The Ahlfors Map and Szegő Kernel for an Annulus

Abstract

In the case of an annulus, it is simple to find an orthonormal basis for the Hardy space. This allows one to write both the Szegő and Garabedian kernel functions as infinite series. These series are classical. The Ahlfors map is a two-to-one branched covering map of the annulus onto the unit disk and is given by the quotient of the Szegő and Garabedian kernels. One of the two zeros of the Ahlfors map arises from the pole of the Garabedian kernel. The other zero corresponds to the zero of the Szegő kernel. In Section 5 it is shown how to use the series for the Szegő kernel to find its zero.The boundary values of the Garabedian kernel are given in terms of those of the Szegő kernel. Hence, knowing the boundary values of the Szegő kernel is tantamount to knowing those of the Ahlfors map. A discovery of Kerzman and Stein provides an efficient numerical method for computing the boundary values of the Szegő kernel for a smoothly bounded, planar domain and hence the boundary values of the Ahlfors map. Since the Ahlfors map is a holomorphic function smooth up to the boundary, the Cauchy integral formula provides the interior values of this map. Unfortunately, this integral formula has a singular nature for interior points near the boundary. In Section 7 it is shown how to alleviate this singular behavior for the annulus, and in Section 8 graphical examples are given for the Szegő and Garabedian kernels and for the Ahlfors map. Preliminaries.Suppose that Ω is a domain in C with C ∞ smooth boundary. Let L 2 (bΩ) denote the space of square integrable functions with respect to arc length measure on the boundary bΩ of Ω, and let H 2 (bΩ) denote the subspace of L 2 (bΩ) consisting of functions that extend to be holomorphic on Ω. An inner product •, • is defined

The computation of zeros of Ahlfors map for doubly connected regions

“…By Newton's method which is a quick and efficient method, we solved the obtained system of equations to find the second zero of Ahlfors map. Analytical method for computing the exact zeros of Ahlfors map for annulus region and a particular triply connected regions are presented in [21] and [22] but the problem of finding zeros for arbitrary doubly connected regions is the first time presented in this paper. The numerical examples presented have illustrated that our method is reliable and has high accuracy.…”

“…For computing the Ahlfors map, some integral equations have been given in [4,18,19,20,21,22]. In [1], Kerzman and Stein have derived a uniquely solvable boundary integral equation for computing the Szegö kernel of a bounded region and this method has been generalized in [18] In [4], Nazar et al extended the approach of Sangawi [11,15,16] to construct an integral equation for solving where is the boundary correspondence function of Ahlfors map of multiply connected region onto a unit disk.…”

Numerical Conformal Mapping via a Boundary Integral Equation with the Generalized Neumann Kernel

“…In this section, we shall consider a numerical example in the annulus 0.1 < |z| < 1 (m = 2). This example have been considered in [16] where the Ahlfors map was computed using the Szegö and the Garabedian kernels. The first zero of the Ahlfors map, a, is assumed to be an arbitrary point in G. The second zero of the Ahlfors map, a 1 , was computed in [16] as the zero of the Szegö kernel.…”

“…The method presented in [1] can be used to compute the Ahlfors map without relying on the zeros of the Ahlfors map. See also [7,15,16].…”

Equivalence problem for annuli and Bell representations in the plane

“…(see [12]). The Ahlfors map with base point ρ is f ρ (z) = S(z, ρ)/L(z, ρ) as in (2.1), which gives a 2-sheeted branched covering of the unit disc U by Ω ρ 2 .…”