Abstract
The generalized Mittag-Leffler function $E_\alpha,\beta(z)$
has been computed for arbitrary complex argument $zınC$
and parameters $\alphaınR^+$ and $\betaınR$ 1.
This function plays a fundamental role in the theory of fractional
differential equations and numerous applications in physics.
The Mittag-Leffler function interpolates smoothly between exponential
and algebraic functional behaviour. A numerical algorithm for its
evaluation has been developed. The algorithm is based on integral
representations and exponential asymptotics. Results of extensive
numerical calculations are presented. We find that all complex
zeros emerge from the point $z=1$ for small $\alpha$. They diverge towards
$-ınfty+(2k-1)\pii$ for $\alpha1^-$ and towards
$-ınfty+2k\pii$ for $\alpha1^+$ ($kınZ$).
All complex zeros collapse pairwise onto the negative real axis for
$\alpha2$. We introduce and study also the inverse generalized
Mittag-Leffler function, and determine its principal branch numerically.
0.5truecm
\noindent
1 R. Hilfer and H.J. Seybold, Integral Transforms and
Special Functions, vol. 17, (2006), p. 637
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