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written in the form

with

Note the appearance of the evaluation on which may cause some anxiety; we
will now address this issue. To see the consequences of this, and before we write down
the complete asymptotic expansion, it is instructive first to solve for i.e.

with the boundary conditions

This second condition involves and to accommodate such terms the asymptotic
expansion must include them; thus we seek an asymptotic solution

and evaluation on is now no longer an embarrassment.
The sequence of equations, generated by using (4.87) in (4.85), starts

The general solution of (4.88a) is

and then we may write (4.88b) as

which itself has the general solution

We seek a solution which is uniformly valid for and but this
latter implies thus we require, for uniformity,