Before revision:  Our method works without any prior knowledge about the number of zeros , or their approximate location , as long as the number of zeros is finite. We give a complete convergence analysis, which relies on results on the caustics of f, and on convergence results for Newton's method.
After revision:  Our method works without any prior knowledge about the number of zeros , or their approximate location . We start by computing all solution of f(z) = \eta with |\eta| sufficiently large and then track all solutions as \eta tends to 0 to finally obtain all zeros of f. Using theoretical results on harmonic mappings we analyze where and how the number of solutions of f(z) = \eta changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular.