TheC2-convergence of circle packings with bounded degree to the Riemann mapping

It is known that hexagonal circle packings can be used to approximate the Riemann mappings. Here, the convergence of the hexagonal circle packings is generalized to that of circle packings with bounded degree. We shall show that the first two derivatives of the bounded degree circle packings converge to the Riemann mapping function's first two derivatives, respectively; moreover we give the explicit estimates for the rates of convergence. §Dedicated to Professor Jian-Ke Lu on the ocassion of his 85th birthday.