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(cf. §2.7)is

where h(X) > 0 is to be determined; then we have the operator identity

Finally, before we use this to transform equation (4.48), we need to decide how to
replace x in the coefficient in equation (4.48): should we use Z or X or both?
Now the important property of the fast scale, (4.49), is that it is zero at the turning
point; thus we elect to write any part of a coefficient which has this same property
in terms of Z, and otherwise use X. With this in mind, we use (4.49) and (4.50) in
(4.48) and then, with we obtain

where, for simplicity, we have written

We seek a (bounded) solution in the form

and hence we obtain the sequence of equations

and so on.
The bounded solution of (4.52a) can be expressed in terms of the Airy function, Ai
(see e.g. Abramowitz & Stegun, 1964). At this stage, however, we have not yet made
a suitable choice for h(X); let us choose

and then we have