The approximate conformal mapping onto simply and doubly connected domains

A new method of approximate conformal mapping of the unit disk onto a simply connected domain and of an annulus onto a doubly connected domain with smooth boundaries is presented. The method is based on integral equations solution which is reduced to a linear system solution and does not require iterations. The mapping function has the form of a Taylor or a Laurent polynomial defined on the unit disk or on the annulus, respectively. The method is easily computable.