On composite conformal mapping of an annulus to a plane with two holes

Abstract A semi-inverse composite conformal mapping is considered for an infinite plane containing two holes. The construction combines a bilinear map from an annulus with a prescribed exterior map for one hole. This produces one boundary controlled by the chosen exterior map and a second boundary that is circular or nearly circular in the physical plane. When the exterior map is available in Laurent series form, the same representation can be used in truncated series solutions of selected harmonic potential problems, such as steady heat conduction and ideal potential flow. The method is illustrated by bilinear–hypotrochoid and bilinear–Schwarz–Christoffel examples.