Approximation of analytic functions with prescribed boundary conditions by circle packing maps

We use recent advances in circle packing theory to develop a constructive method for the approximation of an analytic function $F:\Omega \to \bold C$ by circle packing maps providing we have only been given $\Omega$, $|F'|\big|_{\Omega}$, and the set of critical points of $F$. This extends the earlier result of Ithiel Carter and Burt Rodin [CR] for $F$ with no critical points.