Ultrathin magnetic films are considered where the thickness-induced second order phase transition occurs due to surface magnetic anisotropy changes. By increasing the thickness, the easy magnetization axis state passes through the easy cone state (for film thickness t: t 1 < t < t 2 ) into the easy magnetization plane state. The characteristic material length l c (defined as the ratio of the domain wall energy density to the demagnetization energy) was calculated up to thickness t 2 . The exact determination of stripe domain period, p, was proposed, applying the transcendent Lerch functions to the classical Kooy-Enz model. Near t 2 where l c vanishes, the critical period, p c , was also discussed. Considering real sets of material parameters taken for ultrathin cobalt films, it was found that by increasing the thickness, the equilibrium domain period drastically decreases from practically infinity (many kilometers for t = 0.5 nm) into the nanometer scale near t 2 .The changes of magnetic domain sizes with film thickness and especially in the vicinity of the reorientation phase transition (RPT) are still of great and current interest [1][2][3][4]. For both application and theory it is important to understand the physical mechanisms leading to the formation of domain structures and determining the observed variety of their sizes. A sharp decrease in domain sizes was found experimentally in the vicinity of the RPT [3,4] in Co/Au ultrathin wedges. Small domains in the nanometer scale were observed in the reorientation region.Analysis of equilibrium stripe domain structures was based in the present work on calculations of: (i) the domain period considering the classical parameter, called the material length l c = g w /(m 0 M s 2 ), where M s is the saturation magnetization, and g w is the wall energy, and (ii) l c which is dependent on thickness-induced magnetic anisotropy changes.Let us consider a stripe domain structure in an ultrathin film where the easy axis is perpendicular to the surface. Stripe domain period, p, is usually determined minimizing the total surface energy density of the domain structure -the sum of the domain wall and the demagnetization energies E = E w + E m . The domain wall energy density one can easy calculate as E w = m 0 M s 2 l c x/y, where x = t/l c and y = p/l c . More complicated is the demagnetization energy calculation. The structure can be described by the classical Kooy-Enz theory [5,6] giving E m in the form of infinite series which are slowly convergent, especially for the smallest film thickness. This was the reason for making some approximations, e.g. [2]. We propose to perform exact E m calculations using the transcendent Lerch functions. These functions are easily available in typical program