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 fang N n0 of F: We give asymptotic formulas for fang N n0 and for F ðkÞðzÞ near the support of m; hence the precise growth rates on janj and jF ðkÞj are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of m and K: As a by-product, we also discover new identities of certain infinite products and series