Properties of the Mittag-Leffler relaxation function

The Mittag-Leffler relaxation function, E-alpha(-x), with 0 <= alpha <= 1, which arises in the description of complex relaxation processes, is studied. A relation that gives the relaxation function in terms of two Mittag-Leffler functions with positive arguments is obtained, and from it a new form of the inverse Laplace transform of E-alpha(-x) is derived and used to obtain a new integral representation of this function, its asymptotic behaviour and a new recurrence relation. It is also shown that the fastest initial decay of E-alpha(-x) occurs for alpha =1/2, a result that displays the peculiar nature of the interpolation made by the Mittag-Leffler relaxation function between a pure exponential and a hyperbolic function.