Computing Borel's Regulator

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL($\mathbb{C}$)). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_{3}(GL($\mathbb{C}$)) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= $\mathbb{Q}$(e^{2\pi i/3}) a computation of V_1(F) has been made based on our techniques.