Methods and comparisons for computing the zeros of the Ahlfors map for doubly connected regions

2021 ◽  
Author(s):  
Ali H. M. Murid ◽  
Nur H. A. A. Wahid ◽  
Mukhiddin I. Muminov
Keyword(s):  
Ahlfors Map

Integral Equation for the Ahlfors Map on Multiply Connected Regions

2015 ◽  
Vol 73 (1) ◽  
Author(s):  
Kashif Nazar ◽  
Ali H. M. Murid ◽  
Ali W. K. Sangawi
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Multiple Method ◽  
Boundary Correspondence ◽  
Ahlfors Map ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with  where  is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for  using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of    we solve for the boundary correspondence function  which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

scholarly journals Analytical Solution for Finding the Second Zero of the Ahlfors Map for an Annulus Region

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Nur H. A. A. Wahid ◽  
Ali H. M. Murid ◽  
Mukhiddin I. Muminov
Keyword(s):  
Series Representation ◽  
Mapping Function ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Szegö Kernel ◽  
Szegő Kernel ◽  
Ahlfors Map ◽  
Szego Kernel ◽  
Multiply Connected Region

The Ahlfors map is a conformal mapping function that maps a multiply connected region onto a unit disk. It can be written in terms of the Szegö kernel and the Garabedian kernel. In general, a zero of the Ahlfors map can be freely prescribed in a multiply connected region. The remaining zeros are the zeros of the Szegö kernel. For an annulus region, it is known that the second zero of the Ahlfors map can be computed analytically based on the series representation of the Szegö kernel. This paper presents another analytical method for finding the second zero of the Ahlfors map for an annulus region without using the series approach but using a boundary integral equation and knowledge of intersection points.

scholarly journals The Ahlfors Map and Szegő Kernel for an Annulus

1999 ◽  
Vol 29 (2) ◽  
pp. 709-723 ◽  
Author(s):  
Thomas J. Tegtmeyer ◽  
Anthony D. Thomas
Keyword(s):  
Szegö Kernel ◽  
Szegő Kernel ◽  
Ahlfors Map ◽  
Szego Kernel

scholarly journals The computation of zeros of Ahlfors map for doubly connected regions

2016 ◽  
Author(s):  
Kashif Nazar ◽  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
Yeak Su Hoe
Keyword(s):  
Ahlfors Map
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