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Properties of the Mittag-Leffler relaxation function. (English) Zbl 1101.33015
Summary: The Mittag-Leffler relaxation function, $E_\alpha(-x)$, with $0\le\alpha\le 1$, 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.

33E12Mittag-Leffler functions and generalizations
44A10Laplace transform
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