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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 $$0\leq\alpha\leq 1$$, 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.

##### MSC:
 33E12 Mittag-Leffler functions and generalizations 44A10 Laplace transform
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