Approximations for the Three-Loop Cusp Anomalous Dimension

We analyze the behavior of the three-loop cusp anomalous dimension and its component functions. Its value as the cusp angle $\theta \rightarrow 0$ is analytically determined by utilizing the properties of harmonic polylogarithms and their relation to other functions. Approximations are given in heavy quark speed $\beta$ for both two-loop and three-loop expressions, and are optimized over the range $0.3<\beta<0.8$. Linear approximations are also given in $\theta$ which are very effective for $\theta > 5 $.