Integral Representation of the Fractional Stable Density

This article studies the properties of the characteristic function of fractional stable distribution expressed through the Mittag-Leffler function. It is shown that the existing integral expression for the Mittag-Leffler function is incorrect, and the corrected integral expression is given. New properties of the characteristic function of the fractional stable law are obtained that make it possible to perform the inverse Fourier transformation. As a result, the integral representations are obtained for the density and distribution function of the fractional stable law. Some properties of these representations are studied, and the results of numerical calculations for the probability density and distribution function are presented.