New analytical results are presented pertaining to M. S. Hantush's integral M(u, ¿). This integral is often employed in studies of unsteady flow near partially penetrating wells and is usually evaluated via numerical quadrature. It is demonstrated here that Maclaurin and asymptotic series may be truncated to provide a choice of accurate algebraic approximations to M(u, ¿) over the relevant parameter space. The Maclaurin series lead to formally convergent solutions, although practical evaluation of these series is difficult for some ranges of parameter values. In these cases, the asymptotic expansions provide efficient estimates. Numerical implementation of the new results is straightforward, and guidelines for selection of series truncation limits are provided.