  Given a multivariate generating function F, we determine asymptotics for the
coefficients. Our approach is to use Cauchy's integral formula near singular
points of F, resulting in a tractable oscillating integral. This paper treats
the case where the singular point of F is a smooth point of a surface of poles.
Companion papers will treat singular points of F where the local geometry is
more complicated, and for which other methods of analysis are not known.
