Finite Series Representation of the Inverse Mittag-Leffler Function

Abstract

The inverse Mittag-Leffler functionEα,β-1zis 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 parametersαandβ; specifically,0<α<1/2forβ=1and forβ=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.