Approximations for the Hantush M function

A recent paper (Trefry, M.G., 1998. Analytical series expressions for Hantush's M and S functions. Water Resour. Res. 34, 909–913) provided several interesting results concerning the Hantush M function, an integral that arises often in analytical solutions for groundwater pumping models. It was demonstrated that, although an exact result for this integral form is available, there remain significant difficulties in its numerical evaluation. We provide some simple approximations to the M function, one of which has a relative error of less than 1%, one with a relative error less than 0.1%, and one with a relative error of less than 0.015%. These approximations are likely suitable for all practical purposes.