A significant advance in characterizing the nature of the zeros and organizing the Mittag-Leffler functions into phases according to their behavior is presented. Regions have been identified in the domain of and where the Mittag-Leffler functions have not only the same type of zeros but also exhibit similar functional behavior, and this permits the establishment of an - phase diagram. 1. Introduction The Mittag-Leffler (ML) function defined by is a generalization of the exponential function and plays a fundamental role in the theory of fractional differential equations with numerous applications in physics. Consequently, books devoted to the subject of fractional differential equations (i.e., Podlubny [1], Magin [2], Kilbas et al. [3], and Mainardi [4]) all contain sections on the Mittag-Leffler functions. Despite the inherent importance of Mittag-Leffler functions in fractional differential equations, their behaviour and types of zeros have not been fully characterized. This work resolves this delinquency by identifying regions in the domain of and where the Mittag-Leffler functions exhibit similar behaviour. In this work, is restricted to real numbers. While Mittag-Leffler functions in general exhibit a diverse range of behaviors, ML functions which have the same types of zeros also exhibit many other similar properties. Hence, it is logical to organize the ML functions with similar types of zeros and similar properties into regions of the parameter space ( , ) (restricted to positive real numbers in this work) resulting in a - “phase diagram” for the Mittag-Leffler functions. Information extracted from a review of the literature on the theory of the zeros of ML functions together with the numerical results of this work yields the first depiction of the - phase diagram for the Mittag-Leffler functions shown in Figure 1 for the range . Seven major regions or phases have been identified and descriptively labeled by an ordered pair of symbols where the first symbol of the pair indicates the number of real zeros attributed to the ML function in that phase [i.e., none (0), finite ( ), or infinite ( )] and the second symbol of the pair refers to the number of complex zeros attributed to the Mittag-Leffler functions in that same phase. Figure 1: - phase diagram for . The arrows point to the particular phase into which a Mittag-Leffler function whose parameters lie on a phase boundary belongs. Dotted lines are used to indicate phase boundaries where and are known exactly and solid lines for boundaries where and/or may be determined as accurately as desired
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