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Numerical conformal mapping of doubly connected regions via the Kerzman-Stein kernel. (English) Zbl 1149.30012
Authors’ abstract: Denote by $G$ a ring domain with a smooth boundary $\Gamma= \Gamma_0\cup\Gamma_1$ and by $A=\{w: \mu<|w|< 1\}$ its conformal image. It is known that the mapping $g,g(G)=A$ satisfies the integral equation $$h(z)+ \int_\Gamma A(z,w)h(w)|dw|+ i(1-\mu) \overline{T(z)} \overline{\int_{\Gamma_2} h(w) [(w-z)g(w)]^{-1}\,dw}=0, \tag *$$ where $T(z)$ is the unit tangent vector, $h(z)= \sqrt{g'(z)}$, $H(w,z)= T(z) [2\pi i(z-w)]^{-1}$, $A(z,w)= \overline{H(w,z)}- H(w,z)$, $\Gamma_2=\cases -\Gamma_1, &z\in\Gamma_0\\ \Gamma_0, &z\in\Gamma_1\endcases$. $A(z,w)$ is the Kerzman-Stein kernel. It is smooth and skew-Hermitian. The equation (*) is separated into a system of two integral equations and another equation involving the modulus $\mu$. The discritized integral equation leads to a system of nonlinear equations which is to be solved by an optimization method. An advantage of this approach is that it calculates the boundary correspondence functions and the modulus $\mu$ simultaneously. Some numerical examples are provided.

##### MSC:
 30C30 Numerical methods in conformal mapping theory 45G15 Systems of nonlinear integral equations 65E05 Numerical methods in complex analysis
##### Keywords:
integral equation; Gauss-Newton method