We begin with a brief discussion of singularity analysis, and how Cauchy's integral formula is used to obtain asymptotic estimates on certain univariate gererating functions. We'll look at the difficulties that arise when one attempts to push these ideas to the bivariate case. To handle these difficulties, we will prove a souped-up version of the residue theorem that will allow us to reduce our computation by a dimension, back to that of a contour integral. By using techniques of saddle point integration, the task of asmyptotic computation will once again reduce to the study of finitely many points - this time critical points rather than singular points. Time permitting, we will discuss what this has to do with my current research.