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# 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]

 Eα,β⁢z=∑k=0∞zkΓ⁢α⁢k+β (1)

for complex argument zC and parameters α,βC 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,γ⁢f⁢x=λ⁢f⁢x⁢0<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,γ⁢f⁢x=Ia+γ⁢1-α⁢dd⁢x⁢Ia+1-γ⁢1-α⁢f⁢x⁢0<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 αR+ defined by

 Ia+α⁢f⁢x=1Γ⁢α⁢∫axx-tα-1⁢f⁢t⁢d⁢t⁢with⁢a≤x≤b,α∈R+ (4)

for locally integrable functions fL1a,b. [638.0.3] Recall from Ref. [18, p. 115] that equation (2) is solved for a=0 by

 f⁢x=x1-γ⁢α-1⁢Eα,α+γ⁢1-α⁢λ⁢xα, (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 a (This is a footnote:) aWhile 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 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 α0. [638.2.7] Moreover, the zeros diverge in a complicated fashion to - as α approaches unity from above as well as from below.