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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, \(h(z)= \sqrt{g'(z)}\), \(H(w,z)= T(z) [2\pi i(z-w)]^{-1}\), \(A(z,w)= \overline{H(w,z)}- H(w,z)\), \(\Gamma_2=\begin{cases} -\Gamma_1, &z\in\Gamma_0\\ \Gamma_0, &z\in\Gamma_1\end{cases}\). \(A(z,w)\) 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.

30C30 Schwarz-Christoffel-type mappings
45G15 Systems of nonlinear integral equations
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)