## Cauchy transforms of self-similar measures.(English)Zbl 0959.28006

Summary: The Cauchy transform of a measure in the plane, $$F(z)= {1\over 2\pi i} \int_{\mathbb{C}}{1\over z-w} d\mu(w)$$, is a useful tool for numerical studies of the measure, since the measure of any reasonable set may be obtained as the line 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 compute $$F$$ for the normalized Hausdorff measure on the Sierpiński gasket. Based on this experimental evidence, we formulate three conjectures concerning the mapping properties of $$F$$, which is a continuous function holomorphic on each component of the complement of the gasket.

### MSC:

 28A80 Fractals 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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### References:

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