The Estimation Of Taylor Coefficients Of A Class Of Cauchy Transform Of Hausdorff Measure

In this thesis, we consider mainly a class of special Cauchy transform F(z), and study the asymptotic behavior of its Taylor coefficients . Let the iterated function system(IFS) {S_j}_J=0^q-1 be of the formwhere 0 < ρ ≤ ρ_q(q ≥ 4,ρ_q is defined in [1]). Let K be the attractor of {S_j}_J=0^q-1, and μ be Hausdorff measure of surpport on K. The function F(z) = ∫_K(z - w)^-1dμ(w) is called Cauchy transform of μ. Recently, the paper [2] has studied the Laurent coefficients of F(z) in |z| > 1. In this paper, we first give analytic radius R_q of F(z) in the neighborhood of z = 0: if q = 2m, R_2m = 1-2ρ and if q = 2m + 1,then we study the Taylor expansion of F(z) in |z| < R_q and give asymptotic expression of Taylor coefficients,which is always connected with a multiplicative periodic function. In the other part of the thesis ,we study the properties of these multiplicative periodic functions, getting their analytic scopes and eliminating the measure in integrals and also expressing respectively them as infinite product of a elementary function.