DERIVATIVES OF THE HYPERBOLIC DENSITY NEAR AN ISOLATED BOUNDARY POINT

Suppose that c is an isolated boundary point of a hyperbolic domain Ω in the complex plane, and let λΩ denote the density of the hyperbolic metric on Ω. We show that for each pair of nonnegative integers n and m $\begin{array}{*{20}{c}} {\lim } \\ {\omega \to c} \\ \end{array} {\left( {\omega - c} \right)^n}{\overline {\left( {\omega - c} \right)\,} ^m}\,\left| {\omega - c} \right|\log \frac{1}{{\left| {\omega - c} \right|}}\frac{{{\partial ^{m + n}}{\lambda _\Omega }\left( \omega \right)}}{{\partial {{\bar \omega }^m}\partial {\omega ^n}}} = \frac{1}{2}{c_n}{c_m}$ where c₀ = 1 and cn = ((-1)n/2) 1 - 3 - 5 ... (2n-1) for n = 1, 2, 3, ... Also we find the asymptotic limit of ${\partial ^{m + n}}{\lambda _\Omega }\left( \omega \right)/\partial {{\bar \omega }^m}\partial {\omega ^n}$ as ω → ∞ when Ω is a hyperbolic domain containing a neighborhood of ∞.