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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]

\mbox{\rm E}_{{\alpha,\beta}}(z)=\sum _{{k=0}}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)} (1)

for complex argument z\in\mathbb{C} and parameters \alpha,\beta\in\mathbb{C} with \operatorname{Re}\alpha>0. [637.1.2] Despite the fact that \mbox{\rm E}_{{\alpha,\beta}} 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

\mbox{\rm D}^{{\alpha,\gamma}}_{{a+}}f(x)=\lambda f(x)\qquad 0<\alpha\leq 1,\,\, 0\leq\gamma\leq 1,\quad\lambda\in\mathbb{R} (2)

where \mbox{\rm D}^{{\alpha,\gamma}}_{{a+}} is a fractional derivative of order \alpha and type \gamma with lower limit a [18]. [638.0.1] In eq. (2) the symbol \mbox{\rm D}^{{\alpha,\gamma}}_{{a+}} stands for [18, p. 115]

\mbox{\rm D}^{{\alpha,\gamma}}_{{a+}}f(x)=\left[\mbox{\rm I}^{{\gamma(1-\alpha)}}_{{a+}}\frac{\mbox{\rm d}}{\mbox{\rm d}x}\mbox{\rm I}^{{(1-\gamma)(1-\alpha)}}_{{a+}}f\right](x)\qquad 0<\alpha\leq 1,\,\, 0\leq\gamma\leq 1 (3)

for functions for which the expression on the right hand side exists. [638.0.2] Of course, the notation \mbox{\rm I}^{{\alpha}}_{{a+}}f(x) stands for the right sided fractional Riemann-Liouville integral of order \alpha\in\mathbb{R}^{+} defined by

\mbox{\rm I}^{{\alpha}}_{{a+}}f(x)=\frac{1}{\Gamma(\alpha)}\int\limits _{a}^{x}(x-t)^{{\alpha-1}}f(t)\mbox{\rm d}t\quad\mbox{with}\quad a\leq x\leq b,\qquad\alpha\in\mathbb{R}^{+} (4)

for locally integrable functions f\in L^{1}[a,b]. [638.0.3] Recall from Ref. [18, p. 115] that equation (2) is solved for a=0 by

f(x)=x^{{(1-\gamma)(\alpha-1)}}\mbox{\rm E}_{{\alpha,\alpha+\gamma(1-\alpha)}}(\lambda x^{\alpha}), (5)

where \mbox{\rm E}_{{\alpha,\beta}}(z) 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 \mbox{\rm E}_{{\alpha,\beta}}(z) in physics, and despite a wealth of analytical information about \mbox{\rm E}_{{\alpha,\beta}}(z) 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 \alpha, \beta, z. [638.1.2] Easy numerical evaluation and a thorough understanding of \mbox{\rm E}_{{\alpha,\beta}}(z) as a function of \alpha, \beta, 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 \mbox{\rm E}_{{\alpha,\beta}}(z) for large sets of the parameters \alpha, \beta 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 \mbox{\rm E}_{{\alpha,\beta}}(z). [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 \mbox{\rm E}_{{\alpha,\beta}} coalesce to form a simple pole in the limit \alpha\to 0. [638.2.7] Moreover, the zeros diverge in a complicated fashion to -\infty as \alpha approaches unity from above as well as from below.