**5. Conclusions**

In this paper, integral representations for the probability density have been obtained (Theorem 1) and distribution function (Corollary 1) of a standard (*λ* = 1) strictly stable law with the characteristic function in Equation (1). In the general case *α* -= 1 and *x* -= 0 the probability density and distribution function are expressed in terms of a definite integral. In the case *α* = 1 for any *x* and in the case *x* = 0 for any admissible *α* and *θ* the probability density and distribution function are expressed in terms of elementary functions. Applying the method of numerical integration, the values of the density and distribution function of strictly stable laws with the characteristic function in Equation (1) were calculated. The calculations show that the numerical methods do not have any difficulties in calculating the density and distribution function for the selected parameter values.

However, this does not mean that one can calculate the density and function of distribution for all admissible parameters by using obtained integral representations. Most likely, numerical integration algorithms will have difficulty in calculating the integral for small values *α*, at *α* ≈ 1 and for bigger values of *x*. The results of the works in which integral representations for densities of stable laws with characteristic functions in Equations (5) and (4) were investigated testify to this. An integral representation for a stable law with the characteristic function in Equation (5) was obtained in the work [31] (see also § 2.2 in [32], § 4.4 in [36]). In the work [33], it was pointed out that when values of *α* close to 1 problems arise with the numerical calculation of the integral in this integral representation. An integral representation for the density of a stable law with the characteristic function in Equation (4) was obtained in [33]. In this work, it was emphasized that when calculating the density, calculation difficulties arise at values 0 < |*α* − 1| < 0.02 and at values *α* close to zero. In the works by [38,43] the same problems are mentioned when calculating the integral in the representation obtained in the work [33]. Based on this, it should be expected that, with the above parameter values, calculation difficulties will also arise with the density and distribution functions obtained in the Theorem 1 and Corollary 1. In particular, directly from the expressions in Equations (69) and (113), it can be seen that at *α* close to 1, but not equal to 1, problems may

arise with the numerical calculation of the integral. This is indicated by the exponent *α*/(*α* − <sup>1</sup>). It can be seen that when *α* → 1 this value increases unlimitedly. Most likely, in this case, one will have to look for other ways of calculating the density and distribution function of a strictly stable law.

In conclusion, we would like to point out that the integral representation of the density *g*(*<sup>x</sup>*, *α*, *θ*) formulated in the Theorem 1 was used to calculate the density in Equation (2). To calculate the improper integral in Equation (2) we used the adaptive quadrature Gaussian–Kornord numerical integration algorithm on 15 points. We used the implementation of this algorithm in the library gsl (GNU Scientific Library) version 1.8 [46]. The calculations performed in some cases show the presence of problems of numerical integration. In particular, at *x* close to zero, the calculated density behaves like a periodic function. In addition, in some cases, the integration algorithm generates an integration error. All this indicates the need for additional study of the integrand function in Equation (2) and adapting this expression for numerical integration algorithms. It should be noted that the most likely causes of these difficulties may be the ones described above when calculating the density *g*(*<sup>x</sup>*, *α*, *<sup>θ</sup>*). Therefore, first of all, it is necessary to find a solution to the problems described above. To calculate the density at *x* close to zero and for bigger values *x* the most promising approach is to use an expansion of the strictly stable density in the power series. The method described in the article [43] can be used to calculate the density at *α* → 1. However, the possibility of using this approach requires additional research.

**Funding:** This work was supported by the Ministry of Higher Education and Science of the Russian Federation (project No. 0830-2020-0008).

**Acknowledgments:** The author thanks to M. Yu. Dudikov for translation the article into English.

**Conflicts of Interest:** The author declares no conflict of interest.
