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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.

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\textit{M. N. Berberan-Santos}, J. Math. Chem. 38, No. 4, 629--635 (2005; Zbl 1101.33015)

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