The laminar boundary layer over the semi-infinite rigid plane y = 0, x > 0 is examined for the case when the free-stream velocity takes the form \[ U(x, t) = U_0(x) (1+\alpha_1 \sin \omega t), \] where 0 [les ] α1 < 1 and U0(x) ∝ xn, 0 [les ] n [les ] 1. The corresponding steady solution is the Falkner-Skan boundary layer with zero (n = 0) or favourable pressure gradient (n = 1 corresponds to the stagnation-point boundary layer). The skin friction, and the heat transfer from the wall when that is maintained at a uniform temperature T1 greater than the temperature T0 of the oncoming fluid, are calculated by means of two asymptotic expansions: a regular one for small values of the frequency parameter ε1(x) = ωx/U0(x) and a singular one (requiring the use of matched asymptotic expansions) for large values of ε1. The principal difference between this work and that of earlier authors is that here α1 is not required to be small. Numerical computations are made for three values of n (0, $\frac{1}{3}$, 1), three values of α1 (0·2, 0·5 and 0·8) and (in the case of the heat transfer) two values of the Prandtl number σ (0·72 and 7·1). It is demonstrated that for each n there is a value of ε1 at which the small-and the large-ε1 expansions for the skin friction overlap quite well, and that, near the overlap region, two terms of the small-ε1 expansion provide a more accurate asymptotic representation of the solution than three. It is also shown that there is no region of overlap between the small- and the large-ε1 heat-transfer expansions, except in one case (n = 1/3, σ = 0·72) where the overlap value of ε1 (= 2·0) is the same as for the skin-friction expansions. The question of the existence of eigensolutions in the large-ε1 expansion, where on physical grounds they can be expected to appear, is discussed in the appendix.