  We discuss asymptotic solutions of the kinematic $\alpha\omega$-dynamo in a
thin disc (slab). Focusing upon the strong dynamo regime, in which the dynamo
number $D$ satisfies $|D|\gg1$, 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 $|D|\gg1$ and $|D|\ll1$ 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 $-200< D< -10$ (which is appropriate for
dynamos in spiral galaxies and accretion discs). Finally, we analyse the role
of various terms in the dynamo equations to clarify the fine details of the
dynamo process.
