Analytic continuation of mayer and virial expansions

Functions that under certain assumptions are analytic continuations of Mayer expansions are found. It is shown that there exists a positive number $\rho_1$ satisfying the following conditions: 1) tor any interval of the form $[0,\rho_1(1-\varepsilon)]$, where $0<\varepsilon<1$, there exists a region containing this interval in which there is defined a single-valued analytic single-sheeted function that is the inverse with respect to the analytic continuation $f(z)$ of the Mayer expansion which represents the density as a function of the activity; 2) there does not exist a single-valued analytic function that would be the inverse with respect to the function $f(z)$ in a certain region containing the interval $[0,\rho_1]$. It is shown to be possible to continue analytically the virial expansion along the path $[0,\rho_1(1-\varepsilon)]$, where $0<\varepsilon<1$, and impossible to do this along the pathl $[0,\rho_1]$. An equation that determines a positive number $z_1$ such that $\rho_1=f(z_1)$ is found.