Preprint No.
A-99-06
Rudolf Gorenflo, Ioulia Loutchko, Yuri Luchko
Numerische Berechnung der Mittag-Leffler-Funktion
Ealpha, beta(z) und ihrer Ableitung
Abstract: Recently it turned out that the role of the well-known {\sl
Mittag-Leffler function}
\[
E_{\alpha} (z) := \sum_{n=0}^{\infty} \frac{z^n}{\Gamma
(\alpha n + 1)}, \ \alpha > 0, \ z\in {\bf C}
\]
and the {\sl generalized Mittag-Leffler function}
\[
E_{\alpha,\beta} (z) :=
\sum_{n=0}^{\infty} \frac{z^n}{\Gamma (\alpha n + \beta)}, \ \alpha > 0,
\ \beta \in {\bf C},\ z\in {\bf C}
\]
in different branches of analysis is more essential than it seemed
before. Mainly these
applications are due to connections of the Mittag-Leffler functions with
fractional calculus.
In this paper we first obtain a modification of the Dzherbasyan integral
representation for the generalized Mittag-Leffler function. This representation
and the series representations (for small values of the argument $z$) are then
used for numerical evaluations of the Mittag-Leffler functions for all possible
values of the parameters and
arguments. The estimates of the errors of evaluations are given. A method
for the numerical evaluation of the derivative of the generalized Mittag-Leffler
function is considered.
Keywords: Mittag-Leffler function, integral representations, numerical evaluations
Mathematics Subject Classification (MSC91): 33E20 Other functions defined by series and integrals
, 33C20 Generalized hypergeometric series, $_pF_q$
Language: ENG
Available: Pr-A-99-06.ps
Contact: Gorenflo, Rudolf; Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (gorenflo@math.fu-berlin.de)
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