On the calculation of phase shifts produced by complex potentials

A formula for the complex phase shift, pertaining to the one-turning-point scattering problem with a complex potential, is given in a general form based on a kind of arbitrary-order phase-integral approximation. The accuracy of the formula is illustrated by application to a real Coulomb potential supplemented by an imaginary potential term proportional to $\frac{1}{r^2}$. For that case the arbitrary-order phase-integral expression for the complex phase shift is evaluated in a closed, analytical form, which displays the simplicity and accuracy of the higher-order corrections.