# 1 Introduction

[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]

for complex argument z∈C and parameters α,β∈C
with Reα>0.
[637.1.2] Despite the fact that Eα,β
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

Da+α,γfx=λfx0<α≤1, 0≤γ≤1,λ∈R | | (2) |

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

Da+α,γfx=Ia+γ1-αddxIa+1-γ1-αfx0<α≤1, 0≤γ≤1 | | (3) |

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

Ia+αfx=1Γα∫axx-tα-1ftdtwitha≤x≤b,α∈R+ | | (4) |

for locally integrable functions f∈L1a,b.
[638.0.3] Recall from Ref. [18, p. 115]
that equation (2)
is solved for a=0 by

fx=x1-γα-1Eα,α+γ1-αλxα, | | (5) |

where Eα,βz 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 Eα,βz
in physics, and despite a wealth of analytical information
about Eα,βz 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 α, β, z.
[638.1.2] Easy numerical evaluation and a thorough understanding of
Eα,βz as a function of α, β, z is, however,
a key prerequisite for extending its applications to other disciplines.
[638.1.3] It is therefore desirable to explore the behaviour of
Eα,βz for large sets of the parameters α, β
and complex argument z.

[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
.
[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 Eα,βz.
[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 Eα,β
coalesce to form a simple pole in the limit α→0.
[638.2.7] Moreover, the zeros diverge in a complicated
fashion to -∞ as α approaches unity
from above as well as from below.