If you take unit vectors $e^{ia_j}$, the central term is the tangent of the angle of the vector sum ($Im(\Sigma e^{ia_j}) / Re(\Sigma e^{ia_j})$). The angle of a vector sum is the weighted average of the angles of the vectors. With unit vectors, the central term is $tan \left( \frac{a_1 + a_2 + â€¦ + a_n}{n} \right)$.