  Many important analytic statements about automorphic forms, such as the
analytic continuation of certain L-functions, rely on the well-known rapid
decay of K-finite cusp forms on Siegel sets. We extend this here to prove a
more general decay statement along sets much larger than Siegel sets, and
furthermore state and prove the decay for smooth but not necessarily K-finite
cusp forms. We also state a general theorem about the convergence of
Rankin-Selberg integrals involving unipotent periods, closing a gap in the
literature on L-functions. These properties serve as the analytic basis of a
new method to establish holomorphic continuations of Langlands L-functions, in
particular the exterior square L-functions on GL(n).
  Keywords: Automorphic forms, rapid decay, cusp forms, L-functions,
Rankin-Selberg, integral representations, uniform moderate growth.
