The Cauchy transform of a measure in the plane,
$$
F(z) = \frac{1}{2\pi i}\int_{\C} \frac{1}{z-w} \,d\mu(w)\hbox{,}
$$
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.