[page 637, §1]

[637.1.1] A special function of growing importance
is the generalized Mittag-Leffler function defined
by the power series [2, p. 210]

(1) |

for complex argument and parameters with . [637.1.2] Despite the fact that was introduced roughly 100 years ago [3, 4, 5, 6, 7, 8] its mapping properties in the complex plane are largely unknown.

[637.2.1] Mittag-Leffler functions are important in mathematical as well as in theoretical and applied physics [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. [637.2.2] A primary reason for the recent surge of interest in these functions [page 638, §0] is their appearance when solving the fractional differential equation

(2) |

where is a fractional derivative of order and type with lower limit [18]. [638.0.1] In eq. (2) the symbol stands for [18, p. 115]

(3) |

for functions for which the expression on the right hand side exists. [638.0.2] Of course, the notation stands for the right sided fractional Riemann-Liouville integral of order defined by

(4) |

for locally integrable functions . [638.0.3] Recall from Ref. [18, p. 115] that equation (2) is solved for by

(5) |

where is the generalized Mittag-Leffler function. [638.0.4] Equation (2) shows that the Mittag-Leffler function plays the same role for fractional calculus that the exponential function plays for conventional calculus. [638.0.5] Mittag-Leffler functions and fractional calculus have in recent years become a powerful tool to investigate anomalous dynamics and strange kinetics [13, 18, 19, 20].

[638.1.1] Despite the growing importance of in physics, and despite a wealth of analytical information about its behaviour as a holomorphic function and dependence upon the parameters are largely unexplored, because there seem to be no numerical algorithms available to compute the function accurately for all , , . [638.1.2] Easy numerical evaluation and a thorough understanding of as a function of , , is, however, a key prerequisite for extending its applications to other disciplines. [638.1.3] It is therefore desirable to explore the behaviour of for large sets of the parameters , and complex argument .

[638.2.1] Given this objective the present paper reports
a newly developed numerical algorithm as well as
extensive computations for the generalized Mittag-Leffler function.
[638.2.2] Little will be said in this paper about the algorithm apart from
giving its complete definition
^{a} (This is a footnote:) ^{a}While this work was in progress a simpler algorithm
appeared in [24].
A detailed comparison between the two algorithms
can be found in [25]..
[638.2.3] One should note that the algorithm works not only on the real axis,
but in the full complex plane.
[638.2.4] Rather than discussing details of the algorithm we concentrate here
on exploring the functional behaviour of .
[638.2.5] In particular we study its complex zeros and illustrate
its behaviour as an entire function.
[638.2.6] As an example we find that the zeros of
coalesce to form a simple pole in the limit .
[638.2.7] Moreover, the zeros diverge in a complicated
fashion to as approaches unity
from above as well as from below.