Schwarz-christoffel Mapping of Multiply Connected Domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivity m ≥ 2 thereby extending the results of Christoffel (1867) and Schwarz (1869) for m = 1 and Komatu (1945), m = 2. A formula for f, the conformal map of the exterior of m bounded disks to the exterior of m bounded, disjoint polygons, is derived. The derivation characterizes the global preSchwarzian f ′′ (z) /f ′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on the m boundary circles via the reflection principle, and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a ′′method of images′′ infinite sequence of iterations of reflecting prevertex singularities from the m boundary circles to the whole sphere.