Periodic Schwarz–Christoffel mappings with multiple boundaries per period

Abstract

We present an extension to the theory of Schwarz–Christoffel (S–C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x -direction in an ( x ,  y )-plane, three cases are considered; these differ in whether the period window extends off to infinity as y  →  ± ∞, or extends off to infinity in only one direction ( y  →  + ∞ or y  →  − ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S–C mapping formulae are shown to be expressible in terms of the Schottky–Klein prime function associated with the circular preimage domains. As usual for an S–C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.