Cauchy transforms of self-similar measures: the Laurent coefficients
نویسندگان
چکیده
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 fangn1⁄40 of F : We give asymptotic formulas for fangn1⁄40 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. r 2002 Elsevier Inc. All rights reserved. MSC: primary 28A 80; secondary 30C 55; 30E 20
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