# §33.25 Approximations

Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma_{0}}\left(\eta\right)=\operatorname{ph}\Gamma\left(1+\mathrm{i}\eta\right)$ (see (33.2.10)) for the ranges $0\leq\eta\leq 2$, $2\leq\eta\leq 4$, and $4\leq\eta\leq\infty$. Maximum relative errors range from 1.09×10⁻²⁰ to 4.24×10⁻¹⁹.