  The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet.
This integral converges if certain quantities decay at horizontal infinity, but
can also be summed over periodic images in the horizontally periodic case.
However, non-decaying, non-periodic cases are also of interest, such as the
interaction of periodic wavetrains with non-commensurate periods (i.e.
spatially quasiperiodic solutions), or non-periodic disturbances to periodic
wavetrains. We therefore develop a more general single formula for the
Birkhoff-Rott integral, which unifies and extends the cases of decay and
periodicity. We verify that under some reasonable conditions this new version
of the Birkhoff-Rott integral is the restriction to the vortex sheet of an
incompressible, irrotational velocity field, with continuous normal component
but with a jump in tangential velocity across the vortex sheet. We give a
number of examples of non-decaying, non-periodic sheet positions and sheet
strengths for which our assumptions may be verified. While we develop this in
the case of two-dimensional fluids, the methodology applies equally well to
three-dimensional fluids.
