Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in C. Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {an}n=0∞ of F. We give asymptotic formulas for {an}n=0∞ and for F(k)(z) near the support of μ, hence the precise growth rates on |an| and |F(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.