We present an extension to the theory of SchwarzChristoffel (S-C) mappings by permitting the target
domain to be a single period window of a periodic
configuration having multiple polygonal (straightline) 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 formulas 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 formulas
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.