M. S. Hantush established relationships between the dynamics of groundwater mounding beneath recharge zones and two integral functions, M and S. Exact algebraic expressions for these functions are developed in terms of a formal power series expansion. This expansion may be reordered to provide two independent analytical partial summations involving elementary functions. The convergence characteristics of these two formulae are discussed and compared with numerical quadratures of M and, hence, S. The algebraic expressions are used to generate identities for related integrals. Compact algebraic approximations to M and S can be deduced from the series expansions with essentially arbitrary accuracy, while retaining valuable functional information. For example, a simple two-term truncated sum yields a reasonable approximation to M over a useful range of arguments. The results are amenable for use in further theoretical studies of groundwater percolation and mounding where numerical quadratures may be undesirable. ¿ 1998 American Geophysical Union