Periodic Schwarz-Christoffel mappings with multiple boundaries per period

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.