We discuss asymptotic solutions of the kinematic $αω$-dynamo in a thin disc (slab). Focusing upon the strong dynamo regime, in which the dynamo number $D$ satisfies $|D|≫1$, we resolve uncertainties in the earlier treatments and conclude that some of the simplifications that have been made in previous studies are questionable. Comparing numerical solutions with asymptotic results obtained for $|D|≫1$ and $|D|≪1$ we find that the asymptotic solutions give a reasonably accurate description of the dynamo even far beyond their formal ranges of applicability. Indeed, our results suggest a simple analytical expression for the growth rate of the mean magnetic field that remains accurate in the range $-200