A Fredholm integral equation of the second kind for conformal mapping

摘要

We present a Fredholm integral equation of the second kind for the derivative of the boundary correspondence function θ of the conformal mapping of a Jordan region with piecewise twice differentiable boundary onto the unit disk. One of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind. The integral equation can be solved numerically by applying the Fourier method; the convergence to θ′ in L2 of the solution of the discrete equation so obtained follows by simple theorems. Examples show that the equation often yields better numerical results than both a related method proposed by Henrici and Warschawski's equation.