Abstract
The Mittag-Leffler functions appear in many problems associated with fractional calculus. In this paper, we use the methodology for evaluation of the inverse Laplace transform, proposed by M. N. Berberan-Santos, to show that the three-parameter Mittag-Leffler function has similar integral representations on the positive real axis. Some of the integrals are also presented.
Highlights
The Mittag-Leffler function, introduced in 1902 by Gosta Mittag-Leffler [23], is important in many fields, including description of the anomalous dielectric properties, probability theory, statistics, viscoelasticity, random walks and dynamical systems [9, 10, 11, 14, 19, 25, 26]
Berberan-Santos [2] proposed a new methodology for evaluation of the numerical inverse Laplace transform, without using integration on the complex plane, which was published in 2005, and its methodology was used recently, for instance, to discuss the luminescence decay of inorganic solids, and to obtain an integral representation of Mittag-Leffler relaxation function, a special one-parameter Mittag-Leffler function [3]
With the method for finding inverse Laplace transform without using integration on the complex plane we show that the three-parameter Mittag-Leffler function has integral representations on the positive real axis
Summary
The Mittag-Leffler function, introduced in 1902 by Gosta Mittag-Leffler [23], is important in many fields, including description of the anomalous dielectric properties, probability theory, statistics, viscoelasticity, random walks and dynamical systems [9, 10, 11, 14, 19, 25, 26]. Berberan-Santos [2] proposed a new methodology for evaluation of the numerical inverse Laplace transform, without using integration on the complex plane, which was published in 2005, and its methodology was used recently, for instance, to discuss the luminescence decay of inorganic solids, and to obtain an integral representation of Mittag-Leffler relaxation function, a special one-parameter Mittag-Leffler function [3]. The method to evaluate the inverse Laplace transform without using integration on the complex plane was applied in [6] to find integral representations on the positive real axis for some functions. With the method for finding inverse Laplace transform without using integration on the complex plane we show that the three-parameter Mittag-Leffler function has integral representations on the positive real axis.
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