Placing a diffuser around a wind turbine can increase its power output, but not all mechanisms by which the diffuser alters the aerodynamics have been investigated thoroughly. Here, we concentrate on one such mechanism: the effect of the finite number of blades. In nearly all blade element analyses of wind turbines, finite blade effects are approximated by Prandtl's "tip loss factor" which goes to zero at the blade tip. We argue that this limiting behaviour cannot be correct for the axial velocity in the presence of a diffuser. We provide alternative "finite blade functions" which preserve the finite limit on the axial velocity, but do not alter the conventional limit of zero for the circumferential velocity. In maximizing the power output of a diffuser-augmented wind turbine, the change in the finite blade function for the axial velocity has a large impact on the power-producing region near the tip: it increases both the chord and the power output of an optimized blade. Further, the change appears to make diffuser-augmented turbine power output less sensitive to tip speed ratio than for a bare turbine. (C) 2020 Elsevier Ltd. All rights reserved.