Analytic approximation to 5 dimensional black holes with one compact dimension

We study black hole solutions in R⁴×S¹ space, using an expansion to second order in the square of the ratio of the radius of the horizon, μ, and the circumference of the compact dimension, L. A study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at ϵ=(μ/L)²0.1. At the same value of $ϵ$ the entropies of the uniform black string and of the black hole are approximately equal.