The calculation of the distribution function of a strictly stable law at large X

The paper considers the problem of calculating the distribution function of a strictly stable law at $x\to \mathrm{\infty }$. To solve this problem, an expansion of the distribution function in a power series was obtained, and an estimate of the remainder term was also obtained. It was shown that in the case $\alpha <1$ this series was convergent for any $x$, in the case $\alpha =1$ the series was convergent at $N\to \mathrm{\infty }$ in the domain $\mathrm{\mid }x\mathrm{\mid }>1$, and in the case $\alpha >1$ the series was asymptotic at $x\to \mathrm{\infty }$. The case $\alpha =1$ was considered separately and it was demonstrated that in that case the series converges to the generalized Cauchy distribution. An estimate for the threshold coordinate $x_{\epsilon }^N$ was obtained which determined the area of applicability of the obtained expansion. It was shown that in the domain $\mathrm{\mid }x\mathrm{\mid }⩾x_{\epsilon }^N$ this power series could be used to calculate the distribution function, which completely solved the problem of calculating the distribution function at large $x$.