Difference between revisions of "Mittag-Leffler summation method"
m (added TEX state) |
(texed) |
||
Line 1: | Line 1: | ||
− | {{TEX| | + | {{TEX|done}} |
A [[Semi-continuous summation method|semi-continuous summation method]] for summing series of numbers and functions, defined by a sequence of functions | A [[Semi-continuous summation method|semi-continuous summation method]] for summing series of numbers and functions, defined by a sequence of functions | ||
− | + | \[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \] | |
− | where | + | where $ \Gamma(x) $ is the gamma-function. A series |
− | + | \[ \sum_{k=0}^{\infty} u_k \] | |
− | is summable by the Mittag-Leffler method to a sum | + | is summable by the Mittag-Leffler method to a sum $s$ if |
− | + | \[ \lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s \] | |
and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [[#References|[1]]] primarily for the series | and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [[#References|[1]]] primarily for the series | ||
− | + | \[ \sum_{k=0}^{\infty} z^k . \] | |
− | A Mittag-Leffler 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 | + | A Mittag-Leffler 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 zero and represented by a series |
− | + | \[ \sum_{k=0}^{\infty}a_k z^k \] | |
− | for small | + | for small $z$, then this series is summable by the Mittag-Leffler 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, moreover, uniformly in any closed bounded domain contained in the interior of the star. |
− | For summation methods defined by transformations of sequences by semi-continuous matrices | + | For summation methods defined by transformations of sequences by semi-continuous matrices $ a_k(\omega) $ of the 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, Mittag-Leffler considered the case when | is an entire function, Mittag-Leffler considered the case when | ||
− | + | \[ E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)} \] | |
− | A matrix | + | A matrix $ a_k(\omega) $ with such an entire function is called a Mittag-Leffler matrix. |
====References==== | ====References==== | ||
Line 44: | Line 44: | ||
====Comments==== | ====Comments==== | ||
− | The function | + | The function $E(\omega)$ considered by Mittag-Leffler is called a [[Mittag-Leffler function|Mittag-Leffler function]]. |
Latest revision as of 02:56, 2 June 2013
A semi-continuous summation method for summing series of numbers and functions, defined by a sequence of functions
\[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \]
where $ \Gamma(x) $ is the gamma-function. A series
\[ \sum_{k=0}^{\infty} u_k \]
is summable by the Mittag-Leffler method to a sum $s$ if
\[ \lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s \]
and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [1] primarily for the series
\[ \sum_{k=0}^{\infty} z^k . \]
A Mittag-Leffler 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 zero and represented by a series
\[ \sum_{k=0}^{\infty}a_k z^k \]
for small $z$, then this series is summable by the Mittag-Leffler method to $ f(z) $ in the whole star of the function $ f(z) $ (cf. Star of a function element) and, moreover, uniformly in any closed bounded domain contained in the interior of the star.
For summation methods defined by transformations of sequences by semi-continuous matrices $ a_k(\omega) $ of the 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, Mittag-Leffler considered the case when
\[ E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)} \]
A matrix $ a_k(\omega) $ with such an entire function is called a Mittag-Leffler matrix.
References
[1] | G. Mittag-Leffler, , Atti IV congress. internaz. , 1 , Rome (1908) pp. 67–85 |
[2] | G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181 |
[3] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
[4] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
Comments
The function $E(\omega)$ considered by Mittag-Leffler is called a Mittag-Leffler function.
Mittag-Leffler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_summation_method&oldid=29806