The Faber polynomials for circular lunes (Q1904186)

Let \(D_\alpha\) be the circular lune symmetric with respect to both axis with vertices at \(z = \pm \alpha\) and exterior angle \(\alpha \pi\), \(0 < \alpha \leq 2\). Let \(z = \psi (w) = w + \sum^\infty_{k = 0} b_k w^{-k}\) map \(\{w : |w |> 1\}\) conformally onto the exterior of \(D_\alpha\). The author determines explicitly \(\psi\), its coefficients \(b_k\), and the Faber polynomials \(F_n\) for the set \(D_\alpha\). The zeros of \(F_n\) and of \(F_n'\) are determined as eigenvalues of two matrices involving the \(b_k\), and these are computed for \(\alpha = {1 \over 2}\) and \(\alpha = {3 \over 2}\), for \(n = 30\) and \(n = 31\). The experiments suggest that the zeros of \(F_n\) and of \(F_n'\) are all inside \(D_\alpha\) and are `interlacing'.