We demonstrate how the stripe-domain configuration which is the most stable one in thin ferromagnetic films with perpendicular magnetization can be easily analysed in terms of Bose - Einstein integrals. Exact values for the equilibrium thickness of stripes as well as for the minimal free energy of the domain configuration in ultrathin samples are derived from little-known asymptotic expansions of the Bose - Einstein integrals. It is found that, due to an incidental cancellation, the lowest-order results for the stripe structure are exact also to the next order in the small parameter (the ratio of thickness to domain width). The minimization equation for the determination of the equilibrium quantities of interest is cast in a form that makes it applicable analytically to films which are not necessarily ultrathin and that allows one to control quantitatively the approximations made. The analytic solution, valid to a high order in the small ratio, is also given.