CAUCHY TRANSFORMS OF SELF-SIMILAR MEASURES

The Cauchy transform of a measure in the plane, F(z) = 1/2 pi i integr al(C) 1/z-w d mu(w), is a useful tool for numerical studies of the mea sure, since the measure of any reasonable set may be obtained as the l ine integral of F around the boundary. We give an effective algorithm for computing F when mu is a self-similar measure, based on a Laurent expansion of F for large z and a transformation law (Theorem 2.2) for F that encodes the self-similarity of mu. Using this algorithm we comp ute F for the normalized Hausdorff measure on the Sierpinski gasket. B ased on this experimental evidence, we formulate three conjectures con cerning the mapping properties of F, which is a continuous function ho lomorphic on each component of the complement of the gasket.