In the representation theoretic interpretation of the theory of automorphic forms Fourier transforms at cusps are products of two quantities. The first (under a multiplicity one condition) is a scalar containing all of the arithmetic information. The second is a (generalized) Whittaker model for the representation associated with the form. In this lecture we will analyze the integrals involved in the second part of this factorization. These integrals are paramaetrized by points in a complex vector space and converge and are holomorphic in a half space. The main result gives an algebraic condition that guarantees a holomorphic continuation to the entire space. This result generalizes or implies every known case of a holomorphic continuation of a generalized Jacquet integral.