Integral Representation of the Fractional Stable Density

Two Forms of the Integral Representations of the Mittag-Leffler Function

“…It is convenient to use the representations of such a kind in practical problems as well as for calculating the values of the Mittag-Leffler function. As an example of the use of such integral representations, we can mention the work [24]. In that paper, the inverse Fourier transform of the characteristic function of the fractionally stable law was performed which is expressed in terms of the Mittag-Leffler function.…”

“…To perform the inverse Fourier transform, we used the integral representation of the Mittag-Leffler function which in the current paper is formulated in Corollary 3 item 2 (see (84)). Using this integral representation in the paper [24] expressions of density and distribution function of a fractionally stable law were obtained. Articles [25,26] are another example of usage of the Mittag-Leffler function.…”

The Calculation of the Probability Density and Distribution Function of a Strictly Stable Law in the Vicinity of Zero

“…The latter distribution first came out in the book by V.M. Zolotarev [1] (see formula (2.3.5a)) and later was examined in the works [2,3]. Different representations for stable laws are required to calculate the probability density or distribution function in other cases.…”

“…The use of integral representations is the main method for calculating the probability density and the distribution function of stable laws. This approach is based on the inversion formula (2). There are two possible ways of inverting the characteristic function.…”

“…There are two possible ways of inverting the characteristic function. The first way is to directly calculate the integral in (2). As a result, the probability density is expressed in terms of the integral of the oscillating function [13,14].…”