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 numberρ 1 satisfying the following conditions: 1) for any interval of the form [0,ρ 1(1−ε)], where 0<ɛ<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 continuationf(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 functionf(z) in a certain region containing the interval [0,ρ 1]. It is shown to be possible to continue analytically the virial expansion along the path [0,ρ 1(1−ε)], where 0<ɛ<1, and impossible to do this along the path [0,ρ 1]. An equation that determines a positive numberz 1 such thatρ 1=f(z 1) is found.