A solution formula for regular conformal weldings (Q1924969)

Let \(\Phi : e^{i \theta} \mapsto e^{i \varphi (\theta)} (\theta, \varphi \in \mathbb{R})\) be an orientation preserving homeomorphism of \(C = \{z \mid |z |= 1\}\) onto itself. Assume that \(\Phi\) is analytic with \(\Phi' \neq 0\) in a neighborhood of \(C\). It is known that there exists an analytic Jordan curve \(\Gamma\) with two bijective conformal maps \(F : \text{int} C \to \text{int} \Gamma\), \(G : \text{ext} \Gamma\), \(G (\infty) = \infty\), such that \(F (e^{i \theta }) = G(e^{i \varphi (\theta)})\) \(\forall \theta \in \mathbb{R}\). \({d \over d \theta} \arg F' (e^{i \theta}) = \psi (\theta)\) (A) \(\forall \theta \in \mathbb{R}\), where \(\psi (\theta) = {1 \over 2} [\varphi' (\theta) - 1 - {\partial U \over \partial n} (e^{i \theta})]\). Here \(U\) denotes the solution of the Dirichlet problem for the interior of \(C\) with boundary values \(bg \varphi' (\theta)\), and \(\partial/ \partial n\) indicates the derivative in the direction of the interior normal to \(C\). Relation (A) leads to an explicit formula for \(F(z)\), \(|z |< 1\).