamma(1+\alpha k)}t^k</math>

Then the '''Mittag-Leffler sum''' of ''y'' is given by
:<math>\lim_{\alpha\rightarrow 0}\mathcal{B}_\alpha y( z)</math>
if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows {{harv|Sansone|Gerretsen|1960}}.
Suppose that the Borel transform converges to an [[analytic function]] near 0 that can be [[analytic continuation|analytically continued]] along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the '''Mittag-Leffler sum''' of ''y'' is given by
:<math>\int_0^\infty e^{-t} \mathcal{B}_\alpha y(t^\alpha z) \, dt</math>

When ''α'' = 1 this is the same as [[Borel summation]].

==See also==

*[[Mittag-Leffler function]]

==References==

*{{eom|id=Mittag-Leffler_summation_method|title=Mittag-Leffler summation method}}
*{{Citation | last1=Mittag-Leffler | first1=G. |authorlink=Gösta Mittag-Leffler| title=Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908) | url=http://www.mathunion.org/ICM/ICM1908.1/ | year=1908 | volume=I | chapter=Sur la représentation arithmétique des fonctions analytiques d'une variable complexe | pages=67–86}}
*{{Citation | last1=Sansone | first1=Giovanni | last2=Gerretsen | first2=Johan | title=Lectures on the theory of functions of a complex variable. I. Holomorphic functions | publisher=P. Noordhoff, Groningen | mr=0113988 | year=1960}}

[[Category:Summability methods]]

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en>Tony1 at 02:51, 25 December 2013