Some generalizations of the Schwarz-Christoffel mapping formula

Summary The Schwarz-Christoffel formula for the mapping of a polygon in thez-plane on an upper half-plane (thew-plane) is extended to deal with singlyconnected domains of quite general shape. The mapping problem in the general ease is shown to depend on the solution of an awkward integrodifferential equation and an iterative method of finding this solution is indicated. Two further generalizations are made to the formula; these are (i) the boundary of the singly-connected domain in thez-plane is mapped on to afinite interval of the real axis of thew-plane instead of the whole of it, and (ii) the formula is extended to deal with doubly-connected domains.Paper, read at the first annual general meeting of the Australian Mathematical Society at Sydney, August, 1957.