Faber polynomials

Summary

The Faber polynomials of a compact set $K$ in the plane are polynomials $F_0, F_1, F_2, \dots$, with $F_n$ of degree $n$. They are built from the conformal map of the complement of $K$ onto the exterior of the unit disc. They play for $K$ the role that the powers $z^n$ play for a disc. Functions holomorphic on a neighbourhood of $K$ can be expanded in Faber series $\sum_n b_n F_n(z)$. They are a standard tool of polynomial approximation and of geometric function theory.