Numerical conformal mapping methods for exterior regions with corners
Let \(C\) be a piecewise analytic Jordan curve, and assume that \(f\) maps the exterior of \(C\) conformally onto \(\{w:| w |>1\}\) such that \(f(z)=az+a_ 0+{a_ 1 \over z}+\cdots\). Let \(z=z(\sigma)\) \((0 \leq \sigma \leq L)\) be the representation of \(C\) in terms of arc length, and let \(\theta=\theta(\sigma)=\arg f(z(\sigma))\) be the boundary correspondence between \(C\) and \(\{w:| w |=1\}\). Timman (1952) had given an equation for \(\sigma=\sigma(\theta)\) which the authors transform into a nonlinear integro-differential equation for \(\theta(\sigma)\) involving the curvature of \(C\). This is then solved by iteration, and the new method compared numerically with several other methods.
scientific article; zbMATH DE number 516906