Finite series representation of the inverse Mittag-Leffler function (Q1717951)

Summary: The inverse Mittag-Leffler function \(E_{\alpha, \beta}^{- 1} \left(z\right)\) is valuable in determining the value of the argument of a Mittag-Leffler function given the value of the function and it is not an easy problem. A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parameters \(\alpha\) and \(\beta\); specifically, \(0 < \alpha < 1 / 2\) for \(\beta = 1\) and for \(\beta = 2\). This finite series representation of the inverse Mittag-Leffler function greatly expedites its evaluation and has been illustrated with a number of examples. This represents a significant advancement in the understanding of Mittag-Leffler functions.