The Faber Polynomials for Circular Sectors

The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformai mapping of the complement of that region to the complement of the unit disc. We derive this conformai mapping for a circular sector {;: \z\ < 1, |argz| < i/a}, where a > 1, and obtain a recurrence relation for the coefficients of its Laurent expansion about the point at infinity. We discuss the computation of the coefficients of the Faber polynomials of degree 1 to 15, which are tabulated here for sectors of half-angle 5°, 10°, 15°, 30°, 45°, and 90°, and we give explicit expressions, in terms of a, for the polynomials of degree ^ 3. The norms of Faber polynomials are tabulated and are compared with those of the Chebyshev polynomials for the