ABSTRACT
This chapter aims to determine bounds on the exponential growth, essentially confirming how the first principle of coefficient asymptotics is valid also for higher dimensional objects. It examines the point on the contour of integration that the integrand is maximized, and rewrite the integral to pull out the dominant contribution, with a controlled error subterm. To estimate coefficients univariate meromorphic functions we started with the Cauchy integral, moved the contour past the singularity, and then expressed the coefficient as a sum of residue computations and an error term. Recall in that proof a contour containing singularities in its interior is built and is then rewritten as a sum of contours each containing exactly one pole of the function, and also a contour containing the origin. Rearranging this equation gave an expression for a coefficient as a sum of residue integrals around each pole, plus an error term of exponentially smaller growth.
