Asymptotics of multivariate sequences I. Smooth points of the singular variety

Given a multivariate generating function F(z(1), ..., z(d)) = Sigma a(r1, ..., rd) z(1)(r1) (...) z(d)(rd), 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 treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known. (C) 2001 Elsevier Science.