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120 3. Further applications

The equation for (3.12b), is most conveniently written in terms of the characteristic
variables so that we have the operator identities

Thus (3.12b) becomes

which can be integrated directly to give

where J and K are arbitrary functions. The initial data, (3.14), requires

and

or

where A is an arbitrary constant. These two equations enable us to find J and K as

hence the solution for becomes

where we have written For convenience, let us set

then we have the asymptotic solution (to this order):

This two-term asymptotic expansion, (3.17), contains terms f, H and all of
which are bounded and decay as their arguments approach infinity (because of our
given f(x)), and so there is no non-uniformity associated with these. However, if we