Novel expressions for the one and two dimensional Gaussian Q-functions

The aim of this work is the derivation of a tight closed-form approximation for the one dimensional Gaussian Q-function, Q(x) and an exact series representation for the two-dimensional Gaussian Q-function, Q(x,y;p). The accuracy of the expressions is validated through comparisons with existing approximations as well as with results obtained from numerical integrations. Importantly, the derived relationships have a relatively simple algebraic form and thus, they are easy to handle both analytically and numerically. As a result, they can be considered as useful mathematical tools in various analytic studies in wireless communications.